Convexity (Proceedings of symposia in pure mathematics, Vol.7)

PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS VOLUME VII

CONVEXITY

AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND

1963

Prepared by the American Mathematical Society under Contract Number AF 49 (638) — 964 with the United States Air Force

Copyright 1963 by the American Mathematical Society Printed in the United States of America

All rights reserved except those granted to the United States Government. Otherwise, this book, or parts thereof, may not be reproduced in any form without permission of the publishers

CONTENTS PREFACE

•

ix

fl4TRODUCTION

On systems of linear inequalities in Hermitian matrix variables By RICHARD BELLMAN and

.

.

.

1

Fui

Minimum area of a set of constant width

13

By A. S. BESICOVITCH

On semicircles inscribed into sets of constant width

15

By A. S. BEsIcovITca

A cage to hold a unit.sphere

19

By A. S. BESICOVITCH

On singular points of convex surfaces

21

By A. S. BESICOVITCH

On the set of directions of linear segments on a convex surface By A. S. BESIC0VITCH

The support functionals of a convex set By ERRErr

and R. R. PHELPS

Topological classification of convex sets

37

By HARRY CORSON and VICTOR KLEE

An upper bound for the number of equal nonoverlapping spheres that can touch another of the same size

53

By H. S. M. COXETER

Rotundity By D. F. CumA

73

A characterization of the circle

99

By LUDWIG W. DANZER

Helly's theorem and its relatives

101

By LUDWIG DANZER, BRANKO GRUNBAUM and VICToR KLEE

An extremal problem for plane convex curves

181

By CHANDLER DAViS

Notions generalizing convexity for functions defined on spaces of matrices

187

By CHANDLER DAVIS

Some near.sphericity results

203

By ARYEH DYORETZKY

On the Krein-Milman theorem By On

211

FAN

Lipschitzian mappings of convex bodies

221

By DAVID GALE

Neighborly and cyclic polytopes

225

By DAVID GALE Measures of symmetry

233

for convex sets

By BRANKO GRUNBAUM

V

271

Borsuk's problem and related questions By BRANKO GRUNBAUM

On polyhedral graphs

.

By BRANKO GRUNBAUM and THEODORE S. MOTZKIN

Convex curves of constant Minkowski breadth

291

By PRESTON C. HAMMER

Semispaces and the topology of convexity

305

By PaEs!roN C. HAMMER

On simple linear programming problems

317

By A. J. HOFFMAN

Total positivity and convexity preserving transformations .

.

.

. 329

By SAMUEL KARLIN

Infinite-dimensional intersection theorems

349

By VICTOR KLEE

Endovectors

361

By T. S. MoTZKIN

Representation of points of a set as linear combinations of boundary points

389

By T. S. MOTZKIN and E. G. STIWJS

Support cones and their generalizations

393

By R. R. PHELPS

Convex spaces associated with a family of linear inequalities.

.

. 403

.

. 437

By H. PORITSKY

A combinatorial lemma on the existence of convex means and its application to weak compactness

.

By VLASTIMIL PTAK

Convex cones and spectral theory

451

By HELMUT H. SCHAEFER

The dual cone and Helly type theorems

473

By F. A. VALENTINE UNSOLVED PROBLEMS INDEX OF UNSOLVED PROBLEMS AUTHOR INDEX SUBJECT INDEX

495 501 503

509

PREFACE

Of the thirty-two papers in this volume, seventeen were presented at the Symposium on Convexity and the others were submitted later. (Symposium speakers were Besicovitch, Coxeter, Danzer, Davis, Day, Dvoretzky, Fan, Gale, GrUnbaum, Hammer, Hoffman, Karlin, Klee, Motzkin, Phelps, Pták, Schaefer, and Valentine.) The thirty-third "paper" included here is a report on unsolved problems, based on the Symposium's session devoted to them, on informal discussions during the Symposium, and on later communications from the participants. The papers are arranged alphabetically by author, since this seems most convenient for reference purposes. Interrelationships of the various papers, and their relation to the theory as a whole, are discussed in the Introduction. Since some of the individual bibliographies were so long and in such a state of flux, a common list of references did not seem feasible. However, the Author Index (in conjunction with the individual bibliographies) should be a fair substitute for such a list, and also makes it easy to learn which of the papers cite the work of a given author. There are also a Subject Index and an Index of Unsolved Problems. The editor is indebted to Professors Gale and Grimbaum for their assistance planning the Symposium, to Dr. PtIk and Professors Besicovitch, Coxeter, Day, Fan, and Motzkin for presiding at Symposium sessions, and to Dr. Danzer and Professors Besicovitch, Corson, Firey, Grünbaum, McMinn, and Motzkin for refereeing some of the papers. In particular, the advice and assistance of Branko Grünbaum have been invaluable. The details of publication have been capably handled by Miss Ellen Swanson, Head of the American Mathematical Society's Editorial Department.

Victor Klee

VII

INTRODUCTION

The systematic study of convex sets was initiated by H. Brunn and H. For most of the important notions in the field, at least a germ

Minkowski.

can be found in the latter's collected works (1911). Not only does the theory of convexity play a central role in Minkowslçi's geometry of numbers, but it of elementary number theory. nontechnical also shares some of and of strong intuitive appeal. The are Its basic subject is primarily one \of ideas rather than machinery, and does not lend itself readily to unified treatment. It abounds in attractive special problems, and many mathematicians working mainly in other fields have published one or two papers on convexity. These aspects have accounted for the rapid the theory. but disorganized growth The 1934 by T. Bonnesen and W. Fenchel was an excellent summary of a body of material, and is still a standard source of information in the field. Though in coverage, they cited moçe than 450 references; a current survey of the sAMe degree of completeness be a tremendous undertaking, probably not feasible. More than half of their book emphasized various quantitative notions such as diamet4r, area, volume and mixed volumes.

Since 1934 these same notions have contFnued to play an important role. However, more striking (since less predictable) has been the intensive development of several qualitative aspects of the including the combinatorial geometry associated with intersection and covering properties, the refinement and application (especially in functional analysis and game theory) of such notions as extremal separation properties, the study of convexity in infinite-dimensional spaces, increasing use of convexity as a descriptive tool, and the evolution of various analogues generalizations of convexity. Though several quantitative investigations are included here, the Symposium was intended primarily to emphasize the more qualitative aspects of the theory. In particular, the five aspects listed above are all represented in the present volume. Among the unavoidable omissions, two are especially regretted by the editor. There is nothing here about the geometry of convex surfaces and the associated development of metric methods in differential geometry, carried out by A. D. Aleksandrov and his students in the Soviet Union and in this country by H. Busemann. Also omitted are the important results on infinitedimensional simplexes, boundaries and extremal structure which have been developed in the past few years by G. Choquet and others. In addition to the wide range of topics treated here, there is much variety of approach. Some of the shorter papers treat a single problem in full detail, while at the other extreme are several long papers which include very few proofs but survey broad areas in the field of convexity.

Four of the papers are set in the Euclidean plane BESICOVITCH's first paper gives a short proof of the known fact that a set of given constant width lx

INTRODUCTION

has minimum area when it is a Reuleaux triangle. His second paper solves affirmatively a special case of the following problem: Must a set of constant width w contain a semicircle of diameter w? DANZER gives a short proof of the known result that if C is a dosed convex curve in E2 which does not contain exactly three vertices of any rectangle, then C is a circle. In his first paper, DAVIS characterizes rectangles by means of an extremal area property involving inscribed crosses and also discusses a related conjecture of Ungar on extremal perimeters. HAMMER's first paper is set in an arbitrary Minkowski plane where by

the use of outwardly simple line families he is able to give an analytic representation for all convex curves of constant Minkowski width. He also summarizes his earlier work on diametral lines and associated convex bodies. BESICOVITCH's third paper discusses Coxeter's problem of finding the smallest cage (edges of a convex polyhedron) which will hold a unit-sphere in B3 without permitting it to escape. His other two papers give new proofs of known results concerning smoothness properties of a convex body K in E3 and concerning directions of line segments in the boundary of K. In GALE's first paper he uses the Borsuk-Ulam mapping theorem (involving antipodal points) to prove that if a convex body of width w' in EM is obtained from one of width w by means of a homeomorphism which decreases distances, then w' w. COXETER proposes an exact upper bound for the number of equal nonoverlapping spheres in EM that can touch another of the same size. The dif. Can a rigid ficulty of this problem is indicated by the following quotation: material sphere be brought into contact with 13 other such spheres of the same size? Gregory said 'Yes' and Newton said 'No', but 180 years were to elapse before a conclusive answer was given." His historical survey of the problem in EM extends from a paper by Kepler in 1611 to the latest published works. The problem is treated as the case çS = irI6 of the problem of packing (n — 2)spheres of angular radius # on an (n — 1)-sphere, and the proposed upper bound is attained when the (n — 2)-spheres are inscribed in the cells of a regular polytope {p, 3, - -•, 3}. Though the bound is not fully established, much supporting evidence is given. Some related material is also discussed, such as the growth of the number of spheres as n — oo and the known results for other values of PORITSKY treats a system of linear inequalities of the form x,fi(O) + -•- +

g(O), where g and the f1's are real analytic functions of the real variable 0 ranging over a bounded or unbounded interval 1. He studies the convex region consisting of all points x = (x1, .. ., EM which satisfy the given system of inequalities (for all 0 1), and is especially concerned with describing the region's boundary in terms of the envelope curve C and its tangent and osculating flats of various dimensions, where C is the set of all = points x such that for some 0 I,. for 0 j n — I ((1) indicating the jth derivative). DVORETZKY reviews his earlier results on near-sphericity in EM, one of which asserts that for each e £ JO, 1[ and each positive integer k there exists N(k, e) such that every convex body of dimension N(k, admits a b-dimen-

INTRODUCTION

xi

to within e. He derives new corollaries, including some on orthogonal projections, and discusses some open problems. Two papers treat the facial structure of convex polyhedra. GALE's second paper is concerned with cyclic polytopes in R these being convex polyhedra which are combinatorially equivalent to the convex hull of an n-pointed subset of the moment curve {(t, t', . . ., ta"): t R}. They have the remarkable property of being rn-neighborly in the sense that each m vertices determine a face. He computes the number of (2m — 1)-dimensional faces of such a for convex polytope and this is conjectured to be the maximum which have n vertices. Certain neighborly polytopes are polyhedra in sional section which is

proved to be cyclic, and regular cyclic polytopes are constructed in GRUNBAIJM AND MOTZKIN call an abstract graph k-polyhedral

it is isomorphic with the graph formed by the edges and vertices a k-dimensional convex polyhedron. They prove that each k-polyhedral graph contains as subgraph a refinement of Ck+i, the complete graph with k + 1 nodes. As Gale's result shows, the graph is i-polyhedral whenever 4 j k; however, this and other sorts of ambiguity are excluded for graphs which are 2polyhedral or 3-polyhedral. VALENTINE deals mainly with known results on the intersection properties of convex sets. He obtains refinements and new proofs for many of these, his aim being to show what can be accomplished by systematic exploitation of dual cones. His viewpoint is well expressed by the following quotation:

"•-• since it is a rare coincidence for the proofs of a theorem and its dual to be of equal difficulty, there is a double reason to investigate the dual. One may gain either a simpler proof or a less obvious theorem." Five of the papers are expository surveys of a sort which should be valuable in any field, especially in the field of convexity where so many results

have been rediscovered so many times and where there are so many elementary unsolved problems. Though including few proofs or none at all, they give rather complete descriptions of known results and existing literature in their respective areas. Some of them include new results as well, and most of them discuss many unsolved problems. Since the papers are themselves summaries, it is hardly feasible to summarize them here, but it may be helpful to list their section headings. GRUN BAUM, Borsuk's problem and related questions — reductions of the problem; partial solutions; universal covers; other results on partitions; coverings by translates; finite sets; related problems. GRUNBAUM, Measures of symmetry for convex sets — distance-functions

for spaces of convex sets; invariant points and sets; a property of some measures of symmetry; general methods for geometric definitions of measures of symmetry; known results on special measures of symmetry; some extremal problems which possibly lead to measures of symmetry; an interesting functional; some generalizations. DANZER, GRUNBAUM AND KLEE, Helly's theorem and its relatives — proofs

of Helly's theorem; applications of Helly's theorem; the theorems of Carathéodory and Radon; generalizations of Helly's theorems; common transversals; some covering problems; intersection theorems for special families;

xii

INTRODUCTION

other intersection theorems; generalized convexity. (The last section makes little contact with the others. It contains a rather complete survey of existing generalizations of the notion of convex set.) KLEE, Infinite-dimensional intersection theorems — intersection theorems for infinite families (also in Re); intersection theorems involving the weak topology; intersection properties of metric cells. CUDIA, Rotundity — rotundity and smoothness properties; comparison of properties; product spaces, quotient spaces, and subspaces; duality; geometry and reflexivity. Like those of Cudia and Klee, the papers by BISHOP AND PHELPS and by PHELPS are concerned with the geometry of infinite-dimensional convex sets. The principal result of Bishop and Phelps is that if C is a closed convex

subset of a Banach space, then the support points of C are dense in the boundary of C. They show also that for each bounded closed convex subset C of a Banach space E, the members of the conjugate space E* which attain their maximum on C are dense in E* (norm topology). Several other interesting results are obtained by the same methods. The paper by Phelps treats some of the more technical points which arise when the space is not normable. In particular, he uses supporting cones to give a new proof of the existence of relative extreme points, where a convex cone K with vertex x is said to support the convex set C provided C n K = {x}. CORSON AND KLEE show that the topological classification problem for closed convex bodies in a normed linear space E can be reduced to that for E's unit cell and its closed linear subspaces of finite deficiency. For all dimensional spaces as well as for a wide variety of infinite-dimensional Banach

spaces, the problem is solved by proving that all closed convex bodies in E are homeomorphic with E itself. The main tool is the fact that certain spaces are homeomorphic with their positive cones. Also obtained are some results on uniformly continuous transformations of convex sets. The remaining papers are not so directly concerned with convex sets as such, though in each case some sort of convexity is essential either in the paper itself or for its motivation. Both Karlin and Davis deal with convex

For real intervals X and 1, KARLIN considers the functional transformation T carrying a real function f on Y into the function g = Tf on X given by the formula gx = K(x, the kernel K being a bounded measurable function on the rectangle X x Y. He is especcially interfunctions.

ested in conditions on K which insure that g is convex whenever! is bounded and convex; a similar problem for monotone functions is also considered. The conditions obtained involve the total positivity or sign-regularity of K, where K is said to be sign-regular of order r provided there exists a sequence numbers each either +1 or —1, such that whenever x1 <x, < --< - - < y11,, x, X, Yi Y, and 1 m r, then e.1K(x1, - - -, x1; yj, -. -, y.) 0, Ya where the expression K( ) is the determinant of the matrix which has

y) in the ith row and the jtb column; K is totally positive of order r provided this condition holds with all the e1's equal to +1. Inter-relation-

xiii

INTRODUCTION

ships among given.

various classes of kernels are studied, and many examples are

In his second paper, DAVIS studies various classes of real-valued convex functions (of one or several real variables) where for each class the defining of n x n (real) symmetric matrices. For cx condition involves the class

ample, if I

is

a function of one real variable and the matrix A

has

it is customary to write J(A) = In this way I can be regarded as a function on fin to fin. The function , (1 — .01(A) + Af(B) lot is called matrix-convex provided f((1 — A)A + AR) spectral representation A =

all A e LO, 11 and A, Be H,,, where the ordering is that induced in H,, by agree-

ing that a member of H, is non-negative if and only if it is positive semidefmite. The matrix-conyex functions form a proper subclass of the ordinary convex functions and are closely related to the matrix-monotone functions of Loewner. The paper is devoted to an exposition of Loewner's theory along with related ideas for several variables due to Korányi, Sherman, and Davis himself.

In addition to the paper of Poritsky mentioned earlier, two other papers are included here because of the close connections between convex sets and linear inequalities. BELLMAN AND FAN study systems of linear inequalities in which the variables are Hermitian matrices and the Ordering is defined as in the paper of Davis just mentioned. They find consistency conditions for various systems of inequalities, the conditions being quite analogous to those in the classical situation except that in each case the consistency of an auxiliary

system must be assumed. Also included are several interesting examples, as well as results on the minimum and maximum of the traces of certain matrices related to the systems in question. HOFFMAN supplies a unified approach to some linear programming problems which are amenable to "obvious" solutions. His guide is the observation by

Monge that if unit quantities are to be transported from points X and Y to points. Z and W (not necessarily respectively) so as to minimize the total distance traveled, then the two routes cannot intersect. He defines a Monge sequence to be an ordering of the set -((i,j): 1 i m, 1 j n) and introduces the notion of such a sequence being consonant with a given m x n matrix.

An algorithm is given whereby a solution for the transportation problem associated with a given matrix can be derived from a Monge sequence consonant with the matrix. The warehouse problem of Cahn is transformed into one to which this algorithm is applied and many other problems are mentioned to which the same idea is applicable.

The many new notions in MOTZKIN's paper are treated in 79 theorems distributed among 50 sections. The paper is concisely written and can hardly be summarized here, but we shall describe its basic idea. Let R be a (not necessarily commutative) ring with unit 1 and let V be a left module over R. Let R be the set of all finite sequences A = (A1, - - -, of members of R. When S c V, the vector A is said to be an endovector of S, or S is said to be endo-A, provided S includes the point A,s, for every choice of S

S

A

4.

Since

the family of all endo-A sets in V is intersectional, the A-hull of S is defined

INTRODUCTION

as the smallest endo-A set which contains S. The set A is said to be complete provided for some S, A is the set of all endovectors of S. These and related notions are studied in some detail, where of course the most important cases

are those in which R is the real field and the condition (As, ..-,

A

is

= 1; (iii) A, 0; (tv) equivalent to one of the following: (i) A E R; (ii) — 1 and A1 0. The corresponding endo-1 sets are the linear subspaces (0-flats), the affine subspaces (flats), the positive cones (convex cones with vertex 0), and the convex sets. MOTZKJN AND STRAUS are concerned with representing the points of a set as linear combinations of boundary points. Their principal result asserts a, for 1 that if a, + n, then for very + = 1 and general sets S it is true that each point of S can be represented in the form p= a,rx1 for points x of the outer boundary of S. PTAK presents a unified treatment of several important results on weak compactness, all of which are shown to follow from a combinatorial lemma which gives conditions for the existence of certain convex means. For an infinite set S, let M(S) denote the set of all functions A on S to [0, 001 for which the set N(A) is finite and A(s) = 1, where N(A) = {s S: A(s) > 0). Let be a of subsets of S, and for e > 0 and Hc S let M(H, e M(S) such that N(2) c H and Zsi,ewA(w) < e for all I

W€ (1) M(H,

j

The lemma asserts the equivalence of the following two conaitions: e) = 0 for some infinite H cS and some > 0; (2) there exists a

sequence (se) of distinct points of S and a sequence (W,,) of members of such that {sj, . ., c for all n. With the aid of this lemma he proves that if A is a subset of a complete convex space E and A satisfies a certain double limit condition, then the closed convex hull of A is weakly compact. This includes the well-known theorems of Krein and Eberlein on weak compactness. The same lemma is employed to yield an extensive series of results on weak convergence and weak compactness in locally convex spaces and especially in spaces of continuous functions.

SCHAEFER is concerned with spectral properties in an ordered locally convex algebra A, where this is a locally convex algebra (usually over the complex field) with unit e and with an associated positive cone K 3 such that K is closed, proper, includes the product of any commuting pair of its elements, and is normal in the sense that there is a family of pseudonorms p on E which generate the topology and are such that p(x + y) p(x) for all x, y e K. The principal motivating example of such an A is the algebra of all continuous endomorphisms of a Hubert space, where K is the cone of positive Hermitian operators and the topology is that of either bounded or pointwise convergence. (There are other important examples also.) The paper contains much interesting material on such algebras A, its principal results showing that the spectral behavior of certain members of K is quite analogous to that in the finite-dimensional case. In particular, the members of K whose spectrum is bounded have spectral behavior like that of positive matrices, while those in the unit interval of K (i.e., those a A for which 0 a e diagonal positive matrices in the classical case) behave spectrally like positive Hermitian operators.

xv

INTRODUCTION

FAN's paper is motivated by the Krein-Milman extreme point theorem. He establishes a general lemma which is purely set-theoretical in character, involving neither topological nor vector space concepts, from which the Krein-

(Another lemma, in a sense dual to the first, is shown to imply theorems on filters due to Waliman and Stone.) He then considers a set *P of real-valued functions on a set S, calling a set X C S convex provided X is an intersection of sets of the form {x E S:f(x) a). Since the family of 0-convex sets is intersectional, the 0-hull can be defined in the natural way. The notion of 0-betweenness is defined for points of S and in terms of this the 0-extreme points of subsets of X are defined. These notions appear in several theorems which generalize known results on extreme points and are related to the abstract minimum principal of Bauer. HAMMER's second paper is motivated by his notion of a semispace at a point p in a linear space L, this being a maximal convex subset of L '— {p). He reviews some of the known results on semispaces, including their connection with extreme points and the fact that the semispaces form a minimal intersection base for the convex subsets of L. He then describes his system Milman theorem follows.

of extended topology which arose from an attempt to consider certain processes

and concepts associated with convexity (and especially with semispaces) as topological in character. Many new notions are introduced, complications arising mainly from the fact that in place of the usual topological closure operation he considers an arbitrary expansive function g—i.e., one associating with each set some superset thereof. After discussing the extended topology, he interprets the various notions in terms of convexity, where yX

is the union of X with all the line segments determined by points of X. Several unsolved problems are mentioned. V.K.

ON SYSTEMS OF LINEAR INEQUALITIES IN HERMITIAN MATRIX VARIABLES BY

RICHARD BELLMAN AND KY FAN' 1.

IntroductIon. This paper is concerned with systems of linear inequalities,

in which the variables are Hermitian matrices. The inequalities between Hermitian matrices are to be understood in the sense of positive semidefiniteness or positive definiteness. More precisely, for two Hermitian matrices H and K of same order, we write H K to signify that H — K is positive Similarly, the strict inequality H> K means that H — K is positive definite. Consider the system (2) of linear inequalities, where are arbitrary square complex matrices, B and C1 are Hermitian matrices (all of same order), and c is a real Theorem 1 gives a necessary and sufficient condition for the system (2) LO be consistent, i.e., for the existence of matrices X, satisfying (2). Consistency conditions for more special systems (10), (14), (18) and (22) are also explicitly stated. Then we derive two theorems on minimum and maximum (Theorems 2,3). Theorem 2 asserts that the minimum of the trace of CX,, when {Xj) varies over all solutions of the system (14), is equal to the maximum of the trace of when {.Y1} varies over all solutions of the system (18), provided that the systems (14'), (18') of strict inequalities are consistent.

results are analogous to the well-known theorems on systems of linear inequalities in real variables (see [2; 3]). However, for the case of Hermitian matrix variables, each of our theorems requires an additional hypoth-

esis which is not needed in the case of real variables. Thus, in Theorem 1 we assume that there exist positive definite 1-lermitian matrices

satisfy-

ing (1); in Theorem 2 we assume that the systems (14'), (18') of strict inequalities of the systems (14), (18)) are consistent. These hypotheses are indeed essential (see Examples 2, 3).

The systems studied in \his paper are quite natural, especially in the case = n = 1. For instance, when m = n = 1, system

YA+AtY=-C, where A is aft arbitrary square complex matrix and C, Y are Hermitian. This differs

from the familiar system

Y>0,

YA+A*Y=_l

(I being the identity matrix), which arises in stability problems of differential equations. It a classical theorem of Lyapunov (see [1, Chapter 13; 4, "The twork of the second author was supported by the U. S. Atomic Energy Commission at Argonne National Laboratory.

RICHARD BELLMAN AND KY FAN

Chapter XV, § 5D that there exists a positive definite Hermitian matrix Y satisfying YA + At Y = —I if and only if all eigenvalues of A have negative real parts. All matrices considered here are square matrices with complex elements. Since the matrices considered in a theorem (except in the proof of Theorem 1) are always of same order, the order will often not be mentioned. Through out the paper, A,j are arbitrary square complex matrices which are not are Hermitian matrices. As usual, necessarily Hermitian, B, C,, X1 and

the adjoint of a matrix A is denoted by At, the trace of A is denoted by tr A. We repeat again that, for two Hermitian matrices H, K of same order, H K means that H — K is positive semi-definite, and H> K means that H — K is positive definite. 2. Exletence theorem. We begin with a lemma which will be needed in the proof of Theorem 1. LEMMA. Let H1, H, be p Hermitian matrices of same order. Let denote

the convex cone in the Euclidean fr-sfrace

formed by all points with

(tr ff2, tr H2Z, ..., tr H,Z), when Z varies over all positive semidefinite Hermitian matrices (of the same order as the He's). If there exist real is a closed numbers c1, c2, •, c, such that cd-ft is positive definite, then coordinates

set in PROOF.

Let Z1, Z2, . . ., Z1,--- be a sequence of positive semi-definite Hermitian

tr ]natrices (of the same order as the Hi's) and let We want to prove that (t1, . ., t,)e Let H0 =

= tt (1 k f). CJ-4 be positive

definite, and let a be a positive number such that al H., where I denotes so there exists a the identity matrix. We have trH,Z1 = positive number b such that a- trZ1 b for all i = 1,2,3, . - - Then 0 -c trZ b/a for all i. Since Z1 is positive semi—definite, trZr (bfa)2, which implies that the elements of are bounded. There exists a sul)sequence {Z} such that lim,... Z exists. Then Z is a positive semidefinite Hermitian matrix and tr HkZ = lim,.... tr = tt (1 k P). Hence -

(li,

--,l,)e

EXAMPLE 1.

Let

H2=(° The set

in

.

formed by all points with coordinates (tr H1Z, tr 1hZ), when

Z varies over all positive semi-definite Hermitian matrices of order 2, is not closed. In fact, consists of the origin (0,0) and all points (x, y) with x > 0 and —00 < +00. THEOREM 1. Let A1, be arbitrary matrices, B1, C .be Hermitian matrices (all of same order), and let c be a real number. SupPose that there exist positive definite Hermitian matrices (I)

satisfying

LINEAR INEQUALITiES IN HERMITIAN MATRIX VARIABLES

Then the system

trtCjXj

c

is consistent, i.e., solvable for Hermitian matrices X,, if and only if, for any ,n Positive semi-definite Hermitian matrices the relations

and any non-negative number d,

+ dC1 =0

+

(1

j n)

imply

0.

+ dc

PROOF. Necessity. Assume that X,, X2, •, X,, are Hermitian matrices satisfying (2). For any m positive semi-definite Hermitian matrices A and any non-negative number d, we have •

trZD1B, + dc,

j=1

which can be written 1=1

i=i

+ A1D1) + dC1}X1 i=1

+ dc.

Hence relations (3) imply (4). Sufficiency. Let denote the real vector space of all Hermitian matrices (of the same order as the and let denote the one-dimensional vector space (i.e., the real line). Let . be the direct sum with m summands. In the vector space a vector a = (S1, S2, - -, S,,,, s) is determined by- m Hermitian matrices S. and a real number s. In an inner product is defined in a natural way as follows. For two vectors a = (S1, .. •, S,,,, s) and r = (T1, -. -, T,,., t) in the inner product (a, i-) of a, r is

(a, r)

= trES.T. + st.

We can therefore speak of orthogonality in Let denote the linear subspace of the form + where

...,

(A,,,1X3

formed by au vectors of

+

tr±C1X1)1

Let

=

Dk2,

.

. .,

dk)

(1

k

p)

RICHARD BELLMAN AND KY FAN

be p vectors which span the orthogonal complement E dk e (1 k 1'). (B1, B2, ••, B., c) and bk = —(ôk, Let of ..9' in •', S,., s) S2, is formed by all vectors c = (S1, Then the linear variety 5' — a) (1 5 k p), i.e., satisfying (0k, = bk in (where

(lSkSP). such Let i? denote the set of all vectors (Z1, Z2, ••, Z1., z) in that each Z, is positive semi-definite and z 0. In the Euclidean p-space let

denote the convex cone formed by all points with coordinates of

the form

+ d1z, ", when (Z1, Z2,

•, Z,,1, z) varies in

+

For k = 1,2,

'Ik = diag. {Dkl, Dkl,

• •,

let

Dk.,, dk}

dk are its successive denote the Hermitian matrix such that Dk2, diagonal blocks and all elements outside these blocks are zero. It is clear that Q coincides with the set in formed by all points with coordinates of the form (trH1Z, trH2Z, ..., trH,,Z), when Z varies over all positive semidefinite Hermitian matrices (of the same order as the Ilk's). According to the above lemma, in order to show that Ic" is closed in it suffices to find real numbers c1, c2, ., such that ckHk is positive definite. Now, by hypothesis, there exist m positive definite Hermitian matrices Y1 satisfying (1). So we have

+ =

tr±

+ tr±C1X, +

+ c1}x1

0

for any X, e Thus in the vector 7j = (}'1, 1's, - --, 1'1., 1) is orthogonal to the linear subspace .5'. Therefore is a linear combination of 8,, -, 8,,. We can find real numbers c,, c,, -, C,, such that 1', = c,Dk1 - -

(1 i S m) and I =

Ckd,.

Then

Y,,--,Y1.,1}, i.e., }'1, }',, ..., Y.,, 1 are the successive diagonal blocks in the matrix c,,J-4, and all elements outside these blocks are zero. Hence ckHk is positive

definite and is closed in Assume now that system (2) is inconsistent, which means in view of the characterization (5) of .5' — and our definition of i", (6) amounts to saying that in the point (b1, b,, .. -, b,) is not contained in the

LINEAR INEQUALITIES IN HERMITIAN MATRIX VARIABLES

closed convex cone

which separates there is a hyperplane in strictly. In other words, we can find p real

Hence

and the point (b1, b2,

5

•, b,,)

a, such that

numbers

\

I

p

p

> k=1

for all (Z1, Z2, •••,

As

z)

=

=

dkc

—

we have

+ c)} >0

÷ B,)

for all (Z,,Z,, •••,Z,,,,z)E.y'. Let p

D1

= ZakDk1(1

I

d=

m),

p

k=t

Then

tr for all (Z1, Z,,

+ d(z + c) > 0

D1(Z, +

This implies

•, Z,,,, z)

(7)

and

trZ D1B1 + dc > 0.

(8)

On the other hand, since the vectors

8,

have ;,_1

i=1

we

trZC1X1 =0 1-4

+

,=I

for 1 k

+

Dk,(A1JX, +

are orthogonal to

+ dkCI}XJ =

p and for any arbitrary X, e +

0

Hence

+ dkC, =

0

(1

k p, 1 j

and consequently

Combining this with (7), (8), we infer that the condition stated in the theorem

RICHARD BELLMAN AND KY FAN

6

is not satisfied. Thus the sufficiency of the condition is proved. 3.

ApplIcation to special syatems. We consider some special cases of

Theorem 1. First, in the case C = to the following corollary. COROLLARY 1.

0

(1 j

n) and c =

0,

Theorem 1 reduces

Suppose thai there exist Positive definite Hermilian matrices 1'

satisfying

Then the system

(1 i m)

+

is consistent (i.e., solvable for Hermitian matrices X,) if and only if, for any the relations

m Positive semi-definite Hermitian matrices

+

=0

(1 j

n)

0. As a special case of Corollary 1, we have 2.

Suppose that there exist Positive definite Hermitian matrices Y

satisfying

+

<0

(1

j n).

(1

I

Then the system

m)

is consistent, if and only if, for any m Positive semi-definite Hermitian matrices D4, the relations

0. In § 4, we shall consider systems (14) and (18) simultaneously. The following corollary concerning system (18) differs from Corollary 2 only in notation.

It .is explicitly stated for the convenience of its application in the proof of

LINEAR INEQIJALITIES IN HERMITIAN MATRIX VARIABLES

Theorem 2. COROLLARY 3.

that there exist positive definite Hermit ion matrices X,

satisfying

Then the system

is consistent, if and only if, for any n Positive semi-definite Hermilian matrices E,, the relations

imply

trEE,Ci

0.

We shall also consider system (22) in § 4. By regarding each equation in (22) as two inequalities and by observing that every Hermitian matrix may be expressed as difference of two positive definite (or positive semi-definite) Hermitian matrices, the following corollary follows readily from Corollary 3. CoRoLuRY 4. Suppose that there exist Hermitian matrices X, satisfying

Then the system (1

n)

is consistent, if and only if, for any n Hermitian matrices E1, the relations (23)

(24) REMARK. In Theorem 1, the hypothesis concerning (1) is only needed for the "if" part of the theorem, not for the "only if" part. Similarly, in Corollanes 1—4, the hypotheses concerning (9), (13), (17), (21) are not needed for the "only if" part.

RICHARD BELLMAN AND KY FAN

8

2.

Let

A—'° O\ ki o)' For

a Ilermitian matrix x

B—i' °

k—i o1

we have

=

AX+XA*_B=( 0• x_i) \X+i Z+z

which is never positive semi-definite.

So the inequality

AX+ XA* B (in Hermitian X) is inconsistent. On the other hand, for a Hermitian matrix

D=

we have

\r

DA + A*D DA

0 if and only if $ = 0 and r + ?

0.

D satisfies both D and

0 and

fl=r=O. Hence

imply tr DB = 0. This example shows that the hypothesis concerning (9) in Corollary 1 (therefore also the hypothesis concerning (1) in Theorem 1) and the hypothesis concerning (13) in Corollary 2 are essential. 4. Theorems on minimum and maximum. For the systems (14) and (18), we have THEOREM 2.

Suppose that the systems of strict inequalities

X,>O (1

i

m)

(1

j

n)

and

are consistent. Then the minimum of tr C1X,, when {X1) varies over all solutions of (14), is equal to the maximum of tr B Y1, when { varies over all, solutions of (18). PROOF.

First, if {X1} is a solution of (14) and (YI) is a solution of (18),

then

+

=

+

LINEAR INEQUALITIES IN HERMITIAN MATRIX VARIABLES

of (14) and some solution ( 1')

We have to prove that for some solution of (18), we have

In other words, we are to show that the system

(25)

tc1x1) 0

—

is consistent. Because the systems (14') and (18') of strict inequalities are consistent, there exist positive definite Hermitian matrices Z,, V1, W1 satisfying

Z, +

C, =0

+

(1

Hence Theorem 1 is applicable to system (25). According to Theorem 1, (25) is consistent, if and only if, for any positive semi-definite Hermitian matrices H1, K1, L1, M, and any non-negative number d, the relations

H1 + t(KIAI, +

dC1 =0

(1 j n),

imply

o.

—

In

other words, it remains to verify that, for any positive semi-definite

Hermitian matrices K1, M1 and any non-negative number d, the relations Z(Kii41a +

dC,

+ imply (26).

In case d >

0,

we derive from (27)

(1 (1

n), I m)

RICHARD BELLMAN AND KY FAN

10

d.trZKiB, tr

tr±C1M,,

+

In case d = 0, we use the fact that (14), (18) are consistent. By Corollaries 2, 3 (see also Remark after Corollary 4), relations (27) (with d = 0) 0, and therefore again (26). This M1C1 0 and tr imply tr completes the proof. For the systems (10) and (22), we have THEoREM 3. Suppose that the systems whence (26).

+

i

(1

m)

(22')

Cix,, when {X1} varies over alt are consistent. Then the minimum of tr when solutions of system (10), is equal to the maximum of tr varies over all solutions of system (22). PROOF.

As

in the proof of Theorem 2, it amounts to show that the system

tr(tBiYi — tc1x3) 0 •is consistent. The consistency of the systems (10') and (22') means precisely that the hypothesis concerning (1) in Theorem 1 is satisfied for the present system (28). According to Theorem 1, (28) is consistent if and only if, for Hermitian matrices any arbitrary Hermitian any positive matrices K' and any non.negative number d, the relations

(1

imply

tr(tJi.Bi —

0.

i

m)

LINEAR INEQUALITIES IN HERMITIAN MATRIX VARIABLES

This implication is easily verified (by using Corollaries 1, 4 for the case d = EXAMPLE 3.

0).

Let —

/0

0\

A Hermitian matrix X =

\Z

B — 10

)'/

0\

c

—

(0

1

satisfies

0, AX+ XA* B if and only if x = 0, y 0 and z = 0. Hence, for every solution X of (29), we satisfies have tr CX = 0. A Hermitian matrix V = \w VI

0, VA + A5Y

C

0. So we have tr BY = —1 for 1, wI: u and w + u' every solution Y of (30). This example shows that the hypothesis concerning (14'), (18') in Theorem 2 is essential.

if and only if v =

REFERENCES

1. R. Bellman, Introduction to matrix anaLysis, New York, 1960. 2. K. Fan, On systems of Linear inequaLities, Linear inequalities and related systems, pp. 99-156, Ann. of Math. Studies, No. 38, Princeton Univ. Press, Princeton, N. J., 1956. S. D. Gale, H. W. Kuhn and A. W. Tucker, Linear programming and the theory of

games, Activity analysis of production and allocation, pp. 317-329, Wiley, New York, 1951.

4. F. R. Gantmacher, The theory of matrices, Vol. II, translated by K. A. Hirsch, Chelsea, New York, 1959. RAND CORPORATION AND

NoamwzsTKEN UNIVERSITY

MINIMUM AREA OF A SET OF CONSTANT WIDTH BY

A. S. BESICOVITCH

The area of a set of constant width d is a minimum when the set is a Reuleaux triangle. Like the proofs of Lebesgue [1) and Eggleston [2], the

present one depends on the fact that about any set of constant width d a regular hexagon of the same width can be circumscribed. Let A1, A., .. ., A, be the vertices of the hexagon, and B1, B.,

., B, points of contact of the hexagon

A3

P

A1

FIGURE 1

with the set of constant width where B1 is on side A1A. a.s.o. A point M of

the boundary of the set is defined by the angle 8 between A1A. and the tangent PM at M where P is on A1A.. Let R(8) be the radius of curvature at M so that R(8) + R(8 + ,v) = d, and let MN be the perpendicular to A1A.. We have, for 8 = MN =

sin 8d6, 2

(the sign

I

I

MP1dO

is for area), so that 13

MP = MN/sin

A. S. BESICOVITCH

14

(1)

I

B1B2A,I +

i = —' 2

We have

it,

.

d

nt.R(O) sin OdO

I j, sin ciu.

+

tjo

.1

it,

R(O

+

sin OdO

'I,

R(O ÷

MO] =

sin OdO}

where the sign of equality takes place only if one of the two integrals on the left side is zero; that is, the sum (1) takes its maximum value only when (pr, + ir/3). Similar conR(8) is zero throughout one of the intervals (0, clusions hold for the sums of the other pairs of areas and thus the total area of the part of the hexagon that is outside the set reaches its maximum only when the set is a Reuleaux triangle. REFERENCES

1. T. Bonnesen and W. Fenchel, Th.ori. der konvexen Korper, J. Springer, Berlin, 1934. p. 132.

2. H. G. Eggleston, A proof of Bia.chke's theorem on the Reuieanx triangle, Quart. J. Math. Oxford Ser. (2) 3 (1952), 296-297. UNIVERSITY OF PENNSYLVANLA

ON SEMICIRCLES INSCRIBED INTO SETS OF CONSTANT WIDTH' BY

A. S. BESICOVITCH

V. Klee has set the problem: Can a semicircle of diameter 21 be inscribed into any set 1' of constant width

A partial solution of the problem is given by the THEOREM. Into every simple set 1' of constant width 21 can be inscribed three semicircles of diameter 21.

We call a set of constant width simple if its boundary has a curvature at every point and if the set of centers of curvature is a triangle M'M"M" formed by three inwards convex arcs. The theorem holds under more general conditions and is likely to be true

in the general case. We shall call the boundary of I' also 1'. There is a unique point Ma on M"M" such that' =

—

There are similar points

and

—

on

the other sides of the triangle. Every

normal of 1' is a "double normal," that is it is the normal at each end; it is of length 2!, and it touches the triangle M'M"M" at the point that is the common center of curvature of I' at the end points of the normal. There are these relations between radii of curvature Md'2, MaP4 at different points of 1': + M1P, = —'M"1W2 + Ii1,P,

,

M1P1

+ 1$/laPa

M4Pa = MuNa = 1

similar ones. The other end of the normal at P. P',

and

N,N',...,N,,...

P1,

is

denoted

sothat

NP=N'P'=...

=21.

The end-points above P0N4 are denoted by P, and those below by N. The theorem will be proved by proving that the top half of F bounded from below by NaPa contains the top semicircle on the diameter for which we have

to show that for any point P on the top half of the boundary of r, MaP 1. The line MIIM' meets at most one of the sides M'M", M'M" in a second I am indebted to Dr. Danzer for criticism and suggestions to my first version of this note. 2 The arc on the diagram with end-points A and B, and its length are denoted '—AB. The segment joining points A and B, and its length, are denoted AB, whether the segment itself is on the diagram or not. 15

A. S. BESICOVITCH

P2

/

It,

FIGURE 1

point different from M', so that a line through Ma may touch at most one of the sides M'M", M'M" at a point different from W. Suppose it meets M'M" The proof for this case will also include the case when MaM' does in not meet M'M", M'M" in any point different from M'. Consider the normals of 1' touching M'M"M" at various points M1, M2, M1€

We have =

+ M.P,.

On the other hand M1P1 <M1Ma + MaP1

By (1) and (2) MaP1 > MaPa

=1.

The same is true for M1 e We have + M2P2 = M"Ma + MaPa.

ON SEMICIRCLES INSCRIBED INTO SETS OF CONSTANT WIDTH

On the other hand + M0P. By (3) and (4) M0P2 > MaP6 = 1. The same is true for M2 E M"M'. ÷ M2P2 <

(4)

There is one point or an arc of points of r above curvature is in M'. If P' is any such point we have P'JW'' = (5)

P '114'' + P3M3 —

+

whose center of

'114'''' =

=

'''J%4'a +

JV,,

+ M,,Na.

If the line M3M" does not meet '.—M'M" in a point different from M" then we define the point Q as coincident with M". Otherwise Q is defined as an interior point of '—M'M" such that the line M3Q touches In either case M3Q > M'Q — (6)

P3M3 + M3Q + QM" > PMa

— + But the segments P3M1 and M3Q and the arc QM" (if it differs from the point M") form a convex curve with convexity directed to the point

Hence

(7)

P3M0 + '—M4M" > P3M,, + M0M" > P3M3 + M3Q + '—QM".

By (5), (6), (7)

P3Ma +

>

+ M0N0.

Hence

P3M,, > M0N0 = 1,

which completes the proof that the top semicircle on the diameter N4P0 is included in I'. We have similar results with respect to the semicircles corresponding to the points Mb and M4, which prove the theorem. REMARK. Our theorem has been proved by showing that a simple set always includes one of the semicircies constructed on each normal that is bisected by the center of curvature at end-points of the normal. The question arises whether this is also true in the general case. The answer in the negative is given by the Figure 2. Let ABCDEA (Figure 2) be a pentagon symmetric

with respect to the bisector of the angle CDE and suppose that it is the evolute of a set I' of constant width d. If the sides of the pentagon are linear segments then of course only vertices are centers of curvature. We

shall assume that they are smooth curves of convexity, the direction of convexity being obvious (e.g., AB has its convexity directed upwards). We shall treat the pentagon as linear, but it is obvious that the argument is valid for the case of sides of small convexity. Because of the symmetry of the pentagon the midpoint of AB bisects the normal touching the pentagon at Similarly the points 0,, 0,, 0, such that A01 = A0,, EO, = E0,, DO, = DO4, CO4 = CO (BO, = DO1 because of symmetry) bisect the normals touching other

A. S.

18

B

B

A

FIGURE 2

sides of the pentagon. The endpoint P of the normal rotating about A in Consequently the arc of the angle BAE is distant more than d/2 from the semi-circle center and radius d12 in the angle BAE is inside I'. Hence the opposite arc of the circle is outside I'. Thus only the upper semi-circle may be included in 1'. On the other hand, when the normal rotates about 0 its midpoint describes the circular arc 04003,0 being below and therefore

the top end of the normal through D and 0 is within less than d/2 from that is inside the semicircle. The semi-circle is not included in 1'. OF PENNSYLVANIA

A CAGE TO HOLD A UNIT.SPHERE BY

A. S. BESICOVITCH

Problems on a net to hold a unit-sphere and a box (a polyhedral surface) to hold a unit-sphere have been considered [1; 2]. Coxeter has set a problem on a cage (a frame formed by edges of a convex polyhedron) to hold a unitsphere [3]. The problem is: Find a cage of minimum sum of edges, to hold a unitsphere and not permitting it to slide out. Coxeter expressed a conjecture

that it iS a right triangular prism all of whose edges are equal to i/s, so that = 15.59. This conjecture is false, as follows the sum of all the edges is from the THEOREM. 71w greatest lower bound of the sum of edges of a cage to hold a unit-sphere is

r

= 8ir/3 +

11.88.

The theorem will be proved by showing that for any £ > 0 a cage can be defined for which the total sum of edges is less than r + E. Given a circle of radius r, take the tangent to it at the point M and on the tangent take a segment AB of length 2'1/3r/3 with mid-point at M. From the points A and B draw the second tangents, touching the circle at the points E and C, respectively. We have AE = BC = './3r/3. The length of the curve consisting of the polygonal line EABC and of the arc CDE is equal to + 4n-/3)r. and Divide now the arc CDE into n equal arcs by points P1. P. . .., circumscribe the polygonal line about CDE touching CDE at the points C, P1, P2, -

-, E. Denote its vertices Q1, The polygon ABQIQI -• . , (M, circumscribing the given circle. When n lends to oo the perimeter of the (M, n)-Polygon tends to

will be called + 4ir/3)r.

Take now a great circle of the unit-sphere and a point M on it. Through

the tangent at M draw two planes forming angles a and —a with the great circle, with small a. The planes meet the sphere in two circles of radius r = cos a. Let (M, n)-polygons circumscribing these circles be ... Q1'A. These polygons, together with the -• Q1A and form a polyhedron. The boundaries of the two poly"joints" QQ', gons touch the sphere, while for a fixed n and for a sufficiently small all the joints are outside the sphere. Thus the edges of the polyhedron form a cage and the sphere cannot slide out of it. We shall now fix n so that the perimeter of each of polygons be less than . -. + 4ir/3 + e/3, after which we shall take a small (4V3/3 + 4r/3)r + e/3 < enough for all the joints to be outside the sphere and for their total length to be less than e/3. With these conditions, the total length of the edges of the cage is + e, which proves the theorem. • -

1.9

A. S. BESICOVITCH

20

The question whether r is the greatest lower bound remains open. For finding the lower bound, it is sufficient to consider cages of total length of edges <1?

In such a cage, at least one "face" is within 1/2 of the centre of the

sphere, since otherwise the sphere with the same centre and of radius 1/2 will be contained in the polyhedron, and by Besicovitch-Eggleston theorem (21 the total length of edges will be >12. That means that at least one face circumscribes a circle of radius = = .866 (call it a base). If there were always more than one base, then it would be easy to show that the smallest perimeters belong to the cages of the kind we have described. If this is not

so then, if the base is a triangle, its perimeter is so large that it is easy to show that the perimeter of the whole cage is > r. If on the other hand, the perimeter of the base is not so large than the base is a polygon of 4,5, 6, sides and, because many sides of the cage start from the base, it may be possible to show that in this case the remaining part of the• perimeter is too large; Remember that the perimeter of the circle of intersection of the base = with the sphere is = 5.45 and if the base is a triangle then its

perimeter is >9, if it is quadrilateral then perimeter is >7, pentagon >6.5, hexagon >6.

1. A. S. Besicovitch, A net to hold a apheve, Math. Gaz. 41 (1957), 106-107. 2. A. 5. Besicovitch and H. G. Eggleston, The total length of the edges of a poiyh.Sron, Quart. J. Math. Oxford Ser. (2) 8 (1957), 172-190. 3. H. S. M. Coxeter, review of 2 in MR 20 $1950. UNwsasn'y OF PENNSYLVANIA

ON SINGULAR POINTS OF CONVEX SURFACES BY

A. S. BESICOVITCH

It is well known that the set of singular points on a convex surface in 3space is of a-finite linear measure. For a recent proof and for references to others, see R. D. Anderson and V. L. Klee, Jr. El]. In the present note I give an alternative proof. We shall consider ordered pairs of semi-planes (P, Q), with a common edge PQ, and not forming one plane. The convex hull of (P, Q) will be called the interior of (P. Q). Thus the interior includes the semi-planes. The direction of a semi-plane is defined by the direction of a vector perpendicular to the edge of the semi-plane, with the origin on the edge and the end point on the semi-plane. The angle between two semi-planes either belonging to the same pair or to two different ones is the angle between the two vectors. We take The angle between semi-planes of the same it always to be and pair is always <sr. The distance between two pairs (P. Q) and (P', Q'), A{(P, Q), (P', Q')), is defined by the larger of the angles between the semi-

p'anes P and P' and Q and Q'. Thus 4 is a metric. The set of all pairs (P', Q') for which 4{(P, Q), (P'. Q')} < 8, >0, is the 8-neighborhood of (P, Q). LEMMA 1.

Given a Pair (P, Q) with angle >0, and e > 0 there exists 8 > 0 such

that for any two pairs (P', Q') and (P", Q") of the 8-neighbourhood of (P, Q)

(1) The angle between the edges P'Q' and P"Q" is <e and (ii) if A' is a point of P'Q' interior to (P". Q") and A" a different Point of P"Q" interior to (P', Q') and different from A', A' is on P"Q" then I/ic angle between PQ and A'A" is <e.

(i)

From an arbitrary point M drop the perpendiculars MI" and Mq' on

the planes P' and Q' and MI" and Mq" on the planes P" and Q". The edge P'Q' is perpendicular to the plane M/."q' and the edge P"Q" to M/."q". The angle between the planes Mp'q' and Mp"q" is small for small 8 from which (i) follows.

(ii) Let a' be the angle between P' and Q' (Figure 1). Through the points A' and A" draw planes perpendicular to P'Q' and to P"Q". Their lines of

intersection with P' and Q', P" and Q", are

B'q4,

A"q'. B"A" is the segment of the edge P"Q" between the two planes. The segment B"A" meets one of the semi-planes, say P' in a point C between B" and A" unless P'Q' is parallel to P"Q" in which case P'Q' and P"Q" coincide and there is no problem. Drop the perpendicular CD on As in the right triangle CDB" the angle CB"D is nearly right and the angle CDB" is right, the segment CB" is large compared with B "D and also compared with B"A'. Hence A"B" is large compared with B"A'. Con21

A. S. BESIC0VITC}I

D

Pt

1

the angle B"A"A' is small, which proves (ii). REMARK. On the diagram the points A' and A" are strictly inside the angles respectively. It is clear that the conclusion holds also and when one of them or both are on the boundaries of respective angles. Let 0 < a0 < ir/2. Consider the set of all pairs (P. Q) for which the angle a satisfies the inequalities

p'B

a0

a

iv — a0

Obviously the Heine-Borel argument is applicable to the set and the set can

be represented as the sum of a finite number of neighbourhoods on each of which the Lemma 1 is satisfied for the same £. Hence the set of all pairs N1 of an enumerable set of such o < a < iv, can be represented as the sum neighbour/s oods.

ON SINGULAR POINTS OF CONVEX SURFACES

Let S be a convex surface. With every singular point A of S associate a pair (P, Q) of semi-planes of support at A, not forming one plane, and contain-

ing the whole of S in its interior. A lies on the edge PQ and is inside any pair corresponding to another singular point. Take of the above neighbourand let E1 be the set of all singular hoods a neighbourhood N1 of points of S to which correspond pairs belonging to N1. If A', A" are an arbitrary pair of points of E1 and (P', Q'), (P", Q") the corresponding pairs of N1 so that A' is on the edge P'Q' and in the interior of (P", Q") and A" is on the edge of P"Q" and in the interior (P', Q') then the angle between lie on a rectifiable curve and P,Q, and A'A" is <e. Hence all points of we arrive at the THEOREM. All singular Points of a convex surface lie on an enumerable set of rectifiable curves. COROLLARY.

The set of singular Points of a convex surface is of a-finite linear

measure. DEFINITION.

If there exists a nondegenerate right circular cone with vertex

at a point A of S. containing the whole S, then A is called a conical

The two-dimensional measure of the set of directions of the planes of support at A (that is of their perpendiculars) that do not contain any generator of the cone, is positive and at most at two points such planes

singular Point.

may be parallel. Hence THEOREM 2.

The set of conical singular points of a convex surface is at most

enumerable. REFERENCE

1. R. D. Anderson and V. L. KLee, Jr., Convex functions and upper semi-continuous functions. Duke Math. J. 19 (1952), 349-357. CAMBRIDGE UNIVERSITY

ON THE SET OF DIRECTIONS OF LINEAR SEGMENTS ON A CONVEX SURFACE DY

A. S. BESICOVITCH

The problem on the measure of the set of directions of linear segments on a convex surface has been set by V. Klee [11 and has been solved in 3-space by T. J. McMinn [21. In this note I give an alternative solution. The problem in 4-space remains still open. Let be a set of vectors in the 3-space. Take the set of unit vectors parallel to all vectors of with the origin at the same fixed point. Denote by E the set of end-points, and by the set of directions of vectors of Then the measure of is defined by the measure of E, so that = 4E, } = 4'E, and so on. (i) If is the set of all vectors in a plane then = 2,r. The same result holds if is the set of all vectors parallel to a fixed plane. (ii) Given a trapezoid with bases a and b and height h, if is the set of all vectors meeting each base then < 2(a + b)/h. (iii) If is a cone with the directrix of finite length then < If S is a closed convex set in the 3-space, we shall denote by CS) the of directions of all vectors lying on the surface S. THEOREM The set (S } is of a-finite linear measure. Given two closed convex curves C', C" in the two different parallel planes Z = Z', Z = Z". Denote by 2' the lateral surface of the convex hull of C' UC" (c.h. C' U C") and by 2" the set of all vectors lying in 2' which meet each of the curves C' and C". We shall first prove -

LEMMA.

AZ' is finite.

Consider a variable plane P touching C' and C" and having them on the same side. If p touches C' and C" each at a single point then the segment

joining them lies on I. If p touches one of the curves in a point and the other one in a segment then the triangle with the point and the segment as a vertex and a base respectively lies in 2'. Finally if p touches each C' and C" in a segment then the trapezoid with these segments as bases lies in 2'. We shall consider points of C' and C" as end points of variable vectors M'(t) and M"(t) with origins at the coordinate origin. We define these functions in the following way. Let the plane touch C' and C" at the points M,, MJ'. For 0 I AC' + AC", M'(t) and M"(t) satisfy the conditions AM0'M'(t) + = where M0'M'(t) and M0"M"(I) are the arcs of C' and C" respectively touched by the plane in varying from to p and the points M'(t) and M"(t) lie in the same plane p. It is easy to see that t defines the £

plane p always uniquely. The points M'(t) and M"(t) are also defined uniquely 24

DIRECTIONS OF LINEAR SEGMENTS ON A CONVEX SURFACE

25

unless p meets both C' and C" in segments, in which case we let first M'(t) describe the whole segment of C' keeping M"(I) stationary after which we keep M'(t) stationary and let M"(t) describe the segment of C". For every t, the segment M"(t)M'(t) lies in I, and if I has no trapezoid, these are all the segments of I meeting each curve C' and C". If there are trapezoids then, by (ii) the linear measure of the directions of all the segments with end points on bases of the trapezoids is less than 2(AC' + AC")/(Z" — Z'). It

remains to prove that the linear measure of directions of

all segments M'(t)M"(t) is finite. Denoting by m'(t), m"(t) horizontal components of M'(t)

and M"(t), we see that if we place the origins of the vectors M"(t) — M'(t) on the coordinate origin then the end points of the vectors will describe th curve m"(t) — m'(t) of finite length in the plane Z = Z" — Z'. By (iii), the linear measure of directions of all the segments ±(M"(t) — M'(t)) is finite, which completes the proof of the lemma. We pass now to the proof of the theorem. For all rational r denote by Cr the intersection of the surface of S with the plane z = r. Any vector on the surface of S either parallel to the plane z = 0 or meets for infinitely many values of r. Denote by the set of vectors parallel to z = 0 and by (r', r") the set of vectors of S meeting we have

{(r',r")J 2ir. Every vector of (r', r") belongs to the surface of for if it were inside it would be also inside S. Thus {(r', r")} C {c h 'C,'UC,") and by the lemma

By (i)

C

/t

U

A{(r', r")) < which proves the theorem. REFERENCES

1. V. L. Klee, Research problem No. 5, Bull. Amer. Math. Soc. 63 (1957), 419. 2. T. J. McMinn, On the tine segments of a convex surface in E5, Pacific J. Math. 10 (1960), 943-946. UNIVERSiTY OF PENNSYLVANIA

OF A CONVEX SET

THE SUPPORT BY

ERRETT BISHOP AND R. R. PHELPS The following well-known separation theorem is basic to the considerations of this paper. SEPARATION THEOREM -

Suppose

that A and B are convex subsets of a real

Hausdorff topological vector space E, and that the interior of B is nonempty and disjoint from A. Then A and B can be separated by a hyperplone, that is, there exists a continuous linear functional f 0 on E such that sup! (A) inf f(B). This theorem is a geometric version of the Hahn-Banach theorem. Its proof can be found in any of several texts, for instance in [3, p. 4171. An immediate corollary is the following support theorem.

If C is a convex subset of a real Hausdorff topological vector space E, if x is a point in the boundary of C, and tf the interior of C is nonempty, then there exists a hyperplane which supports C at x, that is, there exists a continuous linear functional f 0 on E such that f(x) = supf(C). We refer to such a functional f as a support functional of C, and x is called a support point. Note that if f is a support functional of C then every positive multiple of f is also a support functional of C. The assumption that C has interior points is a strong one, but some condition is indispensable to the validity of the support theorem. Indeed,

V. L. Klee has shown [6) that there exists a bounded closed convex subset of a dense subspace of a Hilbert space which has no support points. In the same paper Klee asked whether every bounded closed convex set C in a Banach

space has at least one support point. In this paper we answer Klee's question affirmatively: We show that the support points of C are actually dense in the boundary of C. This is shown to be true even if C is not bounded. Still assuming that E is a Banach space, we then prove (if C is bounded) that the support functionals of C are dense in the dual space E*. If C is not necessarily bounded, we show more generally that for each f in E* which is bounded on C and each £ > 0 there exists a support functional g of C with I If — gil <e. In fact it is shown that g can be chosen to strictly separate C from any bounded set X which is strictly separated from C by 1. We also show that every hyperplane which intersects the boundary of C contains a support point of C. Examples are given to show that these theorems fail in certain more general situations. The methods of this paper derive from a previous paper (1], in which a proof was indicated of the fact that the set of support functionals of a bounded closed convex set in a Banach space is dense in the dual space. By extending and simplifying the method of [1], we have been able to improve this 27

ERRETT BISHOP AND R. R. PHELPS

28

result and to obtain proofs of the related theorems mentioned above. Although all our theorems are stated and proved for spaces over the real field, they may easily be formulated and extended to spaces over the complex field by applying them to the underlying real space (obtained by restricting multiplication to real scalars) and to the real parts of complex linear function• als. These formulations are analogous to that given in [31 for the separation theorem.

Throughout this paper we will restrict our attention to normed spaces E and the convex sets under consideration will be assumed to be proper and nonempty. We write U for (x: lixil 1); if JEE*, then 11111 is defined to be supf(U). The proofs of the existence of support points and support functionals are based upon showing the existence of certain support cones for a given closed convex set. We will say that a subset K of E is a convex cone if K is a convex set and Ày e K whenever ye K and A 0. If X is a set containing the point x0, and if K is a convex cone such that K ÷ x0 is disjoint from X then we say that K + x0 supports x at x0. Suppose, now, that K has nonempty interior, that C is convex, and that K + x0 supports C at x0. Then, by the separation theorem, there exists a nontrivial g in E* such that supg(C) infg(K + xo). It is easily verified (since x0 is in both K + x0 and C) that supg(C) = g(x0) = infg(K + x0), i.e., g supports C at x0. Thus, to show the existence of support points and support functionals for C it suffices to find support cones of C which have interior points. The cones which we will use for this purpose are all of the following type: It! is an element of E* of norm one and jfk > 0, then K(f, k) kf(x)).

DEFINiTIoN.

{x: IIxfI

Clearly, K(f, k) is a closed convex cone; furthermore, if k> 1, then the interior of K(f, k) is nonempty. To see this, choose x in E such that II x = 1 and f(x) > Since f and the norm are continuous, and since x
I

I

LEMMA 1. Suppose that X is a complete subset of a normed linear space E, thatf in E* is of norm one and is bounded on X, and that k>O. Ifz€X, then there exists a point x0 in X such that x0 e K(f, k) + z and K(f, k) + x0

supports X at x0.

PROOF. We partially order the set X by means of K; that is, x >- y means This, of course, is equivalent to saying that II x — y II kf(x — y). It is easily seen that if there exists a maximal element x0 in X, then K ÷ x0 supports X at x0. To obtain the conclusion that such an x0 exists for which x — yE K.

x, >- z, it suffices to apply Zorn's lemma to the set 2 of those x in X for which x >- z. Suppose, then, that W is a totally ordered subset of Z. The set {f(x): XE WJ is a bounded monotoriic net of real numbers (using W as

THE SUPPORT FUNCTIONAIS OF A CONVEX SET

our directed index set), hence it converges to its supremum. This implies

that it is a Cauchy net, and since x, y in W implies that

liz — y II

k[f(x) — f(y)J, say, we see that W itself is a Cauchy net in Z. Now, 2= X fl (K + z); since K is closed, Z must be complete and therefore W converges to an element of y in Z. By continuity of I and of the norm, it is simple to verify that y >- x for all x in W, i.e., W has an upper bound in Z. Zorn's lemma then applies and our lemma is proved.

The ideas we have developed so far enable us to prove the density of support points. THEOREM 1. If C is a closed convex subset of a Banach space E, then the support points of C are dense in the boundary of C. >0, choose y in E—C PROOF. If z is a point of the boundary of C and such that It y — z Ii <€/2 and choose (by applying the separation theorem to any convex neighborhood of y which is disjoint from the closed set C) 1 in E* such that 11111 = 1 and supf(C) f(y). By Lemma 1 there exists x0 in C such that x0 e K(f, 2) + z and K(f, 2) + x0 supports C at x0. Applying the separation theorem to C and K + x0 shows that x0 is a support point of C. Furthermore, since x0 — z E K and x0 C, we have I I xo — z ii 2Ef(xo) — f(z)] 2[f(y) 2 — zII < e, which completes the proof. To prove the density of support functionals we first prove that if k is large

and if g is of norm one and is non-negative on K(f, k) then f and g are close together. This result is precisely stated in Lemma 3; the latter follows from a known lemma [7J which is proved again here for the sake of completeness. LEMMA 2.

Suppose that

0, hf II =

1

= ugh, and that ig(x)I €/2 when-

By the Hahn- Banach theorem we can choose h in E * such that sup lg(Uflf'(O))I. Then, by hypothesis, IlhIl e/2. Furthermore, since g — h vanishes on f_I(Ø) there exists a real number a such that g—h =af. Hence hig—afhi = IlhiI €12. Assuming a 0, we will show that hf — gIl e. (Otherwise, the same proof applied to (—a)J would show that Ill + 1, then a' 1 and e.) If a hig—fil = hI(1—a')g+a'(g—af)II 1—a' Also, a = Itafhi — afhI so + 1_&1 + Ilg—afhI)'hlg—afII Hg—alit Hence If 1, then hg—Ill + 11(1 —a)fII = + 1—a = hlg—afhl + — haul which completes the proof. LEMMA 3. Suppose that 0 < e < I, that Ill II = 1 = and that k> 1 + 21€. If g is non-negative on K(f, k), then ill —gil PROOF.

h =g on f'(O) and IIhII =

ERRETT BISHOP AND R. R. PHELPS

30

in E such

lix II = 1 and 1(x) > kt(1 + 2/c), and 2/c. Then lIx±yli suppose that y in E is such that 1(y) = 0 and 0. This implies so x±yeK and henceg(x±y) 1 + 2/c < kf(x) = kf(x±y), Choose x

PROOF.

that

€/2 whenever f(y) = 0 1. Clearly, then, g(y) I that lg( y) I g(x) II x II c or II!— gil c. Choose z and 1, so by Lemma 2, either If + in E such that llzil = 1 and f(z) > c). Then zEK so g(z) Oand it follows that II! + gil (f + g)(z)> c, and our proof iS complete. We could now easily prove our density theorem; to include unbounded sets C, it would he formulated somewhat as follows: If f is bounded on C, then there is a support functional of C which is arbitrarily close to 1. With little care, however, we can do considerably more than this; the result is expressed as follows: If f strictly separates the set C from a bounded set X, then there is a support functional of C which is arbitrarily close to land which strictly separates X and C. More precisely: THEOREM 2. Suppose that C and X are subsets of a Banach space E, that C is closed and convex and that X is bounded and nonempty. If e > 0 and if f -

in E*, 11111 =

=

1,

is such that supf(C) < inf 1(X), then there exist g in

and x0 in C such that 111—gil s and g(x0) =supg(C) < infg(X). PRoof. Let r = supf(C), 8 = inf 1(X) and choose such that r < 8 < 8. Consider the neighborhood V of X defined by X + (8 — 8)U = V. This is a bounded set, and since inf f(U) = —1, we have inff( V) = inf 1(X) —(8— = Let a = 1 + 2/c and choose z in C such that r — f(z) < (2a1'(8 — r). Let M 1,

be larger than 2'(8—r) and sup{lly—zil:ye V} and let k=2aM(8—r)1. (Note that k > a> 1.) Choose, by Lemma 1, a point x0 in C such that K(f,k)+x0 supports Cat x0 and x0—zeK. We will show that VcK+xo. Indeed, if yE V, then fly — xoII iIY — zil + lIxo — all <M + fixo — zil M + kf(xo — a) M+ kEy —f(z)1< r)=2M< 2aM=h(8— kf(y — x0). By the separation theorem there exists g in E*, llgtl = 1, such that supg(C) = g(x0) inf g(K+ x0) inf g( V) infg(X) —(8— $) < infg(X). Since 0 inf g(K) and k> 1 + 2/c, it follows from Lemma 3 that Ill—gil c. A well known variant (and corollary) of the separation theorem substitutes compactness for interior 151: If B and C are disjoint convex subsets of a locally convex space, with B compact and C closed, then there exists a hyperplane which

strictly separates B and C, that is, there exists / in E * such that supf (C) < inf 1(B). This result can be improved if E is a Banach space, giving the following corollary of Theorem 2. COROLLARY 1. If B and C are disjoint convex subsets of a Banach space E, with B compact and C closed, there exist x in C and g in E * such that g(x) = supg(C) < infg(B). An obvious corollary of the above variant of the separation theorem states that a closed convex subset of a locally convex space E is the intersection of all the closed half-spaces which contain it, that is, if xØC then there exists

f in

and a real number c such that the half-space H {y:f(y)

c} con-

tains C but not x. We say that the half-space H supports C if C C H and

THE SUPPORT FIJNCTIONALS OF A CONVEX SET

for som€. y in C. from Corollary 1.

f(y) =

c

COROLLARY 2.

31

The proof of the following corollary is immediate

If C is a closed convex subset of a Banach space, then C is

the intersection of all the closed lzalf.spaces which support it. CORoLLARY 3. If C is a closed convex subset of a separable Banach space E, then C is the intersection of a countable number of its supporting closed half.

spaces.

Let be a dense subset of E '— C and let be the distance to C. By first applying the separation theorem to C and + then applying Theorem 2 to C and U, we can find in E* such + supports C and sup that < for n = 1,2,3, + PROOF.

from

Suppose, now, that x £ E C and let d be the distance from x to C. Choose

<3'd.

d —3'd = (2/3)d. Thus, d, that

(2/3)d and hence liz — > sup g,,(C) and completes the proof.

> x,,

II

which shows

If C is bounded, then every f in E is bounded on C, so we obtain another corollary to Theorem 2. C is a bounded closed convex subset of a Bonach space E, then the support functionals of C are dense in E. If fe E*, we say that f attains its norm provided there exists x in U such that = f(x). Since U is closed and convex, the proof of the following corollary is immediate from the previous one. (This result was first proved in (1).) COROLLARY 5. If E is a Banach space, then the set of f in E* which attain their norm are dense in E

In all the above results we could have dropped the hypothesis that E be a complete normed space provided we assumed that C itself be complete. This follows from the fact that we applied Lemma 1 only to the set C. The following 'result shows, however, that Corollary 4 (and hence Theorem 2) must fail in an incomplete space. THEOREM 3. If E is an incomplete normed linear space, then there exists a bounded, closed convex subset C of E having nonempty interior such that the support functionals of C are not dense in

E a dense subspace of its completion F and identify (in the obvious way) E and Since E * F, there exists x in F'— E such that lix Il = 1. By applying the support theorem to x and the unit ball of F, we can find I in F such that II f II = 1 = f(x). Let D = {y: ye F, ii y II 1 and f(y)=O} and let C' in Fbe the convex hull of D and x, so that C' is the set of all elements of the form z=Ax+(1—2)y, where yeD and Ae[O,1]. It is easily verified (using the compactness of [0, 1]) that the convex set C' is closed; it also has nonempty interior. (If liz — (1/2)xli <1/8, then 0
ERRETT BISHOP AND R. R. PHELPS

32

Define y by a f(z)x + (1 — f(z))y; it is not difficult to verify that y D so that z C'.) Let C = C' n E; then C is convex, closed and has nonempty we will show that interior relative to the (dense) subspace E. Clearly if geE and g(z)= supg(C) for some am c, then hf —gil 1/2. Write a = Ax +(1 — A)y as above; since zeC, we have z * xso that A <1. Hence(since supg(C') = supg(C)) it follows that g(x) g(z) = (1 — A)g(y x) + g(x) and therefore g( y — x) 0. Now 2 II xli + ii y t I I! x — y ii f(x — y) = 1 so that 1 (g — fX y — x) II g — ft II y — x II 2 II g — ft I. This shows that no functional in E5 which is within 1/2 of f can support C, and completes the I

proof.

Note that Theorem 2 may also fail if we drop the assumption that the set X be bounded. Indeed, if E is a separable nonreflexive Banach space, a theorem of James [4] asserts the existence of f in E * such that f(x) < 1 = Let X = {y:f(y) 2) and let C = U; then supf(C) < 11111 for all x in U. inf 1(B), but if g separates C and B, then g must be some real multiple off, and hence cannot support C. Another example (which is closely related to the preceding one) shows that Theorem 2 may fail if we drop the hypothesis that I strictly separates X and C. For, if E is a separable nonrefiexive Banach space, choose I in E5 as and above and Jet C = U and X= fr: hlxfl 1). Then X and Care disjoint bounded closed convex sets having nonempty interior, and sup! (C) = 1 = inf 1(X). Suppose there existed g in Es and x in C such that g(x) supg(C) infg(X). (Thus, g(x) = ugh.) Let A be the convex hull of xand X; since X has nonempty interior, so does A. Furthermore, it is not difficult to see that infg(A) = g(x) = higil. Since XE C we have 1(x) < 1, and we can choose z in C such that 1(z)> 1 — — x). Then y — 1(x)]. Let y = is in the Interior of X; indeed, < 2, while 3'(4 liz II + ii xli) = 5/3 II 1(y) = (4/3)1(z) — (1/3)f(x) > (4/3) — (1/3)[1 —f(x)) — (1/3)f(x) = 1. Since x€ A and y is an interior point of A, we conclude that (3/4)y + (1/4)x z is an interior point of A. But z is also in C, so there must exist a point w in A which is in the interior of C. This is impossible, since if w€ A then g(w) ugh, while g(w) is less than at the interior points w of C. The first example given above shows that the conclusion to Theorem 2 fails if X is unbounded, even though it is assumed that C itself is bounded and X is a linear variety (i.e., a translate of a linear subspace). If we assume that X is a finite dimensional variety, however, we get a valid result (which is actually a corollary to Theorem 2). More generally, we can assume that X is a reflexive variety, that is, X = x + M for some x in E and some subspace M of E, where M is a reflexive Banach space under the induced norm. COROLLARY 6.

Suppose that C is a bounded, closed convex subset of the Banach

space E, that X is a reflexive linear variety in E, that i > 0 and that for I in hILl = 1, we have supf(C) < inf f(X). Then there exists g in E5, <e. II gil = 1, and x0 in C such that sup g(C) = g(x0) < inf g(X) and Ill PROOF.

Since f is bounded below on X, it must be constant on X; writing 0. Let E1 = ElM; under the

X = x + M as above, we then have f(M) =

THE SUPPORT FUNCTIONALS OF A CONVEX SET

usual factor-space norm, B1 is a Banach space and I can be regarded as an Let C1 be the image of C in E1 under the element of norm one in canonical mapping of E onto E1, and assume for the moment that C1 is closed in E1. Regarding X as a point in E1, we have supf(C1) 1(X), so by Theorem 2 there existsg in ugh = 1, and X, in C1 such that supg(C1)=g(Xo)
x0

therefore have nonenipty intersection. Thus, there is a point y in Y such that d(y, C) = 0, which implies that y é C and hence that Ye C,. If we take E to be the Euclidean - 'plane, C to be the convex region bounded

by a branch of a hyperbola, and X to be a translate (which misses C) of an asymptote of C, we see that the above corollary is false if C is not assumed to be bounded.

-

A special case of the above corollary could be fonnulated as follows: If x,, x2, •, X,, are elements of the Banach space E, if s > 0, and if j in Et

then there exists a support functional g of the bounded <e and g(x1) = 0, i = 1,2, , n. — (Simply let M be the linear span of the x x such that supf(C) 0, and if z is an element of the boundary of C such that f,(z) = 0 for each i, then there exists a support point x0 of C = 0, 1 = 1, 2, •.., n. This dual result is valid such that liz — x. II < e and (even without the assumption that C be and is a generalization of Theorem 1. In order to omit reference to the specific functionals f', . . . ,f,,, we will work with the subspace N = = 0, 1 = 1,2, n); N is of finite deficiency, and every subspace of finite deficiency has the same form as N. We first prove a useful lemma. LEMMA 4. Suppose that N is a closed subspace of finite deficiency in E, that C is a convex subset of E, and that x0 isa support point of CflN in the subspace N. Then x6 is a support paint of C. PROOF. Let n denote the deficiency of N; the proof will proceed by induction on n. It is obviously true if n = 0; suppose, then, that n > 0 and assume that the result is true for subspaces of deficiency n — 1. Let g be a nontrivial functional in N5 such that g(x0) = supg(C fl N). Note that if C c N, then any functional which vanishes on N supports C at x,, so we can assume that there exists an element y in C '— N. Let N' be the linear span vanishes on each

closed convex set C such that

I

f

of Nand y, and let C' be the convex hull of CnN' and Then C' has nonempty interior relative to N', C' contains C fl N',

g(x0)).

34

ERRETT BISHOP AND R. R. PHELPS

and x0 is in the boundary of C'. By the support theorem, then, x0 is a support point of C' in N' and therefore it is a support point of C fl N' in N'. By the induction hypothesis, x. must be a support point of C. THEOREM 4. Suppose that N is a closed subspace of finite deficiency in the Banach space E, that C is a closed convex set in E, that e > 0, and that z in of C such N is in the boundary of C. Then there exists a supPort point that x,,eNond JJz—x011 <s. Paoo,. There are several cases to consider. First, if C is contained in a proper closed subspace of E, then any functional which vanishes on this subspace supports C at each of its points, so z itself is a support point of C. Assuming that C is not contained in a proper subspace, we consider whether z is in the boundary (relative to N) of C fl N. If it is, then by Theorem 1

there is a point x0 in N which is a support point of C fl N such that liz — x0 <e. By Lemma 4, x, is a support point of C in E. Finally, suppose z is not in the boundary of C fl N; then there exists a neighborhood (relative

to N) of z which is contained in C fl N. We will show that z itself is a support point of C in E. There exists a point y in E such that the segment this follows is contained in from the fact that N has finite deficiency and that z is in the boundary of C.

Let N' be the linear subspace spanned by N and y, and note that N is a hyperplane in N'. We will show that N supports C fl N' at z, and hence (by Lemma 4) z is a support point of C in E. It suffices to show that the open half-space {x + ry: xe N and r > 0) in N' is disjoint from C. Suppose that C contained a point x + ry of this half-space. Since z is in the interior (relative it such that to N) of C fl N, there would exist a point w of C and A in z Ax + (1 — 2)w. Hence the triangle with• vertices 2, W, and x + ry would a point of [y, 4, a be in C and (as can be easily shown) would contradiction.

The apparent duality between this theorem and Corollary 6 leads one to conjecture that the theorem might still be true if it is merely assumed that E/N is reflexive (rather than finite dimensional). The theorem fails, however, under this weaker hypothesis; there is a well known example (see, e.g., [2, p. 160]) of a compact convex set C in a Hubert space H and a line N in H such that C n N = {ø), but 0 is not a support point of C (even though H/N is reflexive).

All four of our lemmas have valid analogues in more general topological vector spaces, although we do not know whether this is true of the theorems themselves. (These questions will be the subject matter of another paper.) For instance, it is unknown whether a closed convex subset of a complete. locally convex space must have any support points. (For those special classes of closed convex sets C which are known to have support points, it is known that the support points are dense in the boundary of C, e.g., if C has nonempty interior, or if C is locally weakly compact [5].)

THE SUPPORT FUNCTIONALS OF A CONVEX SET BIBLIOGRAPHY

1.

E. Bishop and R. R. Phelps,

A proof :'iat every Bctnach apace is sub reflexive, Bull.

Amer. Math. Soc. 67 (1961), 97-98. 2. N. Bourbaki, Espaces vectoriels topologiques, Chapter V, Hermann, Paris, 1955. 3. N. Dunford and J. Schwartz, Linear operators, Part 1. lnterscience, New York, 1958.

4. R. C. James, Reflexivity and the supremum of linear functional,, Ann. of Math. (2) 66 (1957), 159-169.

5. V. L. Klee, Convex sets in linear spaces, Duke Math. J. 18 (1951), 443-466. 6. , Extremal structure of convex sets. II, Math. Z. 69 (1958), 90-104. 7. R. R. Phelps, A representation theorem for bounded convex sets, Proc. Amer. Math. Soc. 11 (1960), 976-983. UNIVERSITY OF CALIFORNIA, BERKELEY

TOPOLOGICAL CLASSIFICATION OF CONVEX SETS BY

HARRY CORSON' AND VICTOR KLEE2 0. Introduction. This paper contributes to a problem discussed earlier by Keller [15], Stoker [271, and Klee [17; 18; 19; 21; 22]—that of classifying topo-

logically the closed convex subsets of important normed linear spaces. (A forthcoming paper by Bessaga [4] also has some points of similarity with the present work.) Our section headings are as follows: 1. A reduction theorem; § 2. Spaces homeomorphic with their positive cones; § 3. Spaces homeomorphic with their closed convex bodies; § 4. Convexity and uniform continuity; § 5. Additional remarks and problems. A convex body is a convex set which has an interior point. By the principal result of § 1, the topological classification problem for closed convex bodies in a normed linear space E is reduced to that for E's unit cell {x E: II XII i} and E's closed linear subspaces of finite deficiency. A corollary asserts that if E admits (for each finite n) a closed linear subspace of deficiency n which is homeomorphic with its own unit cell, then E is homeomorphic with all its closed convex bodies.

The support of a real-valued function! is the set suppf= {b€dmnf:fb * 0). A subset B of a normed linear space E is a substitutive basis for E provided the following three conditions are satisfied: Si—B is an unconditional basis for E;' S2—whenever x,y,zEEwith IIYII = lizit and suppf5 = suppf c /3—

then IIx+yIt=IIx+zII; S3—IIbII = 1

for all bEB.'

As Dr. Bessaga has kindly informed us, a result of Bohnenblust [5) implies

that if B is a substitutive basis for a normed linear space E of dimension then there is a linear isometry which (for some set X) carries E into the I'X or c0X and carries B onto the canonical substitutive basis for this space. However, in most instances we work directly with the substitutive property rather than using the representation afforded by Bohnenblust's theorem. § 2 shows that if B is an infinite substitutive basis for E and E is either complete or is the linear extension of B, then E is homeomorphic with its positive cone {x E: 0) by means of a transformation which is norm1

Research

supported in part by a grant from the National Science Foundation, U.S.A.

(NSF-G10738). 2

Research

supported in part by a fellowship from the Alfred P. Sloan Foundation

and in part by a grant from the National Science Foundation, U.S.A. (NSF-G18975). 2 That is, each point x C E admits a unique representation in the form z = where is a real-valued function on B and indicates unordered summation. (Of necessity supp is countable.) The normalizing condition S3 is not essential but makes for simpler notation. 37

HARRY CORSON AND VICTOR KLEE

38

preserving and positively homogeneous. It seems probable that every infinite-

dimensional normed linear space is homeomorphic with all its closed convex bodies, though this has been known previously only for reflexive spaces [21]. It is established in § 3 for several other classes of spaces, including separable conjugate spaces, Banach spaces with unconditional bases, and normed linear spaces are homeomorphic spaces. Since all 1231, this settles the topological classification problem for p0-dimensional closed convex bodies. At the same time, our method yields a new proof (independent of the rather complicated isotopy considerations of [17]) of the basic fact that Hubert space is homeomorphic with its unit cell. 1,2, and 3 fail (or their inverses fail) Many of the homeomorphisms of to be uniformly continuous. The results of § 4 show that this is not accidental; in particular, a bounded convex subset of a normed linear space does not admit a uniformly continuous mapping onto any unbounded metric space. § 5 discusses some unsolved problems and indicates still another way of proving that Hubert space is homeomorphic with its unit cell. 1. A reduction theorem. The homeomorphisms which we seek are obtained from the composition of several simple transformations. Recall that for a point p of a convex set C, the characteristic cone of C relative to p is the set cc (C, p) = (x: p + [0, co[(x — p) c C}. This is the union of all rays from p which lie in C, except that cc (C, p) = when there are no such rays.

1.1 PROPOSITION.

are (for c =

Suppose

and E6 are topological linear spaces, W. and Z,

E1, and pt is a point of E, such that p,€int W1c W1cintZ, and cc(W1,p1)=cc(Z,,fr1). Then every homeomora, b) closed convex bodies in

mt phism of OZ, onto OZb can be eltended to a homeomorphism ij of Z. and zeJp.,z]—int W.. such that onto Z6— mt

(Here 0Z1 is the boundary of Z1.) PROOF. For each z E 0Z1, the segment [ps, z] intersects the boundary 0W1 in a unique point w,(z). The assumption about characteristic cones implies that the set Z. — mt W1 is simply covered by the set of segments z let map the segment [wa(z), zJ affinely onto the segment z

carrying the endpoints in the indicated order. Then c and mt WQ biuniquely onto Zb — mt W6. That 7) is a homeomorphism can be established by a straightforward geometric argument. Alternatively, let P1 and v, denote respectively the gauge functionals of W1 and Z1 relative to p4. Then each point x E Za mt W0 admits a unique expression in the [wb(Ez),

carries Za

form x = (1

+ Ag(x/vax) with

—

]0,

11

and we have = Since

(1 —

pb(e(xlMax))

+

all the gauge functionals are continuous and since

-

TOPOLOGICAL CLASSIFICATION OF CONVEX SETS

= (v,,xXp,x —

39

—

it follows that is continuous. Similarly, if' is continuous and the proof of (1.1) is complete. 1.2 PROPOSITION. Suppose M is a closed linear subspace of infinite deficiency in a nornzed linear space E and J is a closed halfsfiace in E whose bounding hyflerJ.ilane H contains M. Then there is a homeomorphism of J .— M onto J which carries H — M onto H.

Let u denote the natural homeomorphism of F! onto its quotient Since HIM is an infinite-dimensional normed linear space, a theorem in [19) guarantees the existence in HIM of a sequence C, of unPROOF.

space HIM.

bounded but linearly bounded closed convex bodies such that 0€ mt C,, always mt C,, C,,+1, and nrc,, = 0. For each n let p,, be a point which is interior to the set tr'C, relative to H. Let z E J — H and for each n let B,, u'C,, + [—1/(n + 1), l/(n + 1)]z. Then the following assertions are easily verified: for each n, fi,, is interior to the closed convex set B,,; always mt B,,

cc(B1,p,,) = /',, + M for all i

flrB,.= 0.

Let e be a positive number such that B0' contains the 2eneighborhood of M and for n 1 let denote the closed em-neighborhood of M. Let = 0 for all n. Then the conditions displayed above for B,, and fi,, are satisfied also by and = M rather than 0. except that Let h0 be the identity mapping on the set E mt B0. Then h0(0B.) = OBO', and from 1.1 it follows that the homeomorphism h0 I can be extended to a homeomorphism h, of B0 — mt B, onto BJ mt which takes points of into H, J—H into E—J. (Recall that Now let B0' = B0.

Continuing in this way, we obtain a sequence of homeomorphisms h0, h,, h2,-.such that h,, carries B,,_, —. mt B,, onto mt h,, extends h,,-, and each of the sets J H, H, and of E J is carried into itself by the transformation h = Since nr B,, = 0 and nr B = M, it follows that h is a homeomorphism of E onto E -.' M which carries J onto J — M and H onto H M. Then h' If — M is the homeomorphism sought for 1.2. We now state 1.3 THEOREM. Suppose C is a closed convex body in a normed linear space E, 0€ mt C, and L = (XE E: (0, oo[x c C}, a closed convex cone with vertex 0.

Then if L is a linear subspace of finite deficiency n in E, there is a ho,neomorphism of C onto the product of L by an n-cell which carries the boundary 0C onto L x If L is not a linear subspace or is a linear subspace of infinite deficiency, there is a homeomorphism of C onto a closed halfspace J in E which carries OC onto the bounding hyperplane 8J; when E is infinite-dimensional, there is also a homeomorphism of C onto the unit cell U = (x E: x S 1) which carries i3C onto the unit sphere 8 U tx E: II x II = 1 }. PROOF. In view of the results of [17J it suffices to consider the case in

40

HARRY CORSON AND VICTOR KLEE

which L is a linear subspace of infinite deficiency in E. Suppose first that L = (0), whence C is linearly bounded, and for each x E 8C let Ex = x/I I x I I e 8U. Then E is a homeomorphism of OC onto OU, and by using 1.1 it is easy to extend e to a homeomorphism of C onto U. Now suppose dim L 1. Let H be a hyperplane thiough 0 which does not contain L and let J be one of the closed haifspaces bounded by H. We shall onto produce a homeomorphism C of C onto f— (L Ii H) which carries He— (L fl H). This will suffice to prove the Theorem 1.3, for by 1.2 and a part of the Theorem already proved [17] we are assured of the existence of (L fl H)) = (L fl H)) = homeomorphisms f and g such that H,gJ= U, and gH=0U. Let z€L—H, so that each point x€E admits a with x1 H and x2 R (the real unique expression in the form x = x, + number space). Let V = {xe E: 1k1 II fl is the given norm in E. For each xeE, let C1x = 11(1 + IIxIDx. Then by a result on p.

is a homeomorphism of E into E and C1K is convex whenever is a bounded convex homeomorph be of It is clear that 0 E mt C,C and (ci C,C) C1C = (0C1C) L. Let the homeomorphism of cI C1C Onto V which, for each ray p from 0, carries the segment p fl cl C1C linearly onto the segment p fl V. Then C?AC = V (L fl 8 V) and CZIOC =8 V L. Let denote the "stereographic projection" of V (z} onto the set J' = H + [0, lIz. Specifically, for each ray p which emanates from z and intersects H, carries the segment p fl V affinely onto the segment p ti I'. Then is a homeomorphism of V (z} onto J' which carries the set C2C1C onto J' (L fl H) and carries W.IOC onto H — (L fl H). For each x = x1 + x2z€J', let C4x = x1 + Then the transhas all the desired properties, and this completes the formation C = proof of Theorem 1.3. Note that two of the three possibilities for L in 1.3 were treated completely on pp. 30-31 of [17]. However, previous treatment of the case in which L is a linear subspace of infinite deficiency depended on the existence of a continuous linear projection of E onto L (17] or at Least the existence of a continuous inverse for the natural homomorphism of E onto E/L (21). By a theorem of Bartle and Graves 12), such an inverse exists when E is complete, but an example in 120] shows that it may fail in general. 42:of [18],

K is convex and 0€ K. Thus the set

1.4 CoRoLLARy.

If C is a closed convex body in an infinite-dimensional

normed linear space E and C contains no line, then C x [0, 11 is homeo,norphic with C x (0, 1].

PROOF. We assume without loss of generality that 0 e mt C.

The space

E x R is normable, C x [0, 1[ is homeomorphic with C x [—1, oo[ and C x [0, 1) with C x [—1, 1]. The sets C x [—1, co[ and C x [—1, 1) are closed convex bodies in E x R which have (0,0) as an interior point. The characteristic cone of C x [—1, oo[ is not a linear subspace and (since C contains no line) that of C x [—1, 1] is either not a linear subspace or is equal to ((0,0)), a subspace of infinite deficiency. Thus the desired conclusion follows from 1.3. The following remark is easily verified:

TOPOLOGICAL CLASSIFICATION OF CONVEX SETS

If F1 and F2 are closed linear subspaces of the same finite deficiency in a topological linear sPace E, there exists a linear of E onto E which carries F1 onto F2. If E is a normed space there is a positively homogeneous norm-Preserving homeomorphism of E onto E which carries F, onto F2. We conclude the section with 1.6 COROLLARY. Suppose the norined linear space E admits (for each finite n) a closed linear subspace of deficiency n which is homeomorphic with its own unit cell. Then E is homeo,norphic with all its closed convex bodies. Pnoor. In view of 1.3, it suffices to establish the following for each n: Se—If F is a closed linear subspace of deficiency n in E, then E F x [0, 1]" means "is homeomorphic with"). Obviously S0 is true. Now suppose Sn-, is known and consider a closed linear subspace F of deficiency n. Choose x€E—F and let G = F+ Rx, a closed linear subspace of deficiency n—i. 1.5 PR0I'oslrIoN.

Let C and D be the unit cells of F and G respectively, whence (using 1.5 F and D G. It follows that

and the principal hypothesis of 1.6) C

[0, 1]

[0, 1)

whence

and

(using the inductive hypothesis) (F x [0, 1]) E G x [0,

F x [0,

x [0,

the proof is complete.

2.

Spaces homeomorphie with their positive cones. When B is an uncondi-

tional basis for a normed linear space E, each point x E admits a unique It will often be convenient to supexpression in the form x = press the distinction between x and f1, writing supp x for for fib, The following stronger "substitutive" property of substitutive bases follows easily from their definition and trivially from the representation theo• rem stated in the Introduction. 2.1 PROPOSITION. Suppose B is a substitutive basis for a normed linear space E, and w,x,y, and z are points of E. If IlwII IIxILIIylI Itzll, and

etc.

(suppw

U suppx) fl (suppy U suppz)

=

0,

then In order to describe an auxiliary transformation which plays an important

role in the proof of 2.3, we define a fan to be a closed convex set F c such that F is symmetric with respect to the line {(r, s): r = s}, and (1, 0), (0, 1')} c Fc cony 0), (1,0), (0, 1), (1, 1)} cony The gauge of F is

the function i-,

on R2 which

s) = inf lm > 0:

Thus tions

(I r

is

defined as follows:

j, IsI) mF}

is a norm for Rt whose unit cell is the union of F with F's reflecthrough the origin and across the two axes. We say that the fan F

HARRY CORSON AND VICTOR KLEE

42

can be spread provided there exists a mapping ço of the quadrant Q = such that the following ((r, s): r 0 s} onto the halfplane {(r, s): $ 01 conditions are satisfied:

ç(1,O)=(1,0)

10

and 9'(O,l)=(—l,O);

ço is positively homogeneous; 3° is norm-preserving (i.e., on Q); = 4° is distance-increasing (i.e., r,(q'x — coy) r,(x — y) for all x, y E Q); — coy) is Lipschitzian (i.e., there exists m < oo such that 5° 2°

— y) for all x, y Q). Each mapping which satisfies these five conditions will be called a spreading of F. çø

We have been unable to determine whether every fan can be spread. However,

the following can be proved:

2.2 PROPoSITIoN. Let F be a fan and P = {(r, s): r,(r, s) = 1). SupPose there exist a subset Zof 1', an arc A in Z and a number m< such that F—Z

is finite, 1' has finite curvature kz at each point z C 2, and sup kA m inf kZ. Then F can be spread. The proof of 2.2 will be omitted, as it is rather tedious. However, some important special cases are easily treated. For example, suppose F is the fan v, is the Euclidean norm for R2. Let lJ, so that + 1(u, v): u 0 ço denote the mapping which sends the point v = (r, 0) (polar coordinates) in the first quadrant into the point v' = (r. 20) in the upper halfpiane. Clearly .p satisfies conditions

1° —

r,(P

for

—

3°, and we claim that q)2

—

q'? 4r,(P —

p and

q in the first quadrant. Because of the homogeneous and rotational nature of the transformation, it suffices to consider the case in all points

which q = (1, 0), and then for p

I + r2— to be established

= (r, 0) the above inequalities reduce to

1+

+4r2—8rcos0,

for all r 0 and 0€ [0, But then the first inequality is equivalent to the assertion that 3(1 — r)2 + 4(1 — cos

obvi-

ous and the second is

(This argument, due to Branko GrUnbaum, is simpler than our original

0.

Two other cases of special interest are those of the fans F for which

r,1(u,v)=IuI+lvI and rr,(u,v)=max(luI,IvD. These fans can be spread as

for 0

follows: For p = (r, 0) in let for

n/2.

the first

quadrant,

and for

let çc'(p) = (r, 20);

(Here r represents the r,,distance from 0 and 0 is

the usual

Euclidean angle.)

From 2.2 we see that the following result applies to the spaces for each infinite cardinal Alternatively, the proof for reduced to

that for

and

can be

by means of Mazur's homeomorphism (241.

2.3 Suppose B is an infinite substitutive basis for a normed linear space E, and E is complete or is the linear extension of B. Then there exists

43

TOPOLOGICAL CLASSIFICATION OF CONVEX SETS

which carries the

a positively homogeneous norm-preserving homeomorphism 0) onto the entire space E. positive cone {x E:

Let PA = {x e E: 0 for all a A). Let us assume that E is complete and denote by L the linear extension of B. We shall define a positively homogeneous norm-preserving

For A c B and xe E, let x4 = LEA

PROOF.

(as well as distance-increasing and Lipschitzian) homeomorphism of L fl PB onto L and then show that it can be extended to carry PB onto E. Let 'p and for x€P{a,b), define be a spreading of F. For a,beB with a +

= where So(xa, Xb) =

P(b}

xb)a + 502(Xa, x6)b

Xb), 502(xg, Xo)) £ R2.

,

is positively homogeneous,

Since

Since 'p is norm-preserving and B is a substithe same must be true of is distancetutive basis, is norm-preserving. Similarly, it is clear that increasing and Lipschitzian. The desired homeomorphism 1) wiLl be described in terms of the transformations We shall discuss only the case in which B is countable since the general case can be handled by a simple extension of this reasoning. be an enumeraLet I denote the set of all positive integers and let tion of the members of B. For each i, let denote the transformation of onto For XE PB, let e1x and then proceed = inductively to define x for I e I (1). We shall write x, for

x L fl PB and k is the largest index for which I k. Thus for x 6 L fl PB, we may define (Although the are defined on all of PB, it is not obvious

x

* 0, then

=

7)kX for all

= that this limit exists when XE PB L.) Since is a positively homogeneous norm-preserving honieomorphism of onto P1o,4-d, it is easy to verify that 'j is a positively homogeneous norm-preserving biunique transformation

of L fl PB onto L. We wish to extend i, to the entire positive cone PB (which is the closure of L fl PB) and then to prove that the extended is a homeomorphism of PB onto E. The following fact is crucial: 10. Suppose XE PB and Then the sequence x,ô. for each n €1. is

a Cauchy sequence.

To prove 10 we consider Xe PB, m €1, and n €1 with m > n, and check that the following equalities and inequalities are valid where A is the Lipschitz constant for so and hence for all II

— 'ix" II = II 'i,.x"' —

I

I

:z

I )?nX" — 'id'

+ —

+

(1),1X — 7JnX )nôn + (m,x

+

—

= II

i-'ni.3

+ 1k" —

+

—

" — _4A J,II Ijmn—IX —

fl\2

W(\ I + k'i,,—,X ),'+i°n+I

*tI II —$AIIX,,+IUa+IIVT,X—X —

I

I

II

+ lix" —

I + X — X11+1 f

I

II

HARRY CORSON AND VICTOR KLEE

whence clearly lim,,,-..., — 'ix" I = 0 and ('iX'),E, must be a Cauchy II Three of the above steps are immediate from the relevant definisequence. I

tions, and the other five can be justified as follows: = 'i,,x", for x? = 0 for i> n. Then use the fact Note first that o o •-• o and each of the transformations is normthat ij, = i,,, o preserving. affects = $ This follows from the facts that certain coordinates are zero, only the ith and (i + 1)th coordinates, and the 'i,'s are produced by composition

of the

Note that x? = x? for 1

i n, and that the transformations

affect

only certain coordinates. and relevant definitions. Refer to 2.1, the Lipschitzian nature of and 1' are alike in all coordinates affected by ij,,—,. Note that Next we claim flu. For each x PB, — I = 0. in fact, (I — 'i,x"

I

=

+

—

=

—

+

+

II ('i,,x —

+

—

II

All

+

—

II

+ lix — x"" II

+ lix — x"'t Ii which clearly converges to 0 as n —.

'ix" e E. We shall establish Now for each x e PB, let 'ix = is a Cauchy sequence in L fl PB, then 'i(Xi),E, is a Cauchy 111°. if sequence in L. Let x = urn1-.., xi PB. Then for each i e / and n e i, ii'ix — vixill

— 'ixiiI + li'ix —

and

U'ix" — 'ixill = lim,....ii'i1x" — il'i1,x — 'i,,xiJl +

'if

'ix — to), — whence 111° follows.

= li'i,,x" — 'i,,xili —

= 0 by the definition of 'ix (which was justified by A"(Ix — xiii, = 0 by 11°, and of course ii'i,,x

From 111° we conclude that 'i (defined now on all of PB) is a continuous transformation of PB into E, and of course 'i(L fl PB) = L. Clearly '7• is norm-preserving and positively homogeneous. Since each transformation increases distances, the same must be true of 'i and consequently 'i is biunique is continuous. Thus to complete the proof of 2.3 it remains only to and show that i'jPB E. This may be done directly or by observing that 'iPB, being homeomorphic with the complete space PB, must be a Ga set in E 126], and that a linear subspace which is a C3 set must be closed [25]. This dis-

poses of 2.3 for the case in which B is countable.

TOPOLOGICAL CLASSIFICATION OF CONVEX SETS

45

Spaces homeomorphic with their closed convex bodies. 3.1 THEoREM. Each of the following conditions on a normed linear space £ insures that E is homeomorphic with all its closed convex bodies; 3.

£ is E admits an infinite substitutive basis B whose linear extension is E; E is a Banach space and some closed linear subspace of E admits a continuous linear transformation onto a Banach space homeomorphic with the space I. PRoOF. In view of 1.6, it suffices to prove that whenever F is a closed

linear subspace of finite deficiency in E, then F is homeomorphic with its own unit cell. With the aid of 1.5, we see the sufficiency of proving that each space £ (as described above) is homeomorphic with its unit cell. In the third case, this follows from the theorem of [21]. For the second case, choose b e B and let H denote the hyperplane {x E: Xb = OJ. Since B — {b} is a substitutive basis for H, it follows from 2.3 that and Since clearly PB—(HflPB)x [O,oo[ it follows that E H x [0, oo[ and then from 1.3 that E is homeomorphic with its own unit cell. (Notice that this argument also applies to the completion of such an E.)

Consider, finally, the case in which E is Let C denote the set of all finitely supported members of the Hilbert space and let U and W be the unit cells of £ and G respectively. Let x. be a basis for E and the natural basis for G. From the second case it follows that C— W.

We know from [23] that E G and wish to show that U

W. Since E is

separable, it can be strictly convexified [6]; that is, there exists in E a symmetric closed strictly convex body U' whose gauge functional is a norm which generates the given topology in E. Of course U U' by means of a simple "radial" homeomorphism. Now Xr, and z0 are B-systems (in the sense of [12; 23]) for E and G respectively, as normed by the gauge functionals of U' and W, so the associated Kadec-Bernstein mapping [12; 231 is a homeomorphism of E onto C which carries U' onto W. Reviewing the assembled facts, we deduce that U W and complete the proof of 3.1. 3.2 COROLLARY. Each of the following conditions on an infinite-dimensional Banach sPace implies that E is homeomorphic with all its closed convex bodies: E is a closed linear subspace of a Banach space which admits an unconditional

basis;

E is reflexive;

E is a

norm-separable subspace of a conjugate space;

E is a closed subspace of finite deficiency in a space CX for some infinite compact Hausdorff space X. PROOF. For the first two cases, let S be an infinite-dimensional separable closed linear subspace of E. Then S is homeomorphic with I by results of Bessaga and Pelczyñski [3] and Kadec [13], so the desired conclusion follows

from 3.1. In the third case, E itself is homeomorphic with I by a theorem of Kadec [14] and Klee [20]. In the fourth case, we may assume (in view of

HARRY CORSON AND VICTOR KLEE

46

cX. Then = = O}, where 1.5) that E= {çoeCX: çpx1 çox, = (adopting a suggestion of M. Jerison) since X is infinite there exists in X an If cY=X infinite sequence of distinct points such that ci is a discrete subset of V. i 2} Otherwise, we choose a then of y10 in the complement of some neighborhood of and subsequence treat in a similar manner. Continuing in this way, we find that either j 1} is discrete. In either i 2) is discrete for some j or the set case, there exists an infinite sequence z0 of distinct elements of X such that is discrete. For each E CX, let vjço = Z. Then of the set Z = U course is a continuous linear transformation of CX into CZ, and it is easily includes every function on Z such that —, 0 and Cx. = 0 verified that for all i. Thus liE has a subspace equivalent to c0, and since c0 is homeomorphic with I by a theorem of Kadec [12], the desired conclusion again folI

lows from 3.1. 4. Convexity and uniform continuity. The basic idea of this section is apparently due to Efremoviè [81 and was suggested to us by John Isbell.

Some related ideas are employed by Edeistein [7). An t-chain in a metric space (M, p) is a finite sequence (Po, of points pt) < e. Such a chain is said to join Po and of M such that always A subset S of M will be p,1, and its length is the number , p.). called quasiconvex provided for each ö > 0 there exists 4'a < oo such that any

two points x and y of S for which p(x, y)> ö can be joined (for each by an i-chain in S of length at most çbip(x, y).

0)

4.1 PRoPosITIoN. Each of the following conditions on a metric set S implies that S is quasiconvex: a°. S is a complete metric space which is metrically convex; b°. S is a convex subset of a normed linear space; c°. S is the boundary of a bounded convex body in a normed linear space; d°. S is a bounded starshaped subset of a normed linear space. PROoF. In cases a° and b°, each two points x and y of S lie together in a subset of S which is an isometric image of the interval [0, p(x, y)]. Thus S is quasiconvex with cl's = 1 for all ö> 0. For case c° we shall employ the following fact from [16]: (t) Suppose Y and Z are bounded closed convex bodies in a normed linear space E, and 0€ mt (Y fl Z). Let p denote the norm of E, rj the gauge functional of Y, C the gauge functional of Z, and (for all x€ E) li*x = max (lix, li(—x)),

Let r denote the radial mapping of E onto E which carries Y onto Z, so that tO = 0 and rx = (iix/Cx)x (or x E 10). Then for all u, tie E, p(vu — rv) Mp(u — v), where Cix = max (Cx, C(—x)).

M=

+ C

p

+

C/\

P

Now suppose C is a bounded convex body in a normed linear space E, and assume without loss of generality that 0€ mt C. With U denoting -the unit cell

TOPOLOGICAL CLASSIFICATION OF CONVEX SETS

47

freE: lixil 1}, there are numbers >0 and < oo such that A1Uc Cc A,L1. Consider two points x and y of the boundary OC. We shall prove that for each e > 0, x and y can be joined in OC by an e-chain of length no greater This is enough for our purpose, though doubtless the than (2 + result can be improved. Let H be a two-dimensional linear subspace of E such that {0, x, y} c IT. By a simple and well-known argument, the plane IT must contain an elliptical disc D centered at 0 such that D c U fl IT c i/iD. Then of course A1D c C c i/TD. Let r denote the radial transformation of IT onto II which carries C onto D. By the result (t) we see that for all u, v e IT it is true that IIru — rvil klIu — vu and

hr the norm in E, k — (2 + VT)/A1,

where liii is and k' = 2 V'TA2 + V2A/A1 Now consider an arbitrary e > 0 and points x and y of OC. Let p denote the

of the elliptical disc D, a norm for if. Clearly rx and ry can be joined in the ellipse OD by an e/k'-chain Po, whose p-length is at most (ir/2)p(rx — ry). But then the points form , an e-chain joining x and y in 0C, and the II Il-length of this chain is gauge

k'

—

Pi)

rIjTX—ryll This completes the discussion of case c°.

For case d°, let P be a point from which S is starshaped and let r = sup,€3 II s —

P11

x,yeS with

< oo. Consider an arbitrary ö > 0.

of length lix — length

I

I P — y II

It is clear that whenever then x and p can be joined in [x,p] by an echain r and p and y can be joined in [p,y] by an e-chain of

P11

r,

—

An e-chain (p0, chain.

y

whence x and y can be joined in S by an e-chain of length II. •,

The proof of 4.1 is complete. consisting of n + 1 points will be called an (n, e)-

4.2 PsoI'osliION. A metric space (M, p) is quasiconvex if and only for each .3> 0 there exists $, < oo such that any two points x and y of S for which p(x, y) >.3 can be joined, for each e e JO, 1(, by an (n, t)-chain in S with ne 198p(x, y) PROOF. Since the length of an (n, is at most ne, the aiffl part is obvious. Now suppose (M, p) is quasiconvex, .3 > 0, x and y are points of M with p(x, y) >.3, and (q0, - - ., q,,) is an e/2-chain in M which joins x and y and is of length (y Let r0 = 0; and having chosen r0 < r1 < let be largest integer k n such that Since , qj < e. (q0, . . •, > e/2 is an e/2-chain, it is clear that r,+1 > r and that

HARRY CORSON AND VICTOR KLEE

48

< n. Lets be such that r= n andletpi=QYL (/'0, , p,) is an (s, e)-chain joining x to y, and since j s — 1, we have when

(s — 1)

j-

,

p)

p(q,_1 ,

Then 1 > e/2 for 1

#ap(x, y),

2çt'ap(x, y) + e (2çbc + 1/ô)p(x, y). This completes the proof of 4.2. A mapping jp of a metric space (M, p) into a metric space (M', p') wilt be called Lipschitzian for large distances [resp. Lipschitzian for small distances) prosuch that p'(çox, coy) Asp(x, y) whenvided for each J> 0 there exists 4 ever x, ye M and p(x, y) > o [resp. p(x, y) < 8]. Of course, 'p is Lipschitzian if and only if 'p is Lipschitzian for both large and small distances. It is evident

whence Sc

Lipschitzian for small distances, then 'p is uniformly continuous. Suppose (M, p) is a quasiconvex metric space and 'p is a 4.3 (M, p) into a metric space (M', p'). uniformly continuous transformation of Then ço is Lipschitzian for large distances and hence carries bounded subsets of

that if 'p

is

M into bounded subsets of M'. PRooF. The uniform continuity of so is not used in its full strength. Indeed, it suffices to assume the existence of a number £ JO, and a number < b whenever p(u, v) e. Consider an arbitrary b < oo such that p'(çou, o >0 and let be as in 4.2. Then if x andy are points of M with p(x,y) > O,x

can be joined to y by an (n, e)-chain (p0, But then p'(çox,'py)

.,

nb

in M with ne

y).

PA p(x,y)

whence ço is Lipschitzian for large distances.

Now consider a subset X of M whose p-diameter is a number d < co an arbitrary pair x and y of points of X. Then when p(x, y) e

Consider

we have p'('px,

soy)

e, p'(çox, coy)

p(x, y)

Consequently the p'-diameter of çoX is at most b max (1, 194/e), and the proof

of 4.3 is complete. 4.4 COROLLARY. ij X is a bounded convex subset of a normed linear space, or is the boundary of such a set, then every uniformly continuous metric image of X is bounded.

(Still open, apparently, is Gorin's question LiOl as to whether Hubert space is uniformly equivalent with some bounded subset of itself.) 4.5 COROLLARY. Suppose Y is a convex cone with vertex 0 in a normed linear space, and ço is a positively homogeneous transformation of Y into a Added in p,oof: Professor Isbell has pointed out the similarity between this result and a theorem of Atsuji 101.

TOPOLOGICAL CLASSIFICATION OF CONVEX SETS

normed linear space. If ip

is

49

uniformly continuous, then ço is Li/.'schitzian.

5. Additonal remarks and problems. Although we conjecture that every infinite-dimensional normed linear space E is homeomorphic with its unit cell C, we cannot prove even that E x [0, ii E x [0, IL. (By 1.4, C X [0, 1)

C x [0, 1[.)

It would be especially interesting to extend Theorem 2.3 to the case of a Schauder basis, though this would probably require new techniques. Bessaga

and Pelczyñski [81 have proved that a separable Banach space is homeomorphic with I if it has a closed linear subspace homeomorphic with 1. They

have pointed out (in a letter) that by a theorem of James [Ii], every nonreflexive separable Banach space E admits a closed linear subspace F with a

Schauder basis such that the positive cone Q generated by this basis is homeomorphic with the positive cone P in 1. Now P 1 by our Theorem 2.3, and thus if 2.3 could be extended to show Q F we could conclude that F I, whence E 1 by [31. In conjunction with Kadec's result [13] for reflexive spaces, this would show that all separable infinite-dimensional Banach spaces are homeomorphic, thus solving the famous .problem proposed by Fréchet [9] and Banach [1]. We repeat the question: If is a Schauder basis for a Banach space E, must E be homeomorphic with the positive cone consisting of all points E for which always 0? Note that by combining 2, part of 3, and the elementary reasoning of [19], we obtain a new proof, independent of the rather complicated isotopy considerations of [171, that Hubert space is homeomorphic with its unit cell We now indicate still another proof of this fact which replaces the reasoning of § 3 and [19] by another argument similar to that in § 2. Let P denote

the positive cone and U the unit cell of the Hilbert space P and let E be the set of all finitely supported members of 12. It follows from 2.3 that P 12, P ii U U, E (1 P E, and E fl P fl U E ii U. To show that P U it suffices to produce a homeomorphism h of E fl P (1 U onto E fl P such that both h and h1 preserve Cauchy sequences, for then h can be extended to a homeomorphism of P fl U onto P and the desired conclusion follows. Since the detailed construction of such a homeomorphism is rather technical, we shall merely describe a simple "unfolding" homeomorphism which carries E fl P fl U onto E (1 P; the same idea can be employed with more care to control the Cauchy sequences as well. A typical stage of the unfolding process is described in the following 5.1 LEMMA. Suppose A, B, and C are distinct spherical cells in a Euclidean space S. concentric at 0 with A c B c C. Suppose H is a hyperplane through 0, p' is a ray from 0 orthogonal to H, and J is the closed halfspace H + p' Suppose K is a pointed closed convex cone in H with vertex 0, and K' = K + p'. Then there is a homeo,norphism of K' fl B onto K' fl C such that Ex = x whenever xe (K' fl A) U (H fl B), and E(Q fl B) = Q fl C for each "quadrant" Q of the form p -f- p' where p is a ray in K emanating from 0. -

Paoop.

Make the appropriate construction for one of the quadrants p + p'

HARRY CORSON AND VICTOR KLEE

50

and then extend it to the others by rotation of the space S about the ray p'. 5.2 PROPOSITION. If X is the space consisting of all finitely supported of 12 and P and U are the positive cone and unit cell of I, then Pfl P. PROOF. It suffices to produce a homeomorphism of P fl 2 U onto P. For n = 1,2, ..', let X,. = fx e X: x' = 0 for i> n} and let p. be the ray {x e X: x" 0, 0 for i * n}. For n 2,3, -••, let be the homeomorphism of X,. A Pa nU onto X,, n Pfl (n + 1)U which is obtained from 6.1 by the following assignment of roles for the principal actors in 6.1: S = X, A =

0), K =

Let g. be the extension of agreeing that if (x', x2, . , f, 0,0, -) dmnf., then {x

X,,:

g,,x

— —

1/

X._1

1

,

.2

,

ii P.

-., X* ,

\ u, • * ', -i- IA A

1= obtained by

.. , X11+1 , X

Finally, let and let hx=lim...,.h,,xforeachxePfl2U. It can be verified that /t is a homeomorphism of P fl 2U onto P.

0. M. Atsuji, Uniform continuity of continuous functions of metric spaces, Pacific J. Math. 8 (1958). 11—16. 1. S. Banach, Theorie dee operations lineaires, Monogr. Mat., Vol. 1, Warsaw, 1932. 2. R. G. Bartle and L. M. Graves, Mappings between function spaces, Trans. Amer. Math. Soc. 72 (1952), 400-413. 3. C. Bessaga and A. Pelczyfiski, Some remarks on homeomorphisms of Banach spaces, Bufi. Acad. Polon. Sci. Sbr. Sci. Math. Astronom. Phys. 8 (1960), 757-761.

4. C. Bessaga, On topological classification of linear metric spaces, Fund. Math. (to appear).

5. F. Bohnenblust, An axiomatec characterization of L,-spaces, Duke. Math. J. 6 (1940), 627-640.

6. J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), 396-414.

7. Michael Edelstein, An extension of Banach's contraction principle, Proc. Amer. Math. Soc. 12 (1961), 7-10. S. V. A. Nonequimorphism of Euclidean and spaces, Lispehi Mat. Nauk 4 (1949), no. 3 (30), 178-179. (Russian) 9. M. Fréchet, Lee sep50.. abatraits, PariB, 1928. 10. E. A. Gorin, On uniforsn-topoiogiov,,Z embedding of metric space. in EucUdean and Hiibert spaces, Uspehi Mat. Nauk 14 (1959), no. 5 (89), 129-134. (Russian) 11. Robert C. James, Reflexivity and the aupremuin of linear functional., Ann. of Math. (2) 66 159-169. 12. M. I. Kadec, On homeoinorphi.sms of certain Banach spaces, Doki. Akad. Nauk SSSR 92 (1953), 465-468. (Russian) 13. , On strong and weak convergence, Dokl. Akad. Nauk SSSR 122 (1958), 13-16. (Russian)

14. , On the connection between weak and strong convergence, Dopovidi Akad. Nauk RSR 1959, 949-952. (Ukrainian) 15. 0. H. Keller, Die Homoiomorphie der keavexen Mengen im Hilbertac/ten

____, TOPOLOGICAL CLASSIFICATION OF CONVEX SETS

Raum, Math. Ann. 105 (1931), 748-758. 16. Victor Klee, On a theorem of Bête

51

Amer. Math. Monthly 9 (1953),

618-619.

17. Convex bodies and periodic ho,neoinorphisma in HiThert space, Trans. Amer. Math. Soc. 74 (1953), 10-43. 18. -, Some topological properties of convex sets, Trans. Amer. Math. Soc.

78

(1955), 30-45.

19. , A note on topological properties of normal linear spaces, Proc. Amer. Math. Soc. 7 (1956), 735-737. 20. , Mappings into normal linear spaces, Fund. Math. 49 (1960), 25-34. 21. , Topological equieolsnce of a Banach space with its unit cell, Bull. Amer. Math. Soc. 67 (1961). 286-290. linear spaces (to appear). 22. , Topological structure of 23. Victor Klee and R. C. Long, On a method of mapping due to Kadec and Bernstein, Arch. Math. 8 (1957), 280-285.

24. S. Mazur, Une remarque ear l'homeomorphie des champs fonctionnels, Studia Math. 1 (1929), 83-85.

25. S. Mazur and L. Sternbach, (Jber die Boretschen Ty pen von Zinearen Mengen, Studia Math. 4 (1933), 48-53. 26. W. Sur lee ensembles corn plots dun espace (D), Fund. Math. 10 (1928), 203-205.

27. J. J. Stoker, Unbounded convex point sets, Amer. J. Math. 62 (1940), 165-179.

AN UPPER BOUND FOR THE NUMBER OF EQUAL NONOVERLAPPING SPHERES THAT CAN TOUCH ANOTHER OF THE SAME SIZE BY

H. S. M. COXETER

Introduction. The problem indicated in the title is regarded as the case zr/6 of the n-dimensional analog of the problem of packing small circles of angular radius 4, on a sphere in ordinary space (Coxeter 1962). Reasons are given for believing that the number of such (n — 2)-spheres that can be packed on an (n — 1)-sphere has the upper bound 1.

4,

2F%_l(a)/F$(a)

where a is given by sec 2a =

sec 24, + n — 2

and the function F is defined

recursively by ir

= F1(a) = 1. This upper bound is attained with the initial conditions when the (n — 2)-spheres are inscribed in the cells of a regular polytope

in which case a = 2. Kepler, Gregory, and Hales. Kep!er (1611) discovered the cubic closepacking of equal spheres in Euclidean 3-space. Their centers, forming the face-centered cubic lattice, may be taken to have integral Cartesian coordinates

with an even sum; then the common radius of the spheres is i/i/2 Each sphere touches 12 others; for instance, the one with center (0,0,0) touches those with centers (0, ±1, ±1), (±1,0, ±1), (±1, ±1,0). These 12 centers are the vertices of a quasi-regular polyhedron, the cuboctahedron, which was described by Plato. The existence of the packing is closely associated with the fact that the circumradius of this polyhedron is equal to its edge-length. Among the unpublished papers of David Gregory, H. W. Turnbull found notes of a conversation with Newton in 1694 about the distribution of stars of various magnitudes. The question arose: Can a rigid material sphere be brought into contact with 13 other such spheres of the same size? Gregory said "Yes", and Newton said "No"; but 180 years were to elapse before r conclusive answer was given. Stephen Hales (1727) made a random close-packing of dried peas, filled the interstices with water, and kept the vessel closed. In his own words, i compressed several fresh parcels of Pease in the same Pot, with a force equal to 1600, 800, and 400 pounds, in which Experiments, tho' the Pease dilated, yet they did 53

H. S. M. COXETER not raise the lever, because what they increased in bulk was, by the great incumbent

weight, pressed into the interstices of the Pease, which they adequately filled up, being thereby formed into pretty regular Dodecahedrons.

Hales presumably reached his conclusion by observing some pentagonal faces on his dilated peas. Modem versions of his experiment have confirmed the prevalence of pentagonal faces. Bit the solid cells could not all be regular dodecahedra. For, since the dihedral angle of this Platonic solid is less than 120°, three specimens with a common edge will leave an angular gap (about 10° 19'). In other words, twelve unit spheres with their centers at the vertices of a regular icosahedron of circumradius 2 (and edge 2 sec 18° = 2.102924- -) will all touch one central unit sphere, but will not touch one another. Gregory may have imagined that, by letting the twelve spheres roll on the central one until the gaps are all concentrated in one direction, there might somehow be enough space for one more, so that the central sphere would touch thirteen others. It may reasonably be argued that there is enough space (actually enough for 13.397 spheres, as we shall see in § 9). The problem is to arrange -

the twelve spheres so as to concentrate the free space and make room for the thirteenth. The impossibility of such a redistribution was eventually proved by R. Hoppe (see Bender 1874), thus settling the Gregory-Newton controversy in favor of Newton. Simpler proofs were given by Gunter (1875), SchUtte and van der Waerden (1953), and Leech (1956). 3. Dirichlet and Barlow. To be precise, an arrangement of equal spheres

is called a Packing if no point of space is inside more than one sphere. In any packing, we may associate with each sphere a Dirichiet region (or Voronoi Polyhedron) consisting of all the points that are as near to the center of that sphere as to the center of any other. Such regions, each surrounding a sphere, fit together to fill the whole space without interstices (Dirichlet 1850; Voronoi 1908). The density of the packing may be defined as the average of the ratio of the volume of a sphere to the volume of the Dirichiet region that surrounds it. For instance, the density of Kepler's cubic close-packing is equal to the ratio of the volume of a sphere to the volume of a circumscribed rhombic dodecahedron (the reciprocal of the cuboctahedron), namely, —

074048

(see, eg., Coxeter 1961, pp. 407-408). W. Barlow (1883) described an equally dense packing in which each sphere still touches twelve others, although the centers do not form a lattice. For this hexagonal close-packing the Dirichlet region is a trapezo-rhombic dodecahedron, which may be constructed by Cut-

ting a rhombic dodecahedron into equal halves by a plane perpendicular to six parallel edges and then sticking the two halves together again after turning one of them through 60° (Fejes T6th 1953, p. 173). 4. Schiafli. Schläfli (1855) studied polytopes in Euclidean n-space and in spherical (n — 1)-space, extending spherical trigonometry from n = 3 to n > 3. He found that an (n — 1)-sphere of radius R in Euclidean n-space has (n — 1)dimensional content (or "surface") and n-dimensional content (or

______________

EQUAL NONOVERLAPPING SPHERES

"volume") JIR*, where —

2K*/t

—

r(f + 1)

—

so that S1 = nj, = S1 = 2,

J1 =

2,

=

2n, S3 = 4r, S4

f2 = ir, f, =

4 J4

23r1,

=

=n8,

S5 =

=

, J3

8

,f,

=

,rt —i—,

He defined (on p. 177) a remarkable function F,1(a), in terms of which a regular simplex of dihedral angle 2a in such a spherical space (with R = 1) has (n — 1)-dimensional content S,F,,(a). When a

= ,r/4, the bounding hyperplanes of the simplex are mutually orthogonal, so that the content is therefore = 1/n!.

When a = n/3, the spherical simplex corresponds by central projection to one of the n + 1 cells of a regular Euclidean simplex; therefore its content is + 1), and

Fn(f) = 2t/(n + 1)!.

(4.3)

The boundary of the spherical simplex decomposes the spherical space into two regions: an "inside" whose dihedral angles are 2a and an "out-ide" whose dihedral angles are 2ir —

2a;

therefore

2'n! F,(a) +

— a)

= 1,

that is, F,,(a) +

(4.4)

— a) = 2k/n!.

In particular, (4.41)

When n =

3,

area 6a —

F3(a) =

(4.5)

When n =

the "simplex" is a spherical triangle with angles 2a, 2a, 2a and therefore, since S3 = 4,r,

2,

ir

—

3

we merely have a circular arc of length 2a; therefore, since

H. S. M. COXETER

56

S1=2ir, (4.6)

Since a regular Euclidean simplex has dihedral angle arcsec n in n dimensions, or arcsec (n — 1) in n — 1 dimensions, and since an infinitesimal spherical simplex is Euclidean, we deduce

a

for

=0

(4.7)

(n — 1).

= ÷ arcsec One of his most brilliant discoveries (Schläfli 1855, pp. 167, 168) is that dF,.(a) =

where

is given by

2a —2. Thus the Schlafli function can be defined recursively by the formula =

sec

(4.8)

ir

sec

sec2P=sec28—2,

(uUsc(.—lflhI

with the initial conditions Fo(a) = F,(a) = 1.

(4.9)

This function has been studied from another standpoint by Ruben (1961, p. 262). 5. fohn, Peechi, Gulnand. Various formulae connecting the angles of the general n-dimensional simplex have been discovered by Poincaré, Som-

merville, Höhn, and Peschl. Their equivalence has been established by Guinand (1959), who observes that, in the symbolic or umbral notation (with standing for a,,), Höhn's formulae become

is the average angle at an (n — k)-cell, expressed as a fraction of the whole angle at an (n — k)-flat. When the simplex is where a,, (symbolically

regular, we have

a,,

=

F,,(a),

so that the formula = becomes

,

' —

, ( F,,(a)=(—1)'Z" .,' F,_,(a).

When r is even, FT(a) cancels, leaving

j.

EQUAL NONOVERLAPPING SPHERES

(—2)'

But

F,_1(a) =

57

(r even).

0

when r is odd, we obtain an expression for F,(a) in terms of simpler

functions:

'

F,(a) =

(5.1)

(r odd).

.,'

From such equations with r =

Fi(a), Fa(a),

1, 3, 5, •••, n, where n is odd, we can eliminate obtaining

-,

=

(5.2)

+

—

--•

—

(n Odd),

where the series ends with the term in Fo(a) and the coefficients are the same as in the expansion tanh x = x

+

—

—

(Schläfli 1855, p. 178).

Alternatively, we can deduce (5.2) from Peschl's formula (2B + a)' = (B + a)'

(r even)

(Guinand 1959, p. 59) or —

1)

i-i

j.

F,1(a) =

j-2

0

or

Since even,

=

0

J.

D B1F,_,(a)

(r even).

when a = (arcsec n)12, we have, for this value of a and n =

+

—

—

(a = (arcsec n)/2, n

For instance, (5.4)

F.(—!2

arcsec 4) = —

a 1

F, (.-i- arcsec 4) 2

farcsec4

—

2 5

—

I

15

even).

H. S. M. COXETER

(Schläfli 1855, p. 182). Other special values that are known explicitly are

f,r\

191

1

(Schläfii 1855, p. 181; Coxeter 1935, pp. 18, 19).

Taking (4.2), (4.3), (4.4), and

(4.7) into consideration, we can now assert that F4(a) is elementary for at

FIGuRE 1

least 13 particular values of a. These results provide a convenient check when numerical computation is applied to (4.8) (with n = 4) or, equivalently, to (5.6)

F4(a)

C—l+3.Ir

ydx,

J—1+(a1c,ecS)Fz

where y is related to x by the symmetrical equation

secirx+seciry+2=O. Figure 1 is a sketch of the curve (5.61) with F(a) indicated as an area in

(5.61)

59

EQUAL NONOVERLAPPING SPHERES

a typical case, actually the case F(ic/4) = 1/24. Figure 2 is a sketch of the curve y = F4(x), with the thirteen "elementary" points marked, namely (with the temporary abbreviation = arcsec k): 4s,

a F(a)

6.

Ifs. - 2\

0

3ks

f:

+= 5)

+:

1

2

191

1

24

15

900

3

409 900

— +s.

8

5

15

8

1/12

s.\

'4599

—

I 2

Minkowukl and Blkhfeldt. We shall return to this function, but first 2 3 5 8

8

is

409 900

1

3

191

900

2 15

0 0

T

ir -i

r

211

FIGURE 2

1!. 2

3i12w 5

3

311

4

H. S. M. cOXETER

we must mention one more historic event: Minkowski (1905, p. 247) proved that the densest packing of equal spheres whose centers form a lattice (in whose centers have" Euclidean n-space) consists of spheres of radius integral Cartesian coordinates with an even sum, not only when = 3 but also when n = 4 or 5. Thus the sphere with its center at the origin touches 2n(n — 1) others, whose centers are given by the permutattons of

In

(±1,±1,0,0)

(n=4)

(±1,±1,0,O,O)

(n=5).

other words, the centers of all the spheres are the vertices of the honey.

and "half measure polytopes" whose cells are cross pol; topes comb hrA (Coxeter 1930, p. 365; 1948, p. 155). When n = 4, this uniform honeycomb is regular, since hr4 = and h85 = {3, 3, 4, 3); the spheres are inscribed in the Dirichiet regions of the lattice, that is, in the cells (3, 4, 3) of the dual honeycomb {3, 4, 3, 3).

Such a packing is rigid for all values of n, but is the densest lattice packing only when n = 3, 4, or 5. When n = 6, 7, or 8 (Blichfeldt 1935) the closest packings are such that each sphere touches 72, 126, or 240 others, respectively. The centers of the spheres are the vertices of the uniform honeycombs ,

331 ,

(Coxeter 1930, pp. 379, 393-397; 1948, pp. 201-205). In other words, for spheres

of radius 3/V2,

convenient

coordinates for the centers are nine integers,

mutually congruent modulo 3 and satisfying the equations

x,+x3+x3=x4+x3+x.=x1+x5+x,=0 Xi+X2+X3+X4 +x-t-x6=x7+x4+x,=0

(n=6), (n=7), (n = 8).

Since Barlow's hexagonal close-packing (in Euclidean 3-space) has the same density as cubic close-packing, namely, = 0.74048..-

it is natural to ask whether some other non-lattice packing may have a greater density. According to Rogers (1958, p. 610), "many mathematicians believe,

and all physicists know" that no such denser packing exists in 3 dimensions. But this has never been rigorously proved, and dense non-lattice packings may reasonably be expected when the number of dimensions is sufficiently great.

Leech has observed that in five dimensions, as in three, there is a nonlattice packing that has the same density as the closest lattice packing. In fact, the three-dimensional honeycomb ho4, formed by the centers in cubic close-packing, may be broken up into a sequence of layers, each consisting of tetrahedra hi3 = a3 and octahedra /93 sandwiched between two tessellations

EQUAL NONOVERLAPPING SPHERES

Hexagonal close-packing is derived by shifting the layers until each triangle {3) of {3, 6) belongs to two tetrahedra or to two octahedra, instead of belonging to one tetrahedron and one octahedron. Somewhat similarly, the five-dimensional honeycomb hl5 may be broken up into a sequence of {3, 6).

layers, each consisting of ht0's and p4-pyramids sandwiched between two fourdimensional honeycombs {3, 3,4, 3). Leech's non-lattice packing is derived by and shifting the layers until each cell {3, 3, 4) of {3, 3, 4, 3) belongs to one

or to two pyramids (the two one pyramid, instead of belonging to two halves of a When the lattice requirement has been abandoned, the packing problem in Euclidean space does not differ greatly from the packing problem in spherical space. The latter begins with the problem of packing small circles on the surface of an ordinary sphere. Meschkowski (1960) states this vividly, as follows: On a planet, say, ten inimical dictators govern. How must the residences of these gentlemen be located in order to get as far as possible from one another? 7. Rankin. Consider a regular polytope inscribed in an (n — 1)-sphere of denote the angle subtended at the radius 1 in Euclidean n-space. Let center by an edge (Coxeter 1948, p. 133). Clearly, the N,, vertices are the packed on centers of N0 nonoverlapping (n — 2)-spheres of angular radius the (n — 1).sphere (e.g., when n = 3, spherical caps on an ordinary sphere). regular simplex afr (1 k is), we have If the polytope is a = it — arcsec k and N,, = k + 1 (Coxeter 1948, p. 295). Rankin (1955) proved that this arrangement is the closest of k + 1(11 — 2)-spheres (1 k is). In other words, letting N(ç6) denote the maximum number of such "caps," of that can be packed on the unit (ii — 1)-sphere, we have angular radius

+1

for ir—arcsecn

it.

Davenport and Hajós (1951)-proved that NW') cannot take a value between n + 1 and 2n, the latter value appearing when the polytope is the cross-poly(Coxeter 1948, p. 121). In other words, tope or "n-dimensional octahedron" although

N(f) = 2n, we have (7.3)

N(#)=n+1

for

Thus NW') is known precisely whenever Rankin

ir/4.

For smaller values of

was content to prove an inequality which yields, for instance,

N(it/10) < 258 when n =

4.

8. The propoeed new bound. It is intuitively obvious that n equal (n — 2)spheres are packed as closely as possible when they all touch one another, SO that their centers are the n vertices of a regular simplex of (angular) edge

H. S. M. COXETER

3

In other words, we may reasonably expect that the density of a packing of at least is "caps" of radius 4' can never exceed 24'.

I-, is the sum of the vertex angles of the simplex, expressed as a fraction of the total angle at a point in n — I dimensions, V is the (n — 1)-

where

dimensional content of such an (ii — 2)-sphere, namely,

v= is the (n — 1)-dimensional content of the simplex. The analogous conjecture for a packing in Euclidean space has been justified by Rogers

and

(1958), with the conclusion that the density cannot exceed

2"(n!)'(n +

arcsec

n).

The truth of the original "intuitively obvious" conjecture has been established

for n = 3 (see Figure 3) by Fejes Tóth (1959, p. 311; see also Coxeter 1962). If it holds universally, we can deduce a bound for N(4') by observing that, since the (n — 1)-sphere is packed with N(4') (as — 2)-spheres of content V, the density is

N(4')j_, and therefore

63

EQUAL NONOVERLAPPING SPHERES

Here 2' is the content of a regular (n — 1)-dimensional spherical simplex and a is the sum whose dihedral angle 2a is a certain function of its edge of the n vertex angles, each equal to the content of a regular (n — 2)-dimensional spherical simplex having the same dihedral angle 2a. By (4.1), 2' = a =n

—

1)!

so that aS,1 — 2F,1_I(a)

2'

—

F,,(a)

-

To express a in terms of & a simple procedure has been suggested by Rogers. Let the vertices of the simplex have coordinates

(c + a, a, -- -, a), (a, c + The angle

a, •-

a),

...,

(a,

a,

c

+ a) -

subtended by two of these points at the origin is given by =

2a(c

+ a) + (n — 2)a'

2ac

— —

+

c+2ac+na2'

whence C

2ac + na

+1

The internal angle 2a between two bounding hyperplanes

{c+(n—1)a}x1—a(x2+-—+x,,)=O and

a(x1+--is given by

cos2a= {c+(n —

2ac+nI

+(n— 1)a2

— c2

+(n

whence

sec2a= 2ac =

Thus our final result is

sec

C

+n—1

+ na + n —2. 2

— lX2ac+ na*)

H. S. M. COXETER

64

sec2a =

(8.1)

+ n —2.

This upper bound is attained whenever the pattern of n spheres touching one another can be continued over the whole spherical space, that is, whenever the centers of the spheres coincide with the vertices of a polytope where p ,r/a. In the notation of Coxeter (1948, pp. 160-162; see also Böhm 1959, 1960) we may conveniently take (n — 2, n) = (sect a)/2,

(j,k)=k—j and (

l(n—1—JXJ-fl,n)—1

(j+1,n—l)

n—2—j

—

whence

(j,n) =

2—j),

—(n

(—1,n)

2

secta — (n

—

1)

(1,n) cos 2a

I —(n—2)cos2a sec

sec 2a — (n

—

2),

as before. For instance, in four dimensions, the regular 600-cell {3, 3, 5) has a = it/S and = 'rhO, so that — —

's.10)

fir\

(2

1

— —

1

—'

The remaining four-dimensional results are epitomized in Figure 4, which shows the smooth curve obtained by plotting the upper bound 2F3(a) F4(a)

for

qS

(it — arcsec

,

sec2a

3)/2, and also the "step function" which expresses the

exact value of for every qS ir/4. For convenience, the accompanying table gives the values of qS and a in degrees instead of radians, so that (4.5) yields 2F;(a) = (a — 30)145:

65

EQUAL NONOVERLAPPING SPHERES

a 2F3(a)

F4(a)

52*

54

18

45

52*

54

36

45

60

72

90

108

3

3

14 15

4 3

26

15 1

1

2

1

3

409 900

8

900

840 — 191

4

1560 409

15 4

212

120

8

5

120 2

15

45

18

135

144

7

38 15

15

5 8

599 900

56 15

2280 599

780

84°

15 14

13 12

It 10 9 8 7

6 5 4

3

2

0 60

12°

180

24°

30°

36°

42°

48°

540

60°

660

72°

90°

96°

FIGURE 4

With only two exceptions (one repeated), every entry in this table is exact. The exceptions are 54j and 52*, which are the approximate numbers of degrees in (arcsec(—3))/2 and (arcsec (—4))/2, respectively. Since F4(a) = 0 when = 0, N(0) 00: the curve for

has the vertical

H. S. M. COXETER

axis as an asymptote. The point (18, 120) is too high to be conveniently shown. The "white" marked points are N(45) = 8, N(521) =

5,

N(54j) =

4,

N(60) = 3, N(90) = 2.

The first three of these, along with N(18) 120, are the only places where and N(#) exactly agree. The "black" marked points help us to plot because, although they have no geometric significance, the curve for they indicate further known cases in which the expression 2F3(a)/F4(a) takes rational values. The lower part of this curve arises because the relation sec2a = +2 makes each value of 4 yield two supplementary values for a, and hence two values for 2F,(a)/F4(a), the larger of which serves as a bound for N(4) while Since the smaller remains in the interval from 3.733 to 3.815 for all < 4

irF4(a) \

= is zero when curve occurs where 9. The cue when

/ sec 2a tan 2cr

ir

the minimum on this nearly straight portion of the = 2r/3.733, which is about 480.

= ir/6.

In any number of dimensions, the centers of three

mutually tangent equal spheres form an equilateral triangle.

Hence the problem indicated in the title of this paper amounts to finding an upper bound for N(ir/6). By (8.1), such an upper bound (attained only when n = 1 or 2) is 2

(9.1)

arcsec arcsec

It is interesting to see how this works out for small values of n. When n = 2, the value

=6 is

attained by the six circles that can obviously touch one of the same size. When n = 3, the bound 2F2(J_ arcsec 3) 2

Fa(+ arcsec 3)

=

6 3 — rfarcsec 3

=

13.39733257

(Coxeter 1958, p. 757), like Rankin's

N=

+ 8 = 16.48528...,

is not strict enough to disprove Gregory's conjecture that 13 spheres in

67

EQUAL NONOVERLAPPING SPHERES

ordinary space could touch one equal sphere. When n = 4, we use (5.3) to compute

\

1

=

\

/1

1

2

=

—

1 farcsec4

2 —

Since, by (4.5),

/1

\= arcsec 4)

arcsec4

1

— —,

it follows that -

/1

2

arcsec 4)

arcsec 4)\

= 5(1 +

\

1

5—2ir/arcsec4 )

= 26.44009910

Though this number is less than Rankin's —

1Y4

10—

3r

—

.

it leaves open the question whether a 3-sphere in Euclidean 4-space can touch only 24 others (as in the closest lattice packing) or 25 or 26. 2a) 10. Leech'8 computation. When n > 4, it is convenient to write for F.(a), so that =

arcsec

x).

In this notation, the formulae (4.6), (4.7), (4.8), (5.2), (5.3) become

f2(x)= arcsec x

f,,(n—1)=0,

,

ir

IA)

—

=

xVx

—1

4-ia .3(x) + —

—

I

(n odd),

7(1) +

—

(n)

(n

even),

and the function (9.1) is simply

Leech helpfully undertook to evaluate this function for by tabulating arcsec(x —

ii

8.

He began

2)

—1

to twelve decimal places for x =

3.5(0.1)8.2,

and integrated by the trapezium

II. S. M. COXF.TER

68

rule with central difference correction at the ends of the ranges. From f4(x) he deduced

=

+

—

and

f.(6) = +1446)

17

—

+

He obtained fe(x) by a similar integration of •

f4(X — 2)

zrxVx'

—1

Finally, he deduced

f,(x) =

17

— 114(X) +

—

and

= +f6(8)

62

+

—

The results are as follows: =

f(6) = 0.00003 02840 12, /6(7) = 0.00010 31285 78, 15(8) = 0.00018 73637 63, /7(7) = 0.00000 14072 22, /7(8) = 0.00000 64990 72, = 0.00000 00531 355,

31255 82, 14(5) = 0.01248 11393 80, /4(6) = 0.01686 59385 17, f4(7) 0.02014 88228 26, 0.02268 05970 96, = 0.00051 25449 97, f3(6) = 0.00129 93981 97, 0.00652

2 13(4)/14(4) = 2

26.440...

= 48.702

2

fs(6)/fG(6) =

85.814...

2 16(7)1/7(7) = 146.570 2

f7(8)/fe(8) = 244.622...

Thus the maximum number of spheres that can touch another of the same

size (in Euclidean

is

N1=2,

where N2

=6, 26, 48, 85,

126

N1

240

N6

146, 244.

N3=12,

EQUAL NONOVERLAPPING SPHERES

69

Notice the remarkably small difference between the upper and lower bounds for Ne. This closeness seems to be a manifestation of the extraordinary "near-regularity" of the honeycomb 5k,, whose vertices are the centers of the spheres in the lattice packing. Any simplicial cell of the honeycomb indicates a set of nine spheres, perfectly packed. Of every 137 cells, 128 are simplexes and only 9 are cross polytopes (Coxeter 1948, p. 204). The same honeycomb appears again in the theory of Cayley numbers (Coxeter 1946, p. 571) and in the related theory of the simple Lie group E8 (Coxeter 1951, pp. 412-414, 420— 426). 11. Spherea In non-Euclidean spacee. The problem that plays the title role in this investigation remains significant when the equal spheres are packed

in a non-Euclidean space (spherical, elliptic, or hyperbolic).

The 3-dimensional

case has already been considered by Fejes Toth (1954), who uses spheres of radius 1 in a space of curvature K. The number of such unit spheres that can touch another unit sphere is evidently is half the angle where of the equilateral triangle of side 2 whose vertices are the centers of three mutually tangent spheres. By the familiar non-Euclideari formula

sinA= sinai/K sin

for the angle A of a right-angled triangle ABC, we have

_sini./R'_

-

—

sin 2i/TI?'

1

2cosifK'

whence + 1.

sec 2çS = sec

By (8.1), an upper bound for the number of spheres that can touch one is 2F,,-,(a)

where a is given by

sec2a=sec2VR'+n—l

-

In the special case when n = 3, we see from (4.5) and (4.6) that 12a 6a —

By (11.2), we have sec 2a —

1 + tant a 1 — tant a

— 1

1

=

+ tant

— 1 —

tan2

sec 2VR'

VR' +

cot2 a + Since

'

+ 1, whence tan2 a

—

tan2

a—

1 —

2.

1

1 —

tan2

VTK'

H. S. M. COXETER

a=

N—26'

this is equivalent to the formula of Fejes Toth (1954, p. 162). 12. An asymptotic bound. How fast does the number of spheres increase when the number of dimensions tends to infinity? According to Rogers (1961), when the number =

sec

2a

n+1

(cf. (11.2)) is bounded,

/1 + nb I I' 2e \.IS y 2 n! ett' k.irnb) When we apply this asymptotic formula to (8.1), we have b ' sec — I in Sec 2qS in the numerator. Thus the asymptotic the denominator, and b upper bound for is

( \n — ii

21

(12 1

\312

\. e

I

This excels, by the factor 2/e, the bound cos

n312

sin'_I

of Rankin (1955, p. 139; our is his a). Setting = in (10.1), we deduce, for the bound named in the title of this paper, the asymptotic expression

REFERENCES

W. Barlow 1883. Probable

nature of the internal symmetry of crystals, Nature 29,

186 188.

C. Bender 1874. Bestimmung der grössten Anzahl gleich grosser Kugein, welehe sick auf eine Ku gel von demselben Radius, wie die übrigen, aufiegen lassen, Grunert Arch.

56. 302 313.

II. E. Illichfeldt 1935. The minimum values of positive quadratic forms in six, seven and eight variables, Math. Z. 39, 1-15. 1959. Simplexinhalt in konstanter Krummung betiebiger Dimen j. nion, J. Reine Angew. Math. 202, 16-51. !nhaltsmessung im

konstanter Krummung, Arch. Math. 11, 298-

II. S. M. Coxeter 1930. The pot ytopes with Trans. Roy. Soc. 1.ondon, Ser. A 229, 329 -425. 1935.

The

vertex figures (1). Philos.

functions of Schtäfii and Lobatschefsky, Quart.

Ser. 6, 13-29. 1946.

integral C'ayley numbers. 1)ukcMath. J. 13,

561 -578.

J.

Math. Oxford

EQUAL NONOVERLAPPING SPHERES

71

1948.

Regular polytopes, Methuen, London; 2nd ed.. Macmillan, New York (to

1951. 1954.

Extreme forms, Canad. J. Math. 3, 391-441.

appear).

Arrangements of equal spheres in non-Euclidean spaces, Acta Math.

Acad. Sci. Hungar. 5, 263-274.

Close-packing and froth, Illinois J. Math. 2, 746-758. Introduction to geometry, Wiley, New York. 1962. The problem of packing a number of equal nonoverlapping circles on a sphere, Trans. New York Acad. Sd. (2) 24, 320-331. H. Davenport and G. Hajós 1951. Aufgabe 35, Mat. Lapok 2, 68. G. L. Dirichlet 1850. (iber die Reduktion der positiven quadratiachen Formen mit drei unbestimmten ganzen Zahlen, J. Reine Angew. Math. 40, 209-227. L. Fejes Tóth 1953. Lagerungen in der Ebene, auf der Ku gel und im Raum, Springer, 1958.

1961.

Berlin.

On close-packings of spheres in spaces of constant curvature, Pub!.

1954.

Math. Debrecen 3, 158-167. 1959. Kugelunterdeckungen mind Kugelüberdeckungen in

konstanter Krumi'nung, Arch. Math. 10, 303-313. S. Günter 1875. Em stereometrieches Problem, Grunert Arch. 57, 209-215. A. P. Guinand 1959. A note on the angles in an n-dimensional stm pIes, Proc. Glasgow Math. Assoc. 4, 58-61. S. Hales 1727.

Vegetable staticks, London.

J. Kepler 1911. Dc nive sexangula, Gesammelte Werke Rd. 4, pp. 259-280, Beck, Munich.

J. Leech 1956. The problem of the thirteen spheres, Math. Gaz. 40, 22-23. H. Meschkowski 1960. UngelUste und unlösbare Problem. der Geometne, Vieweg, Braunschweig. H. Minkowskj 1905. für arithmetische Aquivalenz. J. Reine

Angew. Math. 129, 220-274. R. A, Rankin 1955. The closest packing of spherical cap. in n dimensions, Proc. Glasgow Math. Assoc. 2, 139-144. C. A. Rogers 1958. The packing of equal spheres, Proc. London Math. Soc. (3) 8, 609-620. 1961.

An asymptotic expansion for certain &hldfli functions, J. London

Math. Soc. 36, 78-80.

H. Ruben 1961. A multidimensional generalization of the inverse sine function, Quart. J. Math. Oxford Ser. (2) 12, 257-264. L. Schlafli 1855. Rt1duction d'une integrale multiple, qui corn prend i'arc Se cercie et l'aire du triangle sphrrique comm. Ca. particuliers, Gesammelte mathematische Abhandlungen, Bd. 2, pp. 164-190. (See also pp. 219-258.) IC. Schutte and B. L. van der Waerden 1953. Des Problem dcv dreizehn Kugein, Math.

Ann. 125,

325-334.

G. Voronoi 1908.

134,

Recherches sur tee paralléloèdres primitives, J. Reine Angew. Math.

198-287.

UNIVERSITY OF TORONTO

ROTUNDITY

Ii F. CUDIA The theory of Finsler spaces and, I. Introduction and general more recently, that of spaces which are locally Banach (of arbitrary dimension) has been developed considerably in the last decade. The theory of curvature

in spaces of the latter type has been considered in Ewald and Kelly [1] and in a paper by Lorch [IL For further references to such spaces see Segal IlL Eells [1], Laugwitz [1], and Rund (1).

However in the article (2) H. Busemann observes, "Clearly the step from no geometry to euclidean geometry is incomparably much wider than that from euclidean geometry to Riemannian geometry. The volume problem makes

it more than probable that an analogous situation exists for Finsler spaces. Therefore the study of Minkowskian geometry ought to be the first and main step, the passage from there to general Fin,sler spaces will be the second and simpler step." While there is no obvious analogue to volume and area in a general Banach space, nevertheless it is in the spirit of this quotation that this survey of metric geometry in normed spaces is conceived, namely, as a necessary and important preliminary to the study of infinite dimensional Finsler spaces. In the paper [3) Busemann uses the technique of replacing the unit sphere of a Minkowski space by a more convenient convex body. The technique is useful in general Banach spaces and leads to isomorphism results which are results serve to enlarge the uses of geometry considered in § V.

in the theory of Banach spaces and for that reason are as important as the investigation of the various geometric properties that the norm itself may possess. Although rotundity and smoothness concepts can be defined in general locally convex spaces, the proper domain of these ideas is in a normed space where the theory of linear homeomorphisms can be used to renorm the space without changing the topology so that the geometric theory of the new norm

can be used in the study of the space. Lying within the general framework of infinite dimensional Finsler spaces, then, the purpose of this report is to survey rotundity and smoothness concepts and the theory of linear homeomorphisms in normed linear spaces and especially to provide an up-to-date bibliography of papers which are concerned

with these ideas. The study of non-linear homeomorphisms in infinite dimensional normed spaces is not covered either in the survey or in the bibliography. Some applications to approximation theory are given in § VII but other applications to ergodic theory, fixed point theory, and extreme points are omitted. Aside from the notions of local uniform smoothness and midpoint local uniform smoothness none of the geometric properties considered is original. 73

D. F. CUDJA

74

The pressures of time forced the omission of a detailed study of these two properties. See however Cudia (11 for a discussion of related matters.

In preparing this report the author had the benefit of many stimulating conversations with Dr. M. M. Day whose grasp of the literature on the subject both as to location and importance gave this report any completeness it possesses. The notation and many of the results have been taken directly from the second section of Chapter VII of the book, Normed linear Any misinterpretation of these results or the quoted results of other authors however is entirely due to the present writer. A subset of a normed linear space is called a conrex body provided it is bounded, closed, convex, and has a nonempty interior. A convex body is said to be rotund provided every hyperplane of support of the body has at most one point of contact with the body. It is said to be smooth provided it admits through each boundary point only one supporting hyperplane. The definitions have been phrased to exhibit their dual character. By defining the polar body §81 or Bon of a given convex body in a Euclidean space as in Minkowski the rotundity nesen and Fenchel ti, p. 281 we can observe a duality between or smoothness of a convex body and its polar body. A nornied linear space is said to be smooth or rotund if its unit ball,

U= {x:IIxll 1) is smooth or rotund. For finite dimensional normed spaces the duality is complete because the conjugate space is then congruent to the space itself, under a norm such that the unit balls are mutually polar. A normed linear space B is said to be uni/orrnly rotund, Clarkson Ii), if there exists a function 8 such that 0 < 8(8) if 0 < then llyH = and llx—yU

2 and such that if II

=

1

X

I

—

ii

An

equivalent condition for the rotundity of a normed space is that the

boundary of the Unit ball contain no line segment. The compactness of the surface of the unit ball in a finite dimensional normed space implies the equivalence of uniform rotundity and rotundity for such spaces. Footnote 13), that the It has been observed, Alaoglu and Birkhoff dition that the unit ball have two support planes at the same point is equivalent to the condition that the unit sphere in the conjugate space contain a straight line segment. While this is not true in all Banach spaces, there is complete duality of smoothness and rotundity for reflexive spaces. This footnote is the first observation of the duality between rotundity and ness in infinite dimensional spaces that the author has been able to find. Klee [3j extends the concepts of points of smoothness and points of rotundity

to an arbitrary convex subset of a Hausdorff linear space F in the following manner. We work In terms of a linear space F of linear functionals on F. By an F-hyperplane is meant a set of the form I '(r) = {x:f(x) = r, XE E)

75

ROTUNDITY

for some nonzero fe F and some real number r. The set C is supported at a point ft by the hyperplane H = f'(r) if p C A H, C lies on one side of H, that is, supf(C) r or inff(C) r, and C is not contained in H; p is then said to be an F-supPort point of C, otherwise an F-nonsupport point. A point p of C is said to be a point of F-s,noothness provided all the F-hyperplanes which support C at p have the same intersection with the affine extention of C, a point of F-rotundity provided each F-hyperplane which supports C at ft intersects C only at p, and an F-exftosed point provided C is supported at p by an F-hyperplane H which intersects C only at p. The convex set C is said to be F-smooth provided each of its points is a point of F-smoothness, and to be F-rotund provided each of its points is a point of F-rotundity. There are three especially important choices for F: the space Es of all continuous linear functionals on E, the space of all linear functionals whose restriction to C is continuous, and the space of all linear functionals on E. FOf course when the convex set C has an interior point, rotundity coincide with the previous definitions for each of the above three choices for F. The geometry of Banach algebras has been investigated by Kadison Ill and 'F he latter show that irrespective of the norm Bohnenblust and Karlin the hyperplanes of support at the unit element on the unit sphere of a Banach algebra are inverse images of a total set of continuous linear functionals.

This is equivalent to utter "unsmoothness" at the identity element on the surface of the unit halt; more precisely, to the condition that the intersection of the unit ball with every two-dimensional linear subspace through identity has a corner at the identity. in a C5 algebra the hyperplanes of support at the unit element, i.e., the set of linear functionalsfwhich satisfy the conditions f(zi) = 1, and if ii = I is shown to coincide with the positive linear functionats normalized at u. Thus it is geometrically evident that a linear mapping of norm one of a C5 algebra into another is order preserving if it transforms the unit element into the unit element. Smulian 141 considers differentiability of norms in Banach algebras. The metric geometry (>1 semiordered Banach spaces was first studied by Smulian 131 who obtained a sufficient condition for reflexivity in terms of uniform Fréchet differentiability of the norm on a subset of the positive cone. More recently, Ando 12I, Anieniiya, Ando, and Sasaki Luxemburg 11), Mimes Ill, Nakano (ii, and Yamamuro ii; 21, have studied geometrical questions in modulared vector lattices. A modulared vector lattice is a boundedly complete vector lattice (Day 17, p. 96i) L on which a functional called a nw(Iulur is defined. The least upper bound of elements x and y in L is de-

noted by x V y and the greatest lower bound by x A of x is defined by lxi = (xv 0)— (x A 0).

.

The absolute value

With this notation, the modular m must satisfy the following conditions: (1) (2)

0 ';z(x) if

=

0

for all xeL; 0, then x

for all

0;

D. F. CUDIA

76

(3) (4)

for any xc L there exists a > 0 such that ,n(ax) < oo; for every xe L, ,n(Ex) is a convex function of E > 0, that is, a, >

0

implies 2

1 X) T{m(ax) +

m(x) m(y); implies m(x+y)=m(x)+m(y); monotonically increasing to y implies

(6) xAy=0 (7)

0

XA

m(y) = sup m(xA). A

For any p L, [p] denotes the projection operator associated with p,

Ep](x)=

for each

0.

VXA njpl

For any x€L,

[p](x) = [P][(x v 0) — (—x V 0)] = [p](x

V

0)— [p](—x v 0).

a positive linear operator in L. If P and q are in L then [N and only if for any x 0 in L we have [P1 is

[q) if

[qI(x).

Two norms can be associated with each modular m: tIxII

and

iIIxlIl = mt called respectively the firs! norm and the second (or sometimes ,nodular) norm of m. The modular norm is the Minkowski functional of the set {x: m(x) 1). The spaces (L, II . II) and (L, III'S flJ) can be shown to be (4B)-lattices in the terminology of Day [7, p. 98]. (L, II II) and (L, Ill) are isomorphic in the

sense of Banach [1, p. 180]. In fact, Nakano [1, § 40] proves that lll.*11l

IlxII 2111x111

of the metric geometry in modulared vector lattices cited above is directed toward finding conditions on the modular in order that one or the other of its associated norms be smooth, rotund, or uniformly rotund. We shall mention a few of these results in the appropriate sections of this survey. Most

II. of rotundity and smoothness properties; notation. In this section we list some generalizations and uniformizations of rotundity and smoothness. The abbreviations preceding the conditions will facilitate the statement of theorems in the sequel. In this section B will be a Banach space, U its unit ball; S, the boundary of U, defined by

ROTUNDITY

S=(x:lIxll =1); U', the unit ball of B*, and S' the boundary of U'. (R): B is rotund (R) if and only if S contains no line segments. (MLUR): B is midpoint locally uniformly rotund (MLUR) if and only if given £ > 0 and x0 in S there exists ö(i, x0) > 0 such-that

ttX0_ X±Yll8 whenever Jix—yll i and x,y are in S. (LUR): Lovaglia [1). B is locally uniformly rotund (LUR) if and only if given e > 0 and x0 in S there exists 8(e, x0) > 0 such that

'II

ll8

whenever lix— xolI i and xis in S. (UR): Clarkson [1]. B is uniformly rotund (UR) if and only if given i > 0, there exists 8(1) > 0 such that

whenever ljx—yIi > i, and x andy are inS. (kR): Smulian used (2R) in [4]; Fan and Glicksberg [1). B is k rotund (kR) if and only if every sequence (xe) from B such that

TE"; =1 11

urn

a convergent sequence. If every such sequence is weakly convergent then B is said to be (WkR), weakly k rotund. (WLUR): Lovaglia [1]. B is weakly locally uniformly rotund (WLUR) if is

and only if for each i with 0<1 0 such that for any f in S' with f(xo) = 1, f(x) 1 — whenever

and x is in S. (W*LIJR):

Lovaglia [1]. B* is weak* locplly uniformly rotund (W*LUR) jf

and only if for each i with 0 < 1 and each Jo in S' there is a such that for any x in S with fo(x) = 1 1(x)

1

—

whenever 1—

8

> 0

ii F. CUDIA

78

and f is in S'. (PLUR): Fan and Glicksberg

[21. B is Point locally unifor,nly rotund (PLUR) if and only if whenever a sequence (xe) in B with

=1 has no weak cluster point of norm < 1, and if, for some Xo ill S

then

=0. (S):

B is smooth (S) if and only if at every point of S there is only one

supporting hyperplane of U. (MLUS): B is midpoint locally uniformly smooth (MLUS) if and only if for each e > 0 and each x0 in S there is a 8(e, x0) such that

lko_

X+Yll
y are in S. ([US): B is locally uniform/v smooth (LUS) if and only if for each e and each x0 in S there is a 8(e, x0) such that whenever II x —

> 0

—yfl II

whenever (lxo — xfI (US):

for each

x0) and y is in S.

Day 141 and Nakano 12). B is uniformly e > 0 there is 8(e) > 0 such that

smooth (US) if and only if

i whenever Jix—yll (G):

and x andy are inS.

B is said to be Gateaux differentiable (G) if and only if

G(x,y)=Iim IIx+tylj—iIxIl £

exists for each x and y in S. (UG): Smulian [2]. B is said to be uniformly Gateaux differentiable (UG) if and only if the limit in (G) is approached for each y in S uniformly as x varies over S. (F): B is said to be Fréchet differentiable (F) if and only if the limit in (G) is approached for each x in S uniformly as y varies over S. (UF): B is said to be uniformly Fréchet differentiable (UF) if and only if the limit in (G) is approached uniformly as x and y vary simultaneously over S.

(A): B is (A) if and only if when a sequence

in B converges weakly

79

ROTUNDITY

to x0 and

limlix.lI = lix, II, then

—x0(l =0. Fan and Glicksberg [2]. B is (H) if and only if B is (A) and B is (R). (K): B is (K) if and only if when K is a convex set in B then D(Kri tU), the diameter of K n tU, tends to zero as I decreases toward the distance from K to 0. (D): Smulian [2). B is (D) if and only if for any f in S', when we define (H):

E(f,8)=

1—8) n U,

we have D(E(f, 8)) tending to zero as 8 decreases toward 0. (D*): Smulian [21. is (D*) if and only if for any x in S with E(x, 8) =

{f:f(x)

I

—

8)

fl

U'

we have D(E(x, 8)) tending to zero as 8 decreases towards 0. (DL): (D localized). The conclusion of (D) holds for each f in S' which attains its supremum in U at some point of S. (W*UR): Smulian [21. B* is weak* uniformly rotund(W*UR) if and only if for each £ > 0 and each x0 in S there is a positive 8(e, x,) such that if f' are in S' then the inequality and f1

11>1

implies that —

I

Smulian (21.

(WUR):

f2(x0) I <

£

B is weakly uniformly rotund (WUR) if and only if g) such that if x1 and

and each g in S' there is a positive are in S then the inequality

for each

> 0

÷

>1 —

implies that

ig(x1)—g(x2)i


(0): Klee [5]. B is osculatory (0) if and only if for every x0 in S the set of non-zero x in B for which

>

0

and every

lix — x01i — lxii > s < lix+ x011 — lxii is bounded. is

Suppose that L is a modulared vector lattice with modular ;n. Since a convex function of 0, it is natural to define the strict convexity of

D. F. CUDIA

80

as

follows: m is strictly

convex

if and only if for any x * 0 with

m(ax)
+

+

implies a = Before

defining a conjugate type of strict convexity, we introduce the

following notations:

if

urn m(x) — m((1

if

00

=

urn

m((1 + E)X) — m(x)

m(x) = 00,

if m(x) <

£

if m(x)=oo.

00

The limits cven,

,n(x) <

exist because of the Convexity of m(ex). Now m is said to be — —

e

e>o

for each non-zero XE L, and if lt÷(x) <

o

implies

.7r÷(x) =

implies there exists some projection operator [P1 * 0 such that

and ir..(x) <

LP]

[x)

and <

00

The modular m is uniformly convex if and only if for any positive numbers

and

there exists a 8(7J,

>0

such that

implies

m((

+ for all x *

+

0.

The modular m is uniformly even if and only if for any positive numbers and there exists a 8(ii, c) > 0 such that implies

+

m[C'

+ (a —

81

ROTUNDITY

These definitions for modulars are due to Nakano [1]. Following Efimov and Steèkin [1; 2; 3], we call a set M: (1) An existence set, if for any x X there exists a nearest element Ye £ M such that

for all x *

0.

IIx—yoII

=infllx—ytj. 1€ N

(2) A Chebyshev set, if for any xe X there exists a unique element Ye £ M such that IIx—y011

=inf lix—yll

B is (E) if and only if every closed subspace is an existence set. (C): B is (C) if and only if every closed subspace is a Chebyshev set. In [4], Phelps considers subspaces L having the unique extension properly: Each linear functional on L has a unique norm-preserving extension to the (E):

whole space.

(U): B is (U) if and only if every closed subspace has the unique extension property.

Several well-known Banach spaces are studied in the sequel, and for convenience we shall define them here. If I is an index set, let K(I) be the

Tyhonov cube; that is, the topological product P.€,J1 where each J' is the m(I) is the space of all bounded real functions on I

closed interval [—1, 1]. with

lixil =suplx(i)I. '€1

is the subspace of all those x in m(I) which vanish except on a Countable set. c0(!) is the subspace of those x in m(I) vanishing at infinity on the discrete space I. l,(I) foi p 1 is the set of those real functions x on I for m0(I)

which < oo -

= SE, Given a set X, a a•algebra F of sets in X, and a countably additive, nonnegative set function p defined on F, p is called finite, or a-finite, if there is a set X1, or a sequence of sets X,, in F each of finite p-measure such that every set in F is essentially contained in X1, or in U, X,. Lp(it) is the space of measurable functions on X such that IlfIILp = MCa)

<

is the space of essentially bounded measurable functions on X with

essential supremum for norm.

If Y is a topological space, C( Y) is the space of real-valued, bounded, continuous functions on Y.

In what follows I and sets.

represent countably infinite and uncountable index

D. F. CUDIA

Ill. Comparisons of properties. In this section some of the equivalent statements of the properties listed in § II are given. However the sequential versions, which are easily constructed, are omitted. In addition certain of the implications between these properties in a single Banach space are indicated.

Finally a few theorems on modulared vector lattices are included. In a general Hausdorif linear space with an arbitrary convex subset C and a subspace F of linear functionals, the following assertions are equivalent: C is F-rotund; every F-support point of C is an extreme point of C; every Fsupport point of C is an F-exposed point of C; which support C at distinct points are distinct. En a Banach space B, if C = Uand F B* each of these conditions is equivalent to B is (R). Ahiezer and Krein [11

show that B is (R) if and only if whenever W is a wedge in B in which norm is additive, then W is a half.ray {ix: I 0). B is (R) if and only if every two dimensional subspace of B is (R). Kraékovski and Vinogradov 11) give a condition equivalent to B is (UR): for 0 < <2 there exists > 0 such that if and H, are hyperplanes of support to U at the points x andy of Sand if llx—ylI e, then

1 +8(e).

kuston Ill shows that B is (UR) if and only if for 0 < e

2, there exists such that if 1€ S' and < 8(e) then D(E(f, vj)) < where D(E(f, is the diameter of the set E(f, = {x:f(x) 1 — '?) fl U. If /3 is (UR) then B is (LUR); Lovaglia 111 has shown the converse to be false. If B is (LUR) then B is (MLUR) and if B is (MLUR) then B is (R) and Anderson Ill has shown that neither of these implications is reversible. If B is (UR) then B is (kR) for any k 2 but Fan and Glicksberg [1) showed the converse false. if B is (LUR) then B is (WLUR) and if is (LUR) then 8(e) > 0

e

If is (W*LUR) according to Lovaglia 111. Fan and Glicksberg 121 show that if B is (LUR) then 13 is (PLUR). Fan and Glicksberg (1J show that if B is tkR I(WkR)l then B is ((k ÷ 1)R) I(W(k + 1)R)J for any k 2.

B is (S) if and only if every two dimensional subspace is (S). Mazur 121 showed that B is (S) if and only if 13 is (G). If B is (US) then B is (LUS) which in turn implies that B is (S). Day 141 gives a condition equivalent to 13 is (US): there exists a function 8 positive foi 0 < £ 2 such that lim6(s)=0 while

2—Ijx+yII lIx—yIl

=

and x,y are in S. Nakano 121 shows that B is (US) yll if and only if for any e > 0, there is a 8 > 0 such that if x is in S and whenever Ix — ILvil

v3 then

lix +-yli + II x —yII

2 + iIyIl

ROTUNDITY

B is (US) if and only if B is (UF). Clearly B is (UF) implies B is (UG) or B is (F) either of which implies B is (G). Klee [5] shows that B is (UG) implies that B is (0). It is also true that B is (0) implies B is (G). Smulian 121 indicates that B is (UG) is equivalent to B is (W*UR) and that B* is (UG) is equivalent to B is (WUR). B is (LUR) implies B is (H) and the converse is false according to Ander• son 11), where it is also shown that if B is reflexive and (H) then B is (MLUR). Lovaglia f 1] shows that B is (LUR) if and only if B is (DL) and B is (WLUR).

Fan and Glicksberg [21 note that a condition equivalent to "B is (K)" is obtained by requiring that the defining condition be satisfied only for closed hyperplanes or only for closed half-spaces. Day [7, p. 113] indicates that B is (kR) implies B is (K); and that B is (K) if and only if B is (D). B is (D) if and only if B is (DL) and every supporting hyperplane of U meets S. B is (DL) implies B is (H) according to Day p. 113]. Also, Day (1, p. 113], B is (LUR) does not imply B is (D) but B is (LUR) does imply B is (DL).

Fan and Glicksberg [2] show that B is (PLUR) implies that B is (H) and show that B is reflexive and (PLUR) if and only if B is (K) if and only if B is reflexive and (H). Whence B is (K) implies B is (MLUR) and B is reflexive and (PLUR) implies B is (MLUR). Yamamuro [11 and Ando 12] study conditions on the modular necessary and

sufficient to insure that the modulared vector lattice L be (R) under its first or second norm. L is said to be non-atomic provided for any 0 < z L there exist x,y in L with x> 0, y >0 and x A y = 0 such that Ando 121 proves that in a non-atomic modulared vector lattice L with modular in, (L, II II) is (R) if and only if m is strictly convex and m is infinitely inor — = = creasing, i.e, for every x * 0, = if r < (L, III Ill) is CR) if and only if m is strictly convex and -

-

infm(

x

\UlxIIIJ

(L.

II

-

II) is (S) if and only if in is even, infinitely increasing, and finite, i.e.,

for any x in L.

(L, III 'Ill) is (S) if and only if m is even and finite. m is

uniformly convex if and only if (L, II II) is (UR) if and only if (L, III . III) is

(UR). m is uniformly even if and only if (L, II

II)

is (US) if and only if

CL, III. III) is (US).

Related notions of flatness and uniform flatness are treated in Amemiya, Ando, and Sasaki [1]. IV. Product 8paces, quotient spaces, and subapaces. In this section questions

concerning the extent to which the geometry of the factors influences the geometry of the product of Banach spaces are considered. The remainder of the section deals with the extent to which the geometry of the space is in-

D. F. CUDIA

herited by its quotient spaces and subspaces. Let S be an index set and let X be a Banach space of real-valued functions on S. If for each s in S a normed space N is given, let PEN. be the space of all those functions x on S such that: (1) x, is an element of N. for every s in S, and is the real-valued function defined by (ii) if

= llx,lI for each $ in S, then is in X. We define the norm of x in PEN,, by (I xii = ii If X satisfies the condition that whenever is in X and E(s)

—

for all s

then

is in X and II'?Il

Ii

Eli

PEN. is a Banach space if each of the N, are Banach spaces. We shall by P,N. 1 and Clarkson [II showed that

then

denote

and (UR) if space. Boas 11) generalizes this to

is

a finite dimensional rotund Banach or PP(L,[O, 1)), and to PAN, where

or all of the N, are LQ[O, 1), and p X is L,[O, 1] and all of the N. are and q are greater than 1. The last two product spaces are the spaces of all or Lq[O, 1], respectively, which are functions on 10, 1] with values in integrable in the sense of Bochner [11. Day [2] showed that P2(lg;(lc)) is (UR) if there exist 1 <m M < :n such that m

M

for all I in w, and Day El) shows that otherwise is not even isomorphic to a (UR) space. Finally, Day [3] extends all of these results in showing that is (UR) if and only if X is (UR) and the spaces N. have a common modulus of rotundity. Since (US) is dual to (UR) (see § VI) a similar statement holds for the uniformly smooth product of uniformly smooth Banach spaces. Lovaglia [1] proved that PAB, is (LUR) if Xand all B, are locally uniformly rotund Banach spaces. Fan and Glicksherg [1) show that when I, > 1, and k

is an integer greater than 1, P,B, is (kR) if each of the B. are k rotund 1, P,B, is (MLUR) if each B, is (MLUR) and Day [5] shows that is (R) if X and each B, is (R) and proves a general theorem which gives that for p> 1, P,B, is (S) if each B. is (S). Anderson [1] shows that for p 1, is (A) if each B. is (A) and this combined with the result of Day [5] shows that for p> 1, is (H) if each B, is (H). For a closed linear subspace L of a Banach space B the quotient space B/L is a Banach space whose elements are the translates of L with the norm

Banach spaces. Anderson Eli shows that for p>

ROTUNDITY

infjlx+yII IIx+ LII = VEL [41, Day showed that B is (UR) [(US)] if and only if there is a common modulus of rotundity [smoothness] for all the quotient spaces of B and if and only if there is a common modulus of rotundity [smoothness] for all the two dimensional quotient spaces of B. Day 17, p. 114] observes that a sufficient condition for rotundity [smoothness] of B is the rotundity [smoothnessj of every two dimensional quotient space of B. It is further shown that this is not a necessary condition for rotundity and Day conjectured the same for smoothness. Klee [3] substantiates this conjecture and shows that if L is a reflexive subspace of B then smoothness or rotundity of B is transmitted to the quotient space B/L and that in general no more can be expected. Thus rotundity [smoothness] of every two dimensional subspace of B is necessary but not sufficient for the rotundity [smoothness] of every two In

dimensional quotient space of B. However, in the case that the two dimensional subspaces have a common modulus of rotundity [smoothness] these conditions are equivalent since then B is (UR) [(US)] and hence reflexive. Clearly, if B

is (UR) [(US)] then the subspaces of B have a common modulus of rotundity [smoothness]. V. Isomorphism. Banach [1, p. 180] called two normed spaces isomorphic if they are homeomorphic under a linear function. The general problem considered in this section is: When is a Banach space isomorphic to another Banach space which has more pleasant geometric properties than the first? This problem is equivalent to that of reforming the space without changing the topology in such a way that the new norm has some geometric properties not possessed by the first. Since the theory is still fragmentary this section is divided into two parts; general results, and results valid only for special spaces. The more complete results available for modulared linear lattices are inserted in both parts wherever it seems appropriate. Lower case letters are used as in Day [51, i.e., the phrase, "a normed space B is (q)," means that B is isomorphic to a normed space having property (Q). In the cases where B is isomorphic to a normed space which is both (P) and (Q) we shall write that B is (pq). Clarkson [I] proved that any separable Banach space is (r). Klee [2] showed that every separable reflexive Banach space is (r) and (s). Day [5] improved on both of these by showing that any separable Banach space is (rs). Finally, Klee [3] showed that any separable Banach space B is (rs) with 11* also (R) under the new conjugate norm. This result cannot be strengthened to also obtain smoothness for B * for m(h) = is not (s). In [4] James showed the existence of Banach spaces with any specified number of separable conjugate spaces. Note that if B is a Banach space whose nth conjugate space B " is separable and if n is even then B admits a norm with respect to which B is (R) and (5) and each of the even numbered conjugate spaces from B = B'°' to B IR is (R) and (S) in the norm induced on B" by

In the same manner, the rotund conjugate norm of that in B induces a rotund norm in each of the odd number conjugate spaces from B' = B* to

D. F. CIJDIA

B A similar situation prevails when n is an odd integer. Of course the may not be an nth conjugate norm. James [3] gives an new norm in example of this situation when n 4. Day [5] raised the following question: Is there a nonreflexive, nonseparable

space B such that B is (rs)? Wada (1] answered this in the affirmative by product (p> 1) of C0, the set of all sequences of real taking for B the numbers which converge to zero, and lp(IM) (p> 1). Still unsettled is the question of the existence of a reflexive space which is not (r) or of one that is not (s). Isomorphism questions in moduLared vector lattices can be given almost complete answers. Two modulars in1, ,n2 on a boundedly complete vector

lattice L are said to be equivalent if their modular norms are isomorphic. Every modular is equivalent to one which is infinitely increasing, and continuous, I.e., for any X€ L if m(x) = oo then sup

IriSki;

If the original modular is strictly convex [even], the new one can be also. Suppose that (L, in) is a modulared vector lattice which is complete under the modular norm and that 0 x. monotonically increasing with supA x x

that (L, (II is (r) if and only if (L, III ifi) does not contain a closed vector sublattice linearly isometric and lattice isomorphic to with a continuous norm It can be proven that a complete (x.. going monotonically to 0 implies U II goes monotonically to 0) is (r). According to a result of James [2, Theorem 1) and a result of Lovaglia [I, Theorem 3.1]. if a reflexive Banach space B has a basis then B is (lur) and hence (plur). By the results of III, this last conclusion may be replaced by (k), (d), (plur), or (h). Lovaglia also proves that if B has an unconditional basis and does not have a closed subspace linearly homeomorphic with co(Ie) then B is (lur) 1(wlur)1. Fan and Glicksberg L21 prove that if a B is (H) and Bk is separable then B is (lur). An immediate consequence is that if B is any separable normed linear space such that B is (K) (or (D)) then B is (lur). Kadec [1] shows that in the space CEO, 1], if for fe C[0, 11, rn i112 1 + sup If(r) —f(s)I 11111 = sup If(t)I + 1112] -

LJO

k

i2

Since every separable Banach space is linearly homeoII II is (H). morphic with a linear subspace of CEO, 1], Kadec 11] obtains the result that every separable Banach space is (h). Anderson (1] uses this to improve the theorem of Fan and Glicksberg stated above to say that if B* is separable then B is (lur), and hence every separable reflexive Banach space is (lur). then

That separability of Bk is not a necessary condition for B to be (lur) evident from an example of Phelps [IJ wherein

is

is shown to be (Jur) but

87

ROTUNDITY

is not separable. However, Kadec [2] has asbeing congruent to serted that every separable Banach space is (lur). Klee [4] shows that if B is a Banach space and B* is separable then B* is (h*), i.e., there is a topologically equivalent norm II . II in B* such that (B*, II U) is (R) and whenever U —' If U, and f,, is weak* convergent to is a sequence in B*,f in B*,

f, then lIfe

—f II —GO.

Fan and Glicksberg [1] show that under the same

hypothesis, B is (w2r). The last two statements are true in particular when B is separable and reflexive. Let A and B be normed linear spaces, and define

k(A,B)= iiriist sup inf uTah Then k(A, B)> 0 if and only if A is isomorphic to some subspace of B. [41

shows

Day

that a necessary condition for isomorphism of B or 1? with a

uniformly rotund space is that urn k(11(n), B) = urn k(111(n), B) =

0.

Further considerations concerning the uniform rotundability of normed spaces and modulared vector lattices and the modularibility of complete (AB)-lattices are given in VII. A fundamental question is under what conditions a vector lattice is modularable. Klee [21 proves that continuous linear images of a reflexive Banach space which is (r) [(s)J are also (r) [(s)]. Day [5] gives a related result whose application is not restricted to reflexive spaces: If B0 is a reflexive smooth

space and T is a linear function from B0 onto a dense subset of B, then B is (s) and B* is (r). We shall apply this smoothability theorem to the spaces c0(!) and L1(p) later on in this section. There are examples of nonreflexive nonseparable spaces which are (r) and (s), and nonreflexive spaces which are (r) but not (s), or neither (r) nor (s). There is no example yet known of a Banach space which is (s) and not (r). Ando (21 has shown that if the modulared vector lattice (L, m) is complete under the modular norm then (L, III III) is (s) if and only if (L, III III) does not contain a closed vector sublattice linearly isometric and lattice isomorphic to m(10) or

Day [5) considers isomorphism problems in product spaces. There it is shown that for p 1 the 1,(I) product of rotundable spaces is rotundable, the product of such spaces is (r), and if B is (r), then the space of continuous functions from K(I) into B is (r). If all B, are (s) and if I is countable or if p > 1, then the 4(I) product of the is (s). Criteria for rotundability and sinoothability in terms of subspaces are obtained by Day [5]. There it is shown that a normed space B is (r) if and only if there is a one-to-one and linear map of B into a normed space which

is (r), B is (s) if B is isomorphic to a subspace of a normed linear space which is (s). A related result is that B is (rs) if B is (s) and there is a oneto-one linear map from B into an (rs) space. Ando [2] shows that in a modulared vector lattice (L, m) which is complete under the modular norm,

1). F. CUDIA

88

III) is (ur and us) if and only if (L, Ii III) contains no closed vector sublattices linearly isometric and lattice isomorphic to C0(fe) or

(L, Ill

In a reflexive space B, an isomorphism of either B or B' determines an isomorphism of the other so that in such spaces all the duality results of § VI can be written in lower case letters. For nonreflexive spaces an isomorphism of B' may exist which is not determined by an isomorphism of B, but Klee [2] has observed that if B' is (r) [(s)J and if the new unit ball is weak' closed, then B is (s) [(r)j. For a pair of spaces B and B' in the nonreflexive case there are examples with B and B' both (rs) or B (rs) and B' (r) but not (s) or B (r) but not (s) and B' neither. There is no example yet known with B (s) and B' not (r). (But recall from § IV that Klee has produced a space where

B is (S) but B' is not (R).) Anderson [1) uses a result given in § Vito prove that if B' is (lur) and if the new unit ball is weak' closed, then B is (f). Other isomorphism results can be generated from those of § VI with the extra hypothesis of weak' closure of the new unit ball. As a further example (Anderson (1]), if B' is (f) then B is (mlur).

As an example of an isomorphism duality result in a modulared vector lattice (L, m), Ando (21 observes that L' is (r) if and only if m is equivalent to an even modular (see § VI). Relations between reflexivity and isomorphism results are given in § Vi!.

In the remainder of this section results on special spaces will be listed. These results will facilitate the application of the theory. All of what follows is in Day (51, Klee [3] or Wada [1]. In the following table, a plus (minus)

means that the space in that row has (does not have) the property in that column; a question mark means that the precise nature is not yet known. This table is the same as that in Day (5] but brought up to date in the light of present knowledge. In what follows and represent countably infinite and uncountable index sets. ço is a finite or a-finite measure sufficiently nontrivial that is not finite-dimensional; &' is a finite or a-finite nonseparable measure; A is a non-afinite measure. B separable cO(!R)

(rs) + ?

(r), (s)

(r), not (s)

(s), not (r)

not (r), not (s).

+ + ÷

m0(1.)

+ +

4(l),/>1

+

L1(i')

?

+ +

+

+

+

L1(A)

L,(ie),P>1 M(c)

+

+

M(A)

C(K(h))

+

ROTUWIMTY

The first examples of spaces not rotundable or smoothable were given in is is not (r). There it was shown that m0(I,,) and hence not (s) and if I is infinite, then m0(I) is not (s), so m(I) is not (s). For any is (s) if and only if p is a-finite and, index set 1, c0(I) is (r) and (s). M(p) is (s) only if there do not exist infinitely if and only if M(p) is (r). Day (51.

many disjoint sets of positive p-measure, i.e., only if M(p) is finite dimensional. C(K(f)) is (r). Day [61 showed that every (AL)-lattice is (r). We can apply the smoothability theorem given above and these results to conclude the non-

existence of linear maps of a certain kind. There can be no linear mapping I uncountable, with p > 1 into a dense subspace of of any 1,(I) or

or a dense subspace of m0(I), I infinite, or of M(p), if M(p) is not finitedimensional. However, Klee (3] proves that if I is an infinite set, then the is rotundable in such a manner that every Banach space Banach space having a dense subset of cardinality card (I) is congruent to a quotient with this rotund norm. Wada [1] shows that if X is a metric space, then C(X) is (r)

space of

[(s)]

if and

only if X is separable (compact]. VI. Duality. In reflexive spaces certain pairs of geometric properties are in duality, i.e., the conjugate or the space itself has one of the properties if and only if the space itself or the conjugate has the other. In general, no pair of geometric properties are in complete duality unless each is strong enough to imply reflexivity. For this reason, in the absence of reflexivity there are only fragmentary implications from one to the other of the pair. We shall attempt to collect these implications and show how in the reflexive case, complete duality is øbtained. However, we leave to the reader the problem of using the equivalences between properties which were stated in § HI to obtain further pairs of dual properties that are not given here. The theory of dual properties in modulared vector lattices is also sketched in this section. It will be helpful in organizing the results of this section to center these results around the concept of differentiability of the norm. There are four different kinds of differentiability of the norm: (G) Gateaux differentiability,

(UG) uniform Gateaux differentiability, (F) Fréchet differentiability, and (UF) uniform Fréchet differentiability; and each of these in the reflexive case gives rise to a different level of duality.

Smulian (11 showed that if the unit ball in a Banach space B is weakly compact, then B* is (G) if and only if B is (R); and if every continuous linear

functional attains its supremum on the unit ball of B, then B is (G) if and only if B is (R). Thus if B is reflexive, (G) and (R) are in complete duality. From the equivalence between (G) and (S) in any Banach space it follows that in the reflexive case there is complete duality between (S) and (R). Without reflexivity the best that can be said is given by Klee [21: if B* is (R) then B is (S) and if B is (R). According to the results in is not (s). Thus B can be (R) with B* not is (r) but = § V, (S). In [7, p. 114] Day observed that is (R) ((S)1 if and only if every two-

dimensional quotient space of B is (S) ((R)]. This result coupkd with a

D. F. CUDIA

90

proposition of Klee [3] which states that if B is a separable normed Linear space and L is a nonreflexive closed subspace of B, of deficiency 2, then B admits a norm under which B is (S) but BIL is not (S), shows that B can be CS) but

not (R).

Phelps [4] gives the following theorem: A Banach space B is (U) if and only if every weakt closed subspace in Bt is a Chebyshev set. Thus in a reflexive space, B is (U) if and only if Bt is (C) and B is (C) if and only if Bt is (U). Taylor [1] and Foguel [1], between them, show that B is (U) if and only if 8* is (R). These matters are considered further in § VII. Here we merely wish to note the duality between (U) and (C) in reflexive spaces. In [21 Smulian states that B is (UG) if and only if Bt is (WtUR), and that Bt is (UG) if and only if B is (WUR). If B is reflexive then "Bt is (WtUR)" is (WUR)" so that in such spaces (UG) and (WUR) are is equivalent to dual geometric properties.

Smulian [2] proved that Bt is (F) if and only if B is (D) and that B is (F) if and only if Bt is (Dt). In a reflexive space (D) coincides with (D) so that in such a case, there is full duality between (F) and (D). We saw in § Ill that in a Banach space (D) is equivalent to reflexivity of the space and (PLUR);

hence the properties (F) and (PLUR) are dual when the space is reflexive. B is (PLIJR) if and only if B is (H) when B is reflexive. Thus for reflexive spaces there is full duality between (F) and (H), an observation first made by Anderson LI].

Lovaglia [1] showed that if Bt is (LUR) then B is (F), and if B is reflexive then B is (LUR) implies that B' is (F). However to reverse the implications Lovaglia [1] uses an additional hypothesis: if B' is (F) and B is (WLUR) then is (LUR). B is (LUR); if B is reflexive and B is (F) with Bt (WLUR) then Indeed, Anderson [I] exhibits a reflexive space B such that Bt is (F) but B is not (LUR). Anderson [1] showed that if 8* is (F) then B is (MLUR) and if B is reflexive then B is (F) implies that Bt is (MLUR). But (MLUR) appears "too weak" to yield duality with (F) while (LIJR) is "too strong." As shown in § III, in a reflexive space, (H) is "between" (LUR) and (MLUR) and, as it turns out, is dual to (F) in reflexive spaces. For a more systematic development with proofs of the dual geometric theory of norm differentiability see Cudia [I]. Smulian (21 showed that B is (UF) if and only if Bt is (IJR). Since either

property implies reflexivity, (Smuljan [2], Milman (I] or Pettis [1]) there is complete duality between (UF) and (UR). In [41, Day uniformized the relation between smoothess and rotundity of the conjugate and rotundity and smooth• ness of all two dimensional quotient spaces to obtain complete duality between (UR) and (US). No property dual to (kR) is known.

Let (L, m) be a modulared vector lattice and let Lt and L Lt denote the totality of all linear functionals bounded under the norm and all continuous linear functionals xt which are also universally continuous, i.e., for any XA converging monotonicatly downward to 0, we have

ROTUNDITY

inflx*(xA)I =0 A

On

V the associated modular m* of m is defined by the formula m*(x*) = (x(x) — m(x)}

L*. Nakano [1, § 381 shows that m* satisfies all the modular conditions for given in I. The restriction of m* to L is called the conjugate modular of

m and denoted by th. Then (L, ñz) is a modulared vector lattice called the conjugate space of (L, m). In this section we always assume semi-regularity of L, i.e., for any nonzero

x in L there is i in L such that i(x) *

0. By semi-regularity of L the following formulas are valid (see Nakano El, §39—40; 2, §83)),

sup {i(x)

m(x)

—

L

for xeL; lIxII

= sup Ii(x)I

for all XE L. The first norm and the modular norm of by the conjugate modular ,ñ are denoted by II II and III 1111 respectiveLy. Then we have that

for any ieL,

sup

Ii(x)I

and 1111111= sup

Ando [21 proved that m is strictly convex [even) if and only if m is even [strictly convex]. m is said to be uniformly increasing if

X) supinf

lIIxIPI

Nakano (1, § 51] proved if m is uniformly convex Euniformly even) and uniformly increasing then rn is uniformly even [uniformly convex).

These theorems together with those of § III give conditions on the modular sufficient to insure that the first or second norms by the conjugate modular be rotund, smooth, uniformly rotund, or uniformly smooth. As an example note that (Nakano [1, § 51]) if a modular m is uniformly even and uniformly increasing, then the first norm by the conjugate modular th of m is uniformly rotund.

VII. Geometry and reflexivity. Throughout this section B is a Banach space. We wish to examine some relationships between geometry and reflexivity. Dixmier [1) has shown that if B**** is (R) then B is reflexive. Thus if the fifth conjugate is smooth, B is reflexive. Smulian [2] showed that if B is (UF) or if B* and B** are (F) then B is reflexive. Anderson [1] combines a result of Fan and Glicksberg (2, Theorem 3] with a theorem of

92

D. F. CUDIA

Lovaglia [11 that if B* is (LUR) then B is (F) to obtain the result:

B1

is

(LIJR) implies B is reflexive. In this connection Phelps [lj has exhibited a non-reflexive Banach space whose first conjugate is (LUR). That (ur) implies reflexivity, (see Ringrose El) for a short proof), has long been known. Hence B* is (UR) implies that B is reflexive. It would be interesting to know if (ML1JR) of the third conjugate implies reflexivity. Fan and Glicksberg 11) prove that B is (kR) implies B reflexive and show that a (WkR) space is reflexive if and only if it is weakly complete. They also show the existence of (W2R) Banach spaces which are nonreflexive.

Day El] showed that there are reflexive (AB)-lattices which are not (ur). Ando 12] has shown that the situation is quite different with respect to modulared vector lattices. Such a space with its modular norm is reflexive if and only if it is (ur and us). This shows that not every complete (AB)lattice is mocluLarable, i.e., isomorphic to a modular norm by some modular, a result first obtained by Shimogaki [1). The special character of inodulared vector lattices is further indicated by the example of Ando [2] of a complete (AB)-lattice which is (UR) and (US) but which is not modularable. In the case of modulared vector lattices, if the modular is uniformly convex or uni-

formly even, the space is reflexive when normed by either the first or the second norm.

It is well known that for a reflexive Banach space each continuous linear functional attains its supremum on the unit sphere; i.e., each supporting hyperplane of the unit sphere has a point of support. Klee [1] showed that B is reflexive if each continuous linear functional attains its supremum on the unit sphere of any isomorph of the space, a result first obtained by James (2] for spaces with a basis. James 11) proved that a separable Banach space is reflexive if and only if each supporting hyperplane of the unit ball has a point of contact and then he [5) removed the condition that the Banach space be separable. Phelps [1] and Bishop and Phelps [1) between them show that a Banach space with a smooth conjugate is reflexive if and only if every hyper. plane of support of the unit ball has a point of contact. The normed spaces point of view has been found useful in the theory of approximation. For a general survey of the situation see Buck (2]. We shall be concerned only with those aspects having a direct bearing on rotundity and reflexivity. We shall be interested in characterizing the Chebyshev sets in terms of the geometry of the space. Jessen (I] proved that the class of Chebyshev sets coincides with the class of closed convex sets in an n-dimensional Euclidean space. Busemann [1) characterizes rotund finite dimensional Banach spaces as

those in which every closed convex set is a Chebyshev set. The reverse inclusion characterizes finite dimensional Banach spaces which are smooth. Thus the class of Chebyshev sets coincides with the class of closed convex sets

in a finite dimensional Banach space if the space is smooth and rotund. In this connection see the papers of Efimov and

and Konstantinesku El).

As might be expected the infinite dimensional situation is more delicate. In a reflexive space every closed convex subset is an existence set. An

ROTUNDITY

93

argument of James given in Phelps 141 shows that a space is reflexive if and only if every closed subspace is an existence set. Thus a Banach space is

rotund and reflexive if and only if every closed convex set is a Chebyshev set. This extends the first theorem of Busemann cited above to infinite dimensional Banach spaces. In [5] Klee shows that in a smooth reflexive Banach space, a Chebyshev set must be convex if the associated metric pro jection (carrying each point of the space onto that point of the set which is nearest to it) is both continuous and weakly continuous. In the same paper it is shown that if B is (UR) and (UG) then the class of closed convex sets coincides with the class of weakly closed Chebyshev sets. As can be seen much work remains to be done in characterizing the infinite•dimensional Banach spaces in which every Chebyshev set is convex. In a different vein recall that the rotund spaces are those whose unit spheres contain no line segments. Thus the rotund spaces are precisely those for which every one dimensional subspace is a Chebyshev set. Phelps [4] has obtained a similar characterization for smoothness by considering subspaces M having the unique extension property. A short proof in Phelps (4] gives the following theorem: A closed subspace has the unique extension property if and only if its annihilator is a Chebyshev set in the dual. A normed space is smooth if and only if every point on the unit sphere has a unique hyperplane of support. This condition is equivalent to the requirement that every one dimensional subspace have the unique extension property. Applying the theorem stated above we see that a space is smooth if and only if every weak* closed maximal subspace in the dual is a Chebyshev set. In a Banach space with a smooth conjugate, every maximal closed subspace is a Chebyshev set if and only if every closed subspace is a Chebyshev set.

This complements the corresponding statement fot the unique extension prQperty in any Banach space given in Phelps [4]. BIBLIOGRAPHY

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Integration von Funktionen, deren Werte die El.,nente sines Vektorraumes sind,

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2. Bases and reflexivity of Banach spaces, Ann. of Math. (2) 52 (1960), 518-527. 3. A Banach apace isometric with its second conjugate, Proc. Nat. Acad. Sci. U. S. A. 37 (1951), 174-177. 4. Banack spaces with a specified number of separable conjugate spaces, Amer. Math. Soc. Notices 5 (1958), 680. 5. Characterization of reflexivity. Studia Math. (to appear). Jessen, B. 1. Two theorems on convex point sets, Mat. Tidsskr. B. 1940, 66-70. (Danish); MR 2, 261. Kadec, M. 1. 1.

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12

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UNIVERSITY OP ILLINOIS

OF THE CIRCLE1

A BY

LUDWiG W. DANZER

V. Mizel has asked whether the following is true: Assume is a planar closed convex curve and no rectangle has then 1i is a circle. exactly three of its vertices on A first proof of this theorem was given by A. S. Besicovitch [1] and presented by him at the symposium on convexity in Seattle in June 1961. Since his proof employed rather deep results on linear measures, he invited his audience to look for a more elementary proof. I am glad to accept this invitation.

If c is any chord of and one of the lines (both the lines) perpendicular to c and supporting c also supports then we shall call c a normal (doublenormal) of (L Two points of which are joined by a double-normal will be called anlipodal The first to show is: (1) Every normal AB of is a double-normal. but n does not. This Suppose m, n I AB, A e m, B€ n and m supports and different from B (I is would imply the existence of a point C€ n closed). The point D which makes ABCD a rectangle hence would be in m fl I. Since m is a line of support this would mean DA c I. Now by the line n would be found to considering the midpoints of DA and of have three distinct points in common with I. Hence n supports I (I is convex). Contradiction! As a consequence of (1) we have (2) 1 is a curve of constant width, say of width d. Following from (1) and (2) we get: (3) Every normal of I has length d. Though it has been well known probably for decades that (1) implies (2), there seems to be no older a reference than the paper by Besicovitch men-

I

tioned above.

If A1 e I and two different points X and Y both are antipodal to A1, then the circular arc XY (center A1) lies completely on I and its midpoint A2 is on a unique supporting line, whence A1 is the only point antipodal to A2. Thus we have: (4) There exists a point A on I whose antipodal point A* is unique. From now on we shall keep this point A fixed. Our main purpose is to show, (5) If ABCD is a rectangle with B, C, Del then C = A*. Consider the set Research supported by the National Science Foundation, U.S.A.. under Grant NSFG 18975. 99

L. W. DANZER

100

{A')

and 3B: (Be I and AB .L BC)); it is closed and it is easily seen, that A has a positive distance from and B is different from C. Hence there exists a rectangle for every CE \ =

u {C I Ce

all its vertices on I and such that the open arc AB1C1 of I has no point in common with %. Similarly there is a rectangle n such that = 0. This means c C1C, and hence (5) is equivalent ABICIDI with

= containing PRooF (6). Suppose C1 * A. Consider the closed halfplane B1 and being bound by AC1, and also the semicircle with endpoints A and C1 which contains B1. Then n = {C1) and C1 =

In

cony (s). To see the latter, assume the existence of a point B on I n but outside Draw the line p through B perpendicu'ar to AB. Because of (4) (and As e i)), c

p would not support I. Hence there would be a unique point C p

I

different from B and therefore in From this would follow C in contradiction to the C1 e mt (4ABC) (recall that B was assumed outside convexity of I. at B1 supports Now consider B1. Since it is on the line t tangent to (5,. I is perpendicular to coincide, and of ABICIDI Since the centers of Thus B,D1 and this turns out to be a normal of I. (see(3)) rendering

and

(see(4)). C1=A5 Thus (5) is verified. Similarly one proves A5 But, by the main assumption of our theorem, every point of I may serve forB in (5), and this means (Thales): I coincides with the circle of which AA5 is a diameter. It would be interesting to learn whether this theorem remains true when also nonconvex curves are allowed in the competition. Another question is, how to generalize this theorem in a proper way into higher dimensions. REFERENCE

1. A. S. Besicovitch, A problem on a circle, 5. London Math. Soc. 36(1961), 241-244. UNIVERSITY OF WASHINGTON

HELLY'S THEOREM AND uS BY

LUDWIG DANZER, BRANKO GRUNBAUM, AND VICTOR KLEE

Prologue: Eduard Helly. Eduard Heily was born in Vienna on June 1, 1884. He studied at the University of Vienna under W. Wirtinger and was awarded the Ph. D. degree in 1907. His next few years included further research and study in Göttingen, teaching in a Gymnasium, and publication of four volumes of solutions to problems in textbooks on geometry and arithmetic. His research papers were few in number but contained several important results. The first paper [1] treatçd some basic topics in functional analysis. Its "selection

many applications and is often referred to simply as "Helly's theorem" (see Widder's book on the Laplace transform). The same paper contains also the Helly-Bray theorem on sequences of functions (see principle"

Widder's book) and a result on extension of linear functionals which is mentioned in the books of Banach and Riesz-Nagy. His famous theorem on the intersection of convex sets (also commonly called "Helly's theorem") was discovered by him in 1913 and communicated to Radon. Belly joined the Austrian army in 1914, was wounded by the Russians, and

taken as a prisoner to Siberia, where one of his "colleagues" was T. Rado. He returned to Vienna in 1920, and in 1921 was married and appointed Privatdozent at the University of Vienna. Along with his mathematical research and work at the University, he held important positions in the actuarial field and as consultant to various financial institutions. His paper [3] on systems of linear equations in infinitely many variables was a muchquoted study of the subject. The "Belly's theorem" which is of special

interest to us here was first published by him in

1923 [41

(after earlier

publication by Radon and König), and the extension to more general sets in 1930 [5).

In 1938 the Hellys emigrated with their seven-year-old son to America, where Professor Helly was on the staff of Paterson Junior College, Monmouth Junior College (both in New Jersey), and the Illinois Institute of Technology. He died in Chicago in 1943. For much of the above information we are indebted to his wife Elizabeth (also a mathematician, now Mrs. B. M. Weiss),

who resides at present in New York City. Their son Walter (Ph. D. from M. I. T.) is a physicist with the Bell Telephone Laboratories in New York. 1.

S.-B. Akad. Wiss. Wien 121 (1912),

Uber lineare

265—297.

This work was supported in part by a grant from the National Science Foundation, U. S. A. (NSF-G18975). The authors wish to express their appreciation to the National I

Science Foundation, and also to Lynn Chambard for her expert typing. 101

102

2.

LUDWIG DANZER, BRANKO GRUNBAUM, AND VICTOR KLEE

(Jber Reilwnentwicklungen nach Funklionen eines Orthogonalsyslems, S.-B.

Akad. Wiss. Wien 121 (1912), 1539-1549. 3.

Uber Systeine linearer Gleichungen mit unendlich vielen Unbekannten,

Monatsh. Math., 31 (1921), 60-97. 4. Uher Mengen konvexer KOrper mit gemeinschaftlichen Punkten, Jber. Deutsch. Math. Vereiti. 32 (1923), 175-176. 5. Uber Systeme von abgeschlossenen Mengen mit gemeinschafllichen Punkten, Monatsh. Math. 37 (1930), 281—302. 6. Die nene englische Sterblichkeitsmessung an Versicherten, Assekuranz Jahrbuch, 1934.

Introduction. A subset C of a (real) linear space is called convex if and only if it contains, with each pair x and y of its points, the entire line segment [x, yj joining them. The equivalent algebraic condition is that ax + (1 — a)y e C

whenever x E C, ye C, and a e [0, 1]. At once from the definition comes the most basic and obvious intersection property of convex sets: the intersection of anv famil.v of convex sets is again a convex set, though of course the inter-

section may be empty. The present exposition centers around a theorem setting forth conditions under which the intersection of a family of convex sets cannot be empty. This famous theorem of Eduard Helly may be formulated as follows: THEOREM. Sufrpose is a family of at least n + 1 convex sets in affine

R", and

if each n + 1 members of to all members of .51'.

is

finite or each member of .2 is compact. Then have a common point, there is a Point common

FIGURE

1

Let us inspect two simple examples. Consider first a finite family of compact

convex sets in the line R', each two of which have a common point. Each set is a bounded closed interval. If [a1 , are the sets of the . . .,

THEOREM AND ITS RELAT1VES

family, it is clear that the point mm1 {,9,) is common to all of them, as is the point max1 (a1) and of course all points between these two. Consider next a family of three convex sets in R2, having a common point (see Figure 1). This gives rise to three shaded areas which are the pairwise intersections of

the sets and to a supershaded area which is the intersection of all three. Helly's theorem shows that if a convex set in R2 intersects each of the three shaded areas, then it must intersect the supershaded area. The convex hull cony X of a set X in a linear space is the intersection of all convex sets containing X. Equivalently, cony X is the set of all convex combinaf ions of the points of X. Thus p e cony X if and only if there are (t'1 1 points x1, •, x, of X and positive numbers a:, •, a,,, such that ax,,. Helly's theorem is closely related to the following results and p = on convex hulls: When X c each point of COUV X j5 a CARATHEODORY'S THEOREM. combination of n + 1 (or fewer) points of X. RADON'S THEOREM. Each set of n + 2 or snore Poi,zts in can be expressed point. as the union of two disjoint sets whose convex hulls have a

Consider, for example, a set X c R2. By Carathéodory's theorem, each point

of cony X must be a point of X, an inner point of a segment joining two points of X, or an interior point of a triangle whose vertices are points of X. If X consists of four points, Radon's theorem implies that one of the points lies in the triangle determined by the other three, or the segment determined by some pair of the points intersects that determined by the remaining pair (see Figure 2).

FIGURE

2

The theorem of Carathéodory was published in 1907. Helly's theorem was discovered by him in 1913, but first published by Radon [21 in. 1921 (using Radon's Theorem). A second proof was published by Konig in 1922, and Helly's own proof appeared in 1923 (Helly [1]). Since that time, the three

104

LUDWIG DANZFR,

GRUNBAUM, AND VICTOR KLEF

theorems, and particularly that of Helly, have been studied, applied, and generalized by many authors; especially in the past. decade has there been a

steady flow of publications concerning Helly's theorem and its relatives. Many

of the results in the field (though not always their proofs) would be understandable to Euclid, and most of the proofs are elementary, as is true in most parts of combinatorial analysis which have not been extensively formalized. Some of the results have significant applications in other parts of mathematics,

and there are many interesting unsolved problems which appear to be near the surface and perhaps even accessible to the "amateur" mathematician. Thus

the study of Helly's theorem and its relatives has several nontechnical aspects in common with elementary number theory, and provides an excellent introduction to the theory of convexity. The present report is intended to be at once introductory and encyclopedic. Its principal aim is to supply a summary of known results and a guide to the literature. Contents are indicated by section headings, as follows: Proofs of Helly's theorem; Applications of 1-lelly's theorem; The theorems of Carathéodory and Radon; Generalizations of Helly's theorem; Common transversals; Some covering problems; Intersection theorems for special families; Other intersection theorems; 9. Generalized convexity. Since the report is itself a summary, it seems pointless here to summarize the contents of the individual sections. The emphasis throughout is on comFor interbinatorial methods and hence on finite families of subsets of section properties of infinite families of noncompact convex sets, especially in infinite-dimensional linear spaces, see the report by Klee 16). 1-2 most results are Many unsolved problems are stated, and after stated without proof. The bibliography of about three hundred items contains all papers known to us which deal with Helly's theorem or its relatives in a finite-dimensional setting. In addition, we list many other papers concerning intersection or covering properties of convex sets, and some general references for the study of convexity and generalized convexities. (We are indebted to E. Spanier and R. Richardson for some relevant references in algebraic topology, and to J. Isbell for information about the paper of Lekkerkerker-Boland 111.) Some of the material treated here appears also in books by BonnesenFenchel [II, Yaglom-Boltyanskji [1]. Hadwiger-Debrunner 131, Eggleston [3), Karlin (1), and in the notes of Valentine [8j. In general, these will be referred to only for their original contributions. For fuller discussion of some of the unsolved problems mentioned here (and for other elementary problems), see the forthcoming book by Hadwiger-Erdös-Fejes Tóth-Klee [1). Organization of the paper is such that formal designation of various results as lemma, theorem, etc., did not seem appropriate. However, the most 1.

2. 3. 4. 5. 6. 7. 8.

important results are numbered, both to indicate their importance and for

HELLY'S THEOREM AND ITS RELATIVES

purposes of cross-reference. In order to avoid a cumbersome numbering system, we adhere to a convention whereby 4.6 (for example) refers to the numbered result 4.6 itself, 4.6 if. refers to 4.6 together with immediately subsequent material, 4.6+ refers to material which follows 4.6 but precedes 4.7, and 4.6— refers to material which precedes 4.6 but follows 4.5. Much of our notation and terminology is commonly used, and should be clear from context. In addition, there is an index to important notions and notations at the end of the paper. Equality by definition is indicated by or =:. When used in a definition, "provided" means "if and only if"; the latter is also expressed by "iff". The set of all points for which a statement P(x) is true is usually denoted by Ix: P(x)). However, when P(x) is the conjunction of two statements P'(x) and P"(x), and P'(x) is of especially simple form beginning "x we sometimes write IP'(x): P"(x)} for IX: P(x)} in the 1): = (x: xE E interest of more natural reading. (For example, {x E: II x and lixil 1).) Though much of the material is set in an n-dimensional real linear space the full structure of RW is not always needed. Some of the results are available for finite-dimensional vector spaces over an arbitrary ordered field, while others seem to require that the field be complete or archimedean. We have not pursued this matter. The n-dimensional Euclidean space (with its usual metric) is denoted by is a compact convex set with nonempty interior. It is A convex body in smooth provided it admits a unique supporting hyperplane at each boundary point, and strictly convex provided its interior contains each open segment Jx,v[ joining two points of the body. The family of all convex bodies in is denoted by A flat is a translate of a linear subspace. The group of all translations in is denoted by T", or simply by T when there is no danger of confusion. A positive homothety is a transformation which, for some fixed ye and some real a > 0, sends x R* into y + ax. The group of all positive homotheties 1

in

is denoted by H" (or simply H) and the image of a set X under a

positive homothety is called a homothet of X. Set-theoretic intersection, union, and difference are denoted by fl U, and respectively, + and — being reserved for vector or numerical sums and differences. The intersection of all sets in a family is denoted by For a point xe R", a real number a, and sets X and Y c R", aX: = (ax: x E X}, x Y: = {x+y :v€ YJ, X+ Y: (x fy : XE X,ye YJ, TX is the family of alt

translates of X, and HX is the family of all homothets of X. The interior, closure, convex hull, cardinality, dimension, and diameter of a set X are denoted respectively by mt x, ci X, cony X, card X, dim X, and diam X. The

symbol 0 is used for the empty set, 0 for the real number zero as well as for the origin of R". The unit sphere of a normed linear space is the set {x: lIxIl = 1), while its unit cell is the set (x: IIxII 1). The n-dimensional Euclidean unit cell is denoted by B", the unit sphere (in by S". A cell in a metric space (M, p) is a set of the form {x: p(z, x) c) for some z€ M and > 0. When we are working with a family HC for a given convex

LUDWIG DANZER, BRANKO GRUNBAUM, AND VICTOR KLEE

106

body C

in

these

homothets are also called cells. When p denotes a distance

function and X and Y are sets in the corresponding metric space, p(X, Y): = inf{p(x,y):xe Y,y€ Y}. In general, points of the space are denoted by small Latin letters, sets by capital Latin letters, families of sets by capital script letters, and properties of families of sets (i.e., families of families of sets) by capital Gothic letters. Small Greek letters are used for real numbers, indices, and cardinalities, and sometimes small Latin letters are also used for these purposes. Variations from this notational scheme are clearly indicated. 1.

Proofs of Helly'8 theorem. As stated in the Introduction, Helly's theorem

deals with two types of families: those which are finite and those whose members are all compact. Though the assumptions can be weakened, some care is necessary to exclude such families as the set of all intervals ]O, j9] for > 0 or the set of all half-lines [a, oo[ c R'. For a family of compact convex sets, the theorem can be reduced at once to the case of finite families, for if the result be known for finite families then in the general case we are faced with a family of compact sets having the finite intersection property (that is, each finite subfamily has nonempty intersection), and of course the intersection nf such a family is nonempty. This is typical of the manner in which many

of the results to be considered here can be reduced to the case of finite families. The essential difficulties are combinatorial in nature rather than topological, and whenever it seems convenient we shall restrict our attention to finite families. The reader himself may wish to formulate and prove the extensions to infinite families. (See also Klee [6] for intersection properties of infinite families of noncompact convex sets.)

For a finite family of convex sets, Helly's theorem may be reduced as follows to the case of a finite family of (compact) convex polyhedra: Suppose is

a finite family of convex sets (in some linear space), each n + 1 of

which have a common point. Consider all possible ways of choosing n + 1 members of ,%", and for each such choice select a single point in the intersection of the n + 1 sets chosen. Let I be the (finite) set of all points so

selected, and for each KG let K' be the convex hull of K J. It is evident that each set K' is a convex polyhedron, that each n + 1 of the sets K' have a common point, and that any point common to all the sets K' must lie in the intersection of the original family .2'•. We turn now to some of the many proofs of Helly's theorem, with apologies to anyone whose favorite is omitted. (Some other proofs are discussed in 4 and 9.) It seems worthwhile to consider several different approaches to the theorem, for each adds further illumination and in many cases different approaches lead to different generalizations. Helly's own proof (1] depends on the separation theorem for convex sets and proceeds by induction on the dimension of the space. (Essentially the same proof was given by Konig (Il.) Among the many proofs, this one appeals to us as being most geometric and intuitive. The theorem is obvious ri

for R°. Suppose it is known for R" ', and consider in

a finite family

of

HELLY'S THEOREM AND ITS RELATIVES

107

at least n + 1 compact convex sets, each n + 1 of which have a common point. of is empty. Then there are a subfamily Suppose the intersection

.5t' and a member A of9 such thatir..9 = 0 but ,r(fr 0. Since A and M are disjoint nonempty compact convex subsets of the such separation theorem guarantees the existence of a hyperplane H in that A lies in one of the open halfspaces determined by H and M lies in the other. (To produce H, let ii II be a Euclidean norm for let x and y be pointsof A and Mrespectively such that lix —yli = p(A,M): = inf{lia — mu:

A, me M}, and then let H be the hyperplane through the midpoint (x + y)/2 which is orthogonal to the segment [x, y]. (See Figure 3.) Now let J denote a

the intersection of some n members of {A). Obviously J M, and since each n + 1 members of have a common point, I must intersect A. .

FIGURE

3

Since / is convex, in extending across H from M to A it must intersect H, and thus there is a common point for each n sets of the form G fl H with C e j7 — {A). From the inductive hypothesis as applied to the (n — 1)dimensional space H it follows that M fl H is nonempty, a contradiction completing the proof. Radon's proof [2] is based on the result stated above as Radon's theorem. To prove this result, suppose p1. . ., p,,, are points of R" with m n + 2. Consider the system of n + 1 homogeneous linear equations, .

=0= where

=

in the usual coördinatizatjon of

(1

n),

Since m > n + 1,

108

LUDWIG DANZER, BRANKO GRUNBAUM, AND VICTOR KLEE

the system has a nontrivial solution (r1, •, r,,,). Let U be the set of all i > 0. for which r, 0, V the set of all i for which r, < 0, and c: = completing the proof = LEE Then Lev = —c and of Radon's theorem. we Now to prove Helly's theorem for a finite family of convex sets in observe first that the theorem is obvious for a family of n + 1 sets. Suppose with j the theorem is known for all families of j — 1 convex sets in sets, each n + 1 of which convex a family of j + 2, and consider in n there have a common point. By the inductive hypothesis, for each A — {A}, and by Radon's theorem is a point p.4 common to all members of such that some point and into subfamilies there is a partition of (see Figure z is common to the convex hulisof {p,:FejT} and and the proof is = {B, D)). But then z e = {A, C) and 4, where .

complete.

FIGURE 4

The literature contains many other approaches to Helly's theorem. Some of the more interesting may be described briefly as follows: Rademacher-Schoenberg ni and Eggleston (3] employ Carathéodory's theorem and a notion of metric approximation. Sandgren [11 and Valentine [91 employ Carathéodory's theorem and the duality theory of convex cones; Lannér's use of support functionals [1] leads also to application of the duality theory. Bohnenblust-Karlin-Shapley (1] employ Carathéodory's theorem to prove a result on convex functions from which Helly's theorem follows (see 4.5).

HELLY'S THEOREM AND ITS RELATIVES

Hadwiger [2; 5) obtains Belly's theorem and other results by an application

of the Euler-Poincaré characteristic, which he develops in an elementary setting.

Belly [2] proves a topological generalization by means of combinatorial

topology (see 4.11).

R. Rado [21 proves an intersection theorem in a general algebraic setting and deduces Helly's theorem as a corollary (see 9.4). Levi's axiomatic approach [11 is based on Radon's theorem (see 9.3). Additional proofs of Helly's theorem are by Dukor [11, Krasnosselsky [21, Proskuryakov [1), and Rabin [1). For our taste, the most direct and simple approaches to Belly's theorem are those of Helly and Radon described earlier.

However, each of the many proofs throws some light on the theorem and related matters which is not shed by others. As is shown by Sandgren ['11 and Valentine [9), the duality theory provides efficient machinery for study of Helly's theorem and its relatives. The approach by means of combinatorial topology leads to many interesting problems but remains to be fully exploited. We have indicated the close relationship of Helly's theorem to the theorems of Carathéodory and Radon. In fact, each of these three theorems can be derived from each of the others, sometimes with the aid of supporting hyperplanes and sometimes without this aid, and each can be proved "directly" by means of induction on the dimension of the space (with the aid of the support or separation theorem). Perhaps the inter-r&ationships could best be understood by formulating various axiomatic settings for the theory of convexity,

and then studying in each the interdependence of these five fundamental results: Helly's theorem, Carathéodory's theorem, Radon's theorem, existence

of supporting hyperplanes at certain points of convex sets, existence of separating hyperplanes for certain pairs of convex sets. Levi [1] makes a small step in this direction, and § 9 describes several generalized convexities from which the problem might be approached. This seems a good place to state a sort of converse of Belly's theorem, due provided to Dvoretzky [1). Let us say that a family of sets has the the intersection of the entire family is nonempty or there are n + 1 or fewer sets in the family which have empty intersection. Helly's theorem asserts that each family of compact convex sets in R' has the Of course the does not characterize convexity, for a family of nonconvex sets in R* may have the "by accident" (see Figure 5). However, Dvoretzky's theorem may be regarded as saying that if a family of compact sets in R" has the ON-property by virtue of the linear structure of its members, then all the members are convex. DVORETZKY'S THEOREM. Suppose {K,: e I} is a family of compact sets in none of which lies in a hyperplane. Then the following two assertions are

equivalent:

all the sets K, are convex;

(for each c) J, is affinely equivalent to K,, then the family {J: c 6 fl has the

110

LUDWIG DANZER, BRANKO GRIJNBAUM, AND VICTOR KLEE

FIGURE

5

An example may be helpful. Each of the families 5a, 5b, and Sc in Figure 5 consists of four sets (all but one convex), obtainable by suitable translations because from the members of 5a. The families Sa and 5b have the of their position rather than by virtue of the linear structure of their members. The family Sc lacks the 2. Appllcation8 of Helly's theorem. We may distinguish two types of applications of Helly's theorem, although the distinction is somewhat artificial. It is used to prove other combinatorial statements of the general form: if a certain type of collection is such that each of its k-membcred subfamilies has a certain proPerly, then the entire collection has the property. And it is used to

prove theorems which are not explicitly combinatorial in statement, but in which a property of a class of sets is established by proving it first for certain especially simple members of the class, and then stepping from this special case to the general result by means of Helly's theorem. The same description applies to applications of Carathéodory's theorem. We shall give several examples of applications of Helly's theorem, with references and comments collected at the end of the section. 2.1. Suppose is a family of at least n + I convex sets in R", C is a convex set in R", and .2' is finite or C and of . are Then the existence of some translate of C whiclz intersects [is contained in;

containsl all members of . for each n + I members of

is

guaranteed by the existence of such a translate

PRoOF. For each Ke let K' :(x + C)rK), where r means "intersects" or "is contained in" or "contains". Then each set K' is convex and the above hypotheses imply that each n + I of the sets K' have a common point. By Helly's theorem, there exists a point ze K' and then (z ± C)rK for all Ke A common transversal for a family of sets is a line which intersects every set in the family.

2.2.

Let .9 be a finite family of parallel line segments in R2, each three of

HELLYS THEOREM AND ITS RELATIVES

111

which admit a common transversal. Then there is a transversal common to all members of .5?'. PROOF. We may suppose ..9' to consist of at least three members and that all segments are parallel to the V-axis; for each segment Se .5?' let Ca denote

the set of all points (a, Then each set

ax +

C....

R such that S is intersected by the line y — is cdnvex and each three of such sets have a

common point, whence by Helly's theorem there exists a point (a0 , fto) C fl The line y = a0x + is a transversal common to all members of .5/'. 2.3. If a convex set in R2 is covered by a finite family of open or closed halfspaces, then it is covered by some n + 1 or fewer of these halfspaces. PROOF. Suppose, more generally, that ,sr is a finite family of sets in covering a convex set C, and that for each F€ .5 the set F': = C F is convex. Then (F': Fe is a finite family of convex sets whose inter-

section is empty, so by Helly's theorem there are n + 1 or fewer sets in this family whose intersection is empty. This completes the proof. 2.4. Two finite subsets X and Y of R2 can be strictly separated (by some hyperplane) if and only if for every set S consisting of at ,nost n + 2 points from X U V. the sets S fl X and S fl V can be strictly separated. PROOF. We may assume that X U Y includes at least n + 2 points. For each x: = (x', ..., x") X and y: = (y', , y%) e V. let the open halfspaces f and Q, in R"' be defined as follows: .

/2: =

{'A

= {A: By

= (A,, A,, . .., +

+

:

> o}

tA,y' < o}.

hypothesis, each a + 2 members of the family {J2:xcX} U {Q,:ye Y)

have a common point, and hence by Helly's theorem there exists A e Q,).

Then

fl

the sets X and Y are strictly separated by the hyperplane

(a €R2:

= When u and v are points of a set X c R2, v is said to be visible from u (in

X) provided Eu, vi C X. 2.5. Let X be an infinite comPact subset of and suppose that for each a + I Points of X there is a Point from which all n + 1 are visible. Then the set X is sfarshaped (that is, there is a Point of X from which all Points of X

are visible). PROOF. For each x€ X, let V2 = {y: [x,v) c The hypothesis is that each n + 1 of the sets V2 have a common point, and we wish to prove that V2 * 0. By Helly's theorem, there exists a point ye cony V2, and we shall prove that ye V2. Suppose the contrary, whence there exist XE Xand u e[y, x[ X, and there exists x' X fl [U, with (u,x'[ A X = 0.

Further, there exist w e

x'[ such that II

w

—

x'

= (1/2)p({u}, X), and v

Eu, wJ

LUDWiG DANZER, BRANKO GRONBAUM, AND VICTOR KLEE

112

and x0eX such that fix0 —

= p([u, w], X). Since x0 is a point of Xnearest lies in the closed halfspace Q which misses v and

vu

to v, it is evident that is bounded by the hyperplane through x0 perpendicular to fv, xo]. But then Since p({v}, X) y e cony c Q and Lyx0 ir/2, whence Lx0vy < X) < p({u}, X), it is clear that v * u and hence some point of Lu, z4 is p({w}, closer to x0 than v is. This contradicts the choice of v and completes the proof.

The preceding five results have all illustrated the first type of application of Helly's theorem. We now give three applications of the other sort, in which the theorem is not apparently of combinatorial nature, but nevertheless Helly's theorem is very useful. 2.6. If X is a set in with diam X 2, then X lies in a (Euclidean) cell If X does not lie in any smaller cell, then ci X conof radius [2n/(n ÷ tains the vertices of a regular n-simplex of edge-length 2. PROOF. We present two proofs of this important result. The first is logically simpler and shows how the theorems of Helly and Carathéodory can sometimes substitute for each other in applications. The second is more under-

standable from a geometric viewpoint. By combining aspects of these two proofs, one can arrive at a third which is easily refined to yield the second equality of 6.8 below.

By Heily's theorem (or 2.1), 2.6 can be reduced to the case of sets of n + 1, and for each cardinality + 1. For consider Xc E" with' card = {y: fly — xfl < [2n/(n + If 2.6 is known for sets XEX the cell n+1 the sets have a common point, of cardinality is nonempty by Helly's theorem and the desired conclusion whence follows.

n + 1. Let y denote the center of the

Now suppose X c E" with card X

smallest Euclidean cell B containing X and let r(X) be its radius. With = r(X), let

where m n. It is easily verified that y cony (a0, .. ., a,,) and we assume with without loss of generality that y = 0, whence 0 = 1 and — 2, whence = 2r2 let di,: 0. For each i and always 2(z., z,). For each j, 1

—

a= = r212

and

zi)/2 = r212,

—

summing on j (from 0 to m

(m + 1)r/2, whence

r

n) leads to the conclusion that m

—

2m

m+1

2n

n+1

HE[.LY'S THEOREM AND ITS RELATIVES

Further, equality here implies that m =

= 2 for all i * j, so the

n and

proof is complete. In the above paragraph, the assumption card X n + 1 (justified by Helly's theorem) was used only to insure that the point ye cony X could be expressed

as a convex combination of n + 1 or fewer points of X. But this is also insured by Carathêodory's theorem, so the above proof could also be based on the latter. The second proof of 2.6 is by induction on the dimension. For E', the theorem is trivial. Suppose it is known for E"1 and consider a set Xc with card 1. Let y,B, and r(X) be asabove. IfyeconvZforsome Z X, then the proof is completed by the inductive hypothesis, for B is then the smallest cell containing Z. (It is easily proved that ify cony Z and pe E' —.(y), there exists z€Z with ilz—yll < lIz —P11. This is equivalent tothe lemma mentioned at the end of 9.9 below.)

Hence we may assume that card X= n + 1 andyeintconvX. Now let. denote the family of all (n + 1)-pointed sets Xc E* (with diam X 2) for which r(X) is maximal. By compactness, 2 is nonempty, and clearly Xe 2' implies diam X = 2. Consider a set {x0, - . ., x,,} .2' and suppose min{Iix. — x11I :0 j <j n} = Ilxo —x111 <2. Then it is not difficult to find another point near x0 such that ii x0 — 114 — = i > 1, and ---, Ii <2, lixo — r({x0, - - -,

But of course

.

. -,

e

and

II

>

the contradiction completes

the proof. 2.7. If C is a convex body in there exists a point z cC such that for each chord Eu, v] of C which passes through z, liz — ulI/iiv — ull n/(n + 1). PROOF.

For each point xc C, let

= x + n(n + 1)1(C — x). We claim that

0, and to prove this it suffices (in view of Helly's theorem) to show that if x0, . ., x,, are points of C, then fl C1, includes the point y: = (n + 11' E x,. This is evident, since for each j it is true that (C—x1). n+1 n n +1 Now consider an arbitrary chord Eu, vi passing through the point z e C1. Then z cu + n(n + ([u, v] — ii), whence z = u + n(n + 1)"2(v — u) for some A 10,11 and liz — uII/lIv — uli = An(n + 1)_1 n(n + 1)-', completing the proof. 2.8. Let be an additive family of sets which includes all oPen halfspaces in and suppose p is a function on JV to [0, oo [ which satisfies the following conditions: a°. If X, Ye then p(X U Y) p(X) + p(Y); b°. c°.

if X, Ye.'/ with Xc Y, then p(X) there is a bounded set Bc

such that

(n + IY'p(R1). Then there exists a Point x* for each open halfspace J containing

and p(R1.—B) < such that

(n + 1)1p(R")

114

LUDWIG DANZER, BRANKO GRONBAUM, AND VICTOR KLEE

de1. Let PROOF. We assume without loss of generality that let J' denote and for each Je note the set of all open halfspaces in such that denote the set of all G e the closed halfspace I?" J. Let we see from of Then for any n + 1 members G0, p(G) < (n + p(G1) < 1, whence U and n: * 0. (Similarly, a° that p(U has fewer than n -f- 1 members.) It follows from the theorem is trivial if has nonempty Helly's theorem that every finite subfamily of {G': Ge

intersection. Now with B as in c°, it is easy to produce a finite subfamily F' is bounded and contains B. such that the intersection of we have F c R" — F' c — B, whence from b° and For each Fe Thus c° it follows that /L(F) p(R" B) <(n + 1)' and consequently Fe is a family of closed sets with the finite intersection property, {G': G e F'. Hence there and each set in the family intersects the compact set G' and this completes the proof of 2.8. exists a point e *

*

*

*

*

*

*

*

From the eight examples above, it can be seen that Helly's theorem is not only one of the most interesting results but also one of the most important tools in the study of finite-dimensional convexity. For other applications of

Helly's theorem, see Yaglom and Boltyanskii [11, Rademacher-Schoenberg (1], Hadwiger-Debrunner [1; 3], and also some of the references .below.

Theorem 2.1 is due to Vincensini 12) and Klee [2], a related result to It is a generalization of Helly's theorem, since the latter results when C consists of a single point and r means "intersects" or "is contained in". For various covering problems, 2.1 is useful when the family C consists Edeistein [1].

of one-pointed sets. (See 2.6 and 6.2.)

The result 2.2 is due to Santaló [2] and is discussed also by RademacherSchoenberg [1]. In a similar manner, Helly's theorem for RR yields the follow.. ing "fitting theorem" of Karlin-Shapley [1]: Suppose are real-valued

functions on a linear space E; x, •, x., are points of E; a1, - •, a,,, are real numbers; and e,, real numbers Then the existence of a linear €,,, combination of the which fits each point (x1 , within (i.e, I çox — e,) is guaranteed by the existence of such a fitting for each n + I points (x1 , as). The same paper of Karlin and Shapley contains a result similar to 2.3.

(For other approximation theorems obtained with the aid of Helly's

theorem, see Snirel'man (1].) Theorem 2.4 is due to Kirchberger [ii and the above proof to RademacherSchoenberg [1]. (See also Shimrat [1].) The original proof, which did not employ Helly's theorem, is nearly 24 pages long. Theorem 25 is due to Krasnosselsky [1] and related results to Molnfir [5] and Valentine (8]. Theorem 2.6 originated with Jung [1], while the first proof above is ally Gustin's reformulation (2) of Verblunsky's proof [1). Blumenthal-Wahlin [1) were apparently the first to apply Helly's theorem to this sort of problem, which is studied in more detail in § 6. Carathéodory's theorem was employed by Eggleston [3).

HELLY'S THEOREM AND ITS RELATIVES

Theorem 2.7 was treated by Minkowslci 11] for n = 2 and n = 3, and in the general case by Radon [1). However, these authors prove more, namely that

the centroid of C has the stated property. The above proof by means of Helly's theorem is apparently due to Yaglom-Boltyanskii [1]. Grünbaum

6

of [16]) treats this and related matters in his discussion of Minkowski's measure of symmetry. Theorem 2.8 was established by B. H. Neumann (1] for R2 and under more restrictive hypotheses on The above proof is an improvement of Rado's reasoning in [1]. Griinbaum [13] has given a different proof based on Helly's theorem; however, for validity of his reasoning the mass-distribution should

be assumed continuous. See GrUnbaum [16] for additional references and related results. 3. The theorema of Carathéodory and Radon. We referred earlier to the close relationships among the theorems of Carathéodory, Helly, and Radon. As an additional illustration, we include the standard derivation of Carathéodory's theorem from Radon's. Consider a set X c and a point y cony X. There are points x, of X and positive numbers such that = 1, E' a;x; = y, and y cannot be expressed as a convex combination of fewer than m points of X. Suppose m n + 2. Then by Radon's theorem there exist numbers -, a,,,, not all zero but with zero sum, such that = 0. Let V = {i: /3, <0) and let j E V be such that a,//3, for all i e V. Then -

we have

y=

+

where the coefficients are all their sum is 1, and the coefficient of x, is 0. Thus y is a convex combination of m — I points of X. The contradiction shows that m n + 1 and establishes the theorem of Carathéodory. For other proofs of Carathéodory's theorem, see Carathéodory [1], Steinitz [1], Rémès [1], (1], Eggleston [3], Ptak t2], and others. For some recent reñnements and related material, see Motzkin 14; 5], Motzkin-Straus [1], and some of the references below. Carathéodory's theorem has many interesting applications (e.g., 2.6, 4.5, 6.8). Pták [1] applies Carathéodory's theorem in a beautiful proof of Haar's theorem on the uniqueness of best approximations, and other applications in approximation theory are those of Rérnès [1], Rademacher-Schoenberg [1], Mairhuber [1], and Motzkin [5]. (See also Snirel'inan [1).) An especially simple and useful consequence of Carathéodory's theorem is 3.1. In the convex of a compact set is compact. Paooi'. Let X be a compact subset of and let A denote the set of aLL points a: = (ao, , a,,) such that = 1 and always 0. For each - -

point

(a, x): =

let f(a, x): =

f(A x

((ao

,

-

eAx Continuous and A x

. •, a,,), (x,, ,

•, x,,))

Since f is compact, the set compact. But byCarathéodory'stheorem,f(A x

The following result is due essentially to Steinitz [1] arid has also been

116

LUDWIG DANZER, BRANKO GRUNBAUM, AND VICTOR JCLEE

proved (in various forms) by Dines-McCoy Eli, C. V. Robinson [21, Gustin [li, Gale [1]. and others. The proof below by means of Carathéodory's theorem is due to Valentine and Grünbaum. 3.2. If a Point y is interior to the convex hull of a set Xc interior to the convex hull of some set of 2n or fewer froints of X.

then y is

To simplify the notation, we assume without loss of generality that a finite subset Y of y is the origin 0. With 0€ mt cony X, there is of cony X such that 0€ mt cony Y and we conclude the existence of a finite set V c X with 0€ mt cony V. Let J denote the set of all linear combinations of n — 1 (or fewer) points of V. Since clearly J * R', there exists a line L through 0 such that L fl J = (0). Let w1 and w2 be the two points of L which are boundary points of cony V and let denote a hyperplane which supports By Carathéodory's cony V at w1. Clearly 0€ w e cony (V fl theorem and the choice of can be expressed as a convex corribination of some n points V1, v, of V fi but cannot be expressed as a linear combination of fewer than n points of V. It follows that w1 is interior to the set cony relative to •.-, and since 0€ 1w1, we see that 0€ mt cony

--•

-

- -,

This completes the proof. Rademacher-Schoenberg [11 deduce from Steinitz's theorem a relative of the theorem 2.4 of Kirchberger [11. Some recent work of Bonnice-Klee [1J exhibits the theorems of and Steinitz as manifestations of the same underlying result. For a subset Z of and an integer j between 0 and n, define the j-interior as the set of all points y such that for some j-dimensional flat F c 1?', y is interior to Zn F relative to F. Then of course mt0 z = 2' and int' 2' mt Z. The theorem of Bonnice and Klee is as follows: 3.3.

If X c R' andy int, cony X, then y

ing of at most max {2j, n + 1)

mt1 cony

Y for son'ie set Y consist-

of X.

Now for a positive integer j and a set X in a Linear space, let I-f,(X) denote the set of all convex combinations of j or fewer elements of X. Then of course cony X = H,(X). On the other hand, the convex hull cony X can also be generated by iteration of the operation H1 for fixed j > 1. That is, cony X = H(X), where H(X): = It is natural to ask how many times the operation H, must be iterated to produce the convex hull of a set in 1?". The case .1 n + I is settled by Carathêodory's theorem. The case j = 2 was treated by Brunn [11 and in subsequent years by several other authors (Hjelmslev (1), Straszewicz [1], Bonnesen-Fenchel [1], Aix -Kubota-Yoneguchi [I]), However, the question

becomes trivial (modulo Carathêodory's theorem) in view of the following

almost obvious fact: 3.4.

= H,k(X).

HELLY'S THEOREM AND ITS RELATIVES

From this it follows (as noted by Bonnice-Klee [1]) that if X C R" and J&fz = c.onv X; conversely, if X is the set of all n ÷ 1, then H,1(H,2 (H,,1(X)) •) * cony X. vertices of an n-simplex and f112 j1 n, then H,L(Hjz (Related problems on the generation of affine hulls are much more complicated. See Klee [8].)

Carathéodory's theorem is the best possible in the sense that if the number n + 1 is reduced, the theorem is no longer true for all sets X C R". However, the theorem can be sharpened when attention is restricted to special classes

of subsets of R", such as those which are connected (3.5) or have certain is said to be convexly connected symmetry properties (3.6). A set in such that the set misses H but provided there is no hyperplane H in intersects both of the open halfspaces determined by H. Results of Fenchel [1] may be stated as follows: is the union of n connected sets or is compact 3.5. SupPose the set X c and the union of n convexly connected sets. Then each Point of cony X is a convex combination of n or fewer Points of X. [1], the conclusion of 3.5 may fail even in As is noted by R2 if the set X is assumed merely to be convexly connected and to be bounded or closed but not compact. For Xc R2, let = H2(X) 111(X), the set of all points of cony X which do not lie in X or in any line segment joining two points of X. Then for various bounded convexly connected subsets X of R2, the set may have any finite cardinality and it may be countably or uncountably infinite (the latter proved by P. Erdös, using well-ordering). It is easy to produce closed convexly connected subsets X of R2 for which consists of one or two points. Danzer has given a complicated example in which consists of three points. Are there other possibilities? The following result of Fenchel [4] reduces to Carathéodory's theorem when m= 3.6. Suppose G is a group of linear isometries of onto itself, m is the dimension of the set M of all G-invariant points of and X is a subset of E" which is mapped into itself by each member of G. Then each point of M fl cony X lies in a set cony g Y for some set Y consisting of at most m + 1 points of X. It would be interesting to know just how much 3.3 can be sharpened when the set X is subjected to various restrictions. From 3.4 and 3.5 it is evident that cony X = !I(X) when X is as in 3.5 and j' n —' 1. We turn now to some results and problems which were inspired by Radon's [11, Stoelinga [1], Bunt [1], and

theorem. For each pair of natural numbers n and r, let f(n, r) denote the

smallest k such that every set of k points in

can be divided into r pairwise disjoint sets whose convex hulls have a common point. It follows from Radon's theorem that f(n, 2) = n + 2. The function f has been studied by R. Rado [6] and Birch [I], whose results are as follows: 3.7. For each n and r,f(n, r) (n + 1)r — with equality when a = I and

n=2.

LUDWIG DANZER, BRANKO GRONBAUM, AND VICTOR KLEE

118

The recursive inequality and determination of f(1, r) are due to Rado and the other results to Birch. Birch's argument employs Carathéodory's theorem, a fixedpoint theorem, and the result 2.8 on measures in R". Combining the recursive inequality with the fact that f(2, r) = 3r — 2, we see that always a bound which is sometimes' better than rn(n + 1)—n2 —

n

+ 1. Other minor

improvements are possible, but all known results are far from Birch's conjecture that perhaps always f(n, r) (n + 1)r — n. The above results imply 17, but the exact values of f(3,3) and 11 and 13 that 9 f(3, 4) are unknown. Several other problems are implicit in the paper of Rado [6], and a result related to one of these was obtained recently by Birch. For natural numbers n and r with r < n, let b(n, r) denote the smallest number k (if one exists) there is an r-dimensional flat which such that for each set of k points in contains r + 1 of the points and intersects the convex hull of the remaining points. Birch's result and its proof, not previously published, are as follows: 2r+2,b(7I,r) does not exist. 3.8. For n 2r+ 1,b(n,r)= n +2; for By are points of •, PROOF. Suppose first that n 2r + 1 and x1, Radon's theorem, there are real numbers a,,, not all zero but with zero sum, such that ax1 = 0. At least n — r + 1 of the a1's have the same sign, are all non-negative and so we may assume that the numbers a,, a > 0, we have at least one is positive. With s: tZ—r+1

=

.s42

(—a,/s)x,

1-4

this point lies simultaneously in the set cony {x1: 1 I n — r + 1) and in n + 2}. It follows that the smallest fiat containing {x,: n — r + 2 j b(n, r) n + 1. For the case n 2r + 2 it suffices to check that if c a A: = {a, a2, and

then A has the remarkable property that for any r + I points x0, ., of A misses the convex hull of (with r n/2 — 1), the r-flat through {x0, . . ., A

above was first given by Carathéodory 12]. It

established the following fact, later rediscovered by Gale 13] and Motzkin [31:

n, R" contains a convex polyhedron with k 3.9. For 2 m k and 2m vertices such that each m of these vertices determine a face of the polyhedron.

For further information on polyhedral graphs, see the papers by Gale [5], Griinbaum-Motzkin [2], and others listed by them. For an application of 3.9,

see 8.2+. We should mention also another interesting unsolved problem which involves choosing subsets of a finite set in such a way that certain convex hulls

HELLY'S THEOREM AND ITS RELATIVES

119

n < r, let h(n,r) denote the smallest number k such that in each set of k points in general position in R", there is an r-pointed set X which is convexly independent (no point of X lies in the convex hull of the remaining points of X). It is obvious that have empty intersection. For natural numbers 2

h(n, n + 1) = n + 1 and easy to verify that h(n, n + 2) = n + 3. Erdos-Szekeres

[1] mention a proof by E. Makai and P. Turan that h(2, 5) = 9, and they establish upper bounds for h(2, r) which are far from their conjecture that h(2, r) = + 1. In a later paper (Erdös-Szekeres [2)) they prove that h(2, r)

of his generalization of Helly's theorem is based on a De Santis's proof generalization of Radon's theorem which applies to sets of flats in a linear space. We shall state a slight extension of the latter result, but this requires further notation. For each finite or infinite sequence Ta of integers with 0 r2 ..., let e(ra) denote the smallest integer k for which the following is true: whenever FG is a sequence of flats in a real linear space, each

being of deficiency r, then the set of all indices i can be partitioned into

jr,) fl complementary sets I and J such that the intersection (cony (cony contains a flat of deficiency k. When no such k exists, define = oo• The following is a slight improvement of De Santis's generalization of Radon's theorem.

3.10. If m is the smallest integer for which = 00• this fails to apply,

< m, then e(ra) =

When

It may not be obvious that 3.10 is a generalization of Radon's theorem. To Each point may see that it is, consider a sequence x1, •, of points in be regarded as a flat of deficiency n, so the corresponding sequence r1, has always = n. When k n + 1, then m (in the statement of 3.10) is equal provided this is defined; otherwise E(Ta) to n + 1, and thus = It follows that when k n + 2 the set of indices (1, , k} can be partitioned into complementary sets I and J such that the intersection cony (xi: 1€ lJ fl cony : j JJ contains a flat of deficiency n; that is, the intersection is non.

empty.

By combining the ideas of 3.10 and 3.8, one may obtain further zations of Radon's theorem. However, these appear to be of only marginal interest. 4.

Generalizations of Helly's theorem. This section contains some generali-

zations and other relatives of Helly's theorem, all in except for a few supplementary comments. (See § 9 for generalizations in other settings.) At the end of the section we formulate some general problems which may serve as a guide to further research in this area. Division of material between § 3 and § 4 is somewhat artificial in view of the close relationships between the theorem of Helly on the one hand and those of Carathéodory and Radon on the other. Nevertheless, it seems to be justified in that the two different lines of investigation are different in spirit and lead to different generalizat ions.

LUDWIG DANZER, BRANKO GRUNBAUM, AND VICTOR KLEE

120

Helly's theorem may be regarded as saying something about the "structure" of convex sets in RN for which = 0. Specifiof certain families . are compact, then cally, it says that if 2' is finite or all members of such there is a subfamily .7 consisting of at most n + 1 members of that = 0. This suggests the attempt to find theorems which say someof convex sets in R' for which thing about the structure of every family = 0, and which has Helly's theorem as a consequence. The known partial results in this direction are summarized in a separate report by Klee [6], since they seem more akin to the infinite-dimensional considerations.

How-

ever, the relevant papers are listed also in our bibiliography (Gale-Klee Lii, Karlin-Shapley [1], Klee [2; 4), Rado 151, Sandgren [1)).

of convex sets, Helly's theorem asserts As applied to certain families . The set {x) may the existence of a point x common to all members of be regarded, for j = 0, in any of the following six ways: as a

f-dimensional convex set which JIS contained Zfl} each member of .2'. f-dimensional flat Lintersects

(j + 1)-pointed set In each of the six cases, we may ask for conditions on a family which assure the existence of such sets corresponding to other values of j. Two of these questions are redundant, for there exists a f-dimensional convex set intersecting all members of if and only if there exists a f-dimensional flat intersecting all members of and (since the members of are convex) for j I there exists a (j + 1)-pointed set which is contained in all members of if and only if there exists a 1-dimensional convex set which is contained in all members of The remaining four questions have all led to generalizations of Helly's theorem. The most recent of these is the .

following theorem of Grünbaum [18):

41.

Let

g(n,O)=n+1,g(n,l)=2n,g(n,j)=2n—j for 1<j
is a finite family of at least g(n, j) convex sets in RN + 1. If and each have an at-least-f-dimensional intersection, then j) members of the intersect ion is at least f-dimensional. n

.

When j 0, this is merely Helly's theorem. Examples show that for each choice of j and n (with 0 j n), g(n, j) is the smallest integer which has the stated property. The following result is due to De Santis [11: 4.2. If is a finite family of at least n + 1 — j convex subsets of R* and the intersection of each n ÷ 1 — f members of .2' contains a f-dimensional fiat, then

contains a f-dimensional flat.

This also becomes Helly's theorem when j = 0, and the integer n + 1 — j is the smallest which has the stated property. The proof of De Santis is based on an analogue of Radon's theorem (see 3.10), and Valentine [91 has given a proof by means of the duality theory. As De Santis has noted, his theorem

HELLY'S THEOREM AND ITS RELATIVES

121

may be stated in the following form which is independent of the dimension of the space: Suppose .2r is a finite family of convex sets in a real linear space E (which may even be infinite-dimensional) and the intersection of each j contains a members of contains a flat of deficiency <j in E. Then fiat of deficiency
theorem, since in each an intersection property of the entire family follows from the assumption of that same property for certain subfamilies of

Now, turning to another of the four questions mentioned above, suppose

we wish to produce for some j

1

a f-dimensional flat which intersects all

members of .51-. This is the problem of common transversals discussed in § 5.

Simple examples show that the existence of a transversal common to all members of

not assured merely by assuming this for certain subfamilies

of .51'. Indeed, for each m 3 there are m congruent line segments in the plane such that each m — 1 of them admit a common transversal but no line intersects all m of them (Santaló [1J). (For other examples see the references cited in § 5.) The only true generalization of Helly's theorem in the direction of common transversals is the following result of Horn [1) and Klee [1): n + 1 and a family .X of at least j compact 4.3. For integers 1 j convex sets in the following three assertions are equivalent: have a common point; a°. each j members of b°. each flat of deficiency j — 1 in R* admits a translate which intersets all members of ..5r; c°. each flat of deficiency j — 2 in lies in a flat of deficiency j — 1 which intersects all members of The two-dimensional case had been considered earlier by Horn-Valentine [1). For the general case, Horn (I] (and later Karlin-Shapley [1]) proved that a° implies c°, and the rest is due to Klee [1]. Alternate proofs are given by

Klee's proof depends on the following Hadwiger [4) and Valentine which is "surrounded" by 5 + 1 open convex description of the "hole" in sets C0, •- -, C, such that each 5 have a common point but no point is commisses mon to all j + 1: there is a flat F of deficiency j such that the set F but intersects every half-flat bounded by F in a flat of deficiency 5 — 1 containing F. * 0, For 5 = n + 1 each of the conditions b° and c° above asserts that and thus Helly's theorem is obtained. For j = 2 = n, the theorem asserts

that if there is a common point for each two members of a family .51- of compact convex sets in R2, then each line L in R2 is parallel to some line L' which intersects all members of .51' and each point xeR2 lies on a line X which intersects all members of .51' (see Figure 6). Regarding the lines parallel to a given line as the pencil of lines through a certain point at infinity, we see that the real projective space is a natural setting for 4.3. The theorem may also be approached by means of spherical convexity (Horn [1]). (See 9.1 and 9.2 for a discussion of spherical and projective convexity.) Now consider the fourth of the questions mentioned above: for 5 > 1, what

122

LUDWIG DANZER, BRANKO GRUNBAUM, AND VICTOR KLEE

L'

FIGURE 6

will assure the existence of conditions on a family of con".rex sets in a f-pointed set which intersects each member of (Cf. the discussion of Gallai-type problems in 7.4+ and near the end of this section.) It is convenient to employ the terminology of Hadwiger-Debrunner [21, in which a i-Partition of . is a partition of . into j subfamilies each of which has nonempty intersection. Under certain stringent geometric restrictions on the is assured by the existence of partitions for existence of a i-partition for certain subfamilies of (see §7). However, an examp'e of HadwigerDebrunner (3) shows that for arbitrarily large integers m there exists a family of m plane convex sets such that each proper subfamily of admits a 2-partition but itself does not. It seems probable that such a family exists for each m 3. Hadwiger-Debrunner [21 approach the problem of jpartitions by means of the following theorem. 4.4.

For

r

j(n

—

1)

+ 2, the existence of a f-partition for a finite family

.91' of convex sets in is implied by the condition that among each r + f — 1 sets in some r have a common point. The same conditton does not imfrly the existence of a (j — 1)-partition.

When j =

and r =

1

n

+ 1, this reduces to Helly's theorem. The result

may be formulated somewhat differently, as follows: for integers r, s, and n, let J(r, s, n) denote the smallest integer j for which a f-partition is admitted by each finite family of convex sets in such that among each s members of some r have a common point. Then 4.4 asserts that .

HELLY'S THEOREM AND ITS RELATIVES

123

J(r,s,n)=s—r+ 1 whenever

and nr_(n—1)s +(n +1). It is noted by Hadwiger-Debrunner [2] that J(r, s, n) = co when s n and that always J(r, s, n) s — r + 1. There remains the problem of completely determining the numbers J(r, s, n), which is solved by 4.4 for n = 1 but not even for n =2. In fact, the above result yields only J(4, 3,2) 2, while examples of Danzer (in Hadwiger [12], Hadwiger-Debrunner [3]) and GrUnbaum [9] show that J(4, 3, 2) 3. Danzer's example consists of six congruent triangles, while Grlinbaum's consists of nine translates of an arbitrary centrally symmetric strictly convex body in R0. It is unknown even whether J(4, 3, 2) is finite. (For a generalization of the functions J, see 7.4+.) Yet another generalization of Helly's theorem arose in connection with the theory of games. It is due to Bohnenblust-Karlin-Shapley (1], and various applications to Karlin-Shapley [1]. 4.5.

Suppose C is a compact convex set in R" and 0 is a finite family of

continuous convex functions on C such that for each x C there exists p €0 with ço(x) > 0. Then there are positive numbers a,, , a with j n and members that all xeC. To deduce theorem from 4.5, let be a finite- family of compact convex sets in R", C a compact convex set confaining their union, and for each Ke and xe C let = p({x}, K). Then if the intersection is empty, the set of functions 0: = {%oR: Ke satisfies the hypotheses of 4.5 and hence there exist positive a1, - •, a- with j n such that fticOK,(X) > 0 for all x C. This implies that = 0 and Belly's theorem follows. A well-known topological theorem of Knaster-Kuratowski-Mazurkiewicz Li] has the following corollary, of which an elementary proof was given by Klee [ii: If j + 1 closed convex sets in have convex union and each j of them have a common point, then there is a point common to all. Using this fact as a lemma, Levi [2] proved:

46. Suppose

is a finite family of at least n closed convex sets in R" and has the following two properties: the union of any n + 1 members of .2 has connected complement in R"; each n members of have a common point.

Then there is a point common to all members of

-

This may be regarded as a generalization of Helly's theorem, since L41 is obviously implied (for n> 1) by the condition that the members of are bounded and convex and each n + 1 have a common point. The above lemma was extended in another way by Berge [2] and Ghouila-

Houri [1], though their results do not formally imply Belly's theorem.

than stating in full the result of Ghouila-Houri Ii],

we

Rather

state its two most

interesting corollaries as 4.7. Suppose C1, C,,, are closed convex sets in a topological linear space and each k of the sets have a common point, where 1 k < m. If U" c, is convex, then some k + 1 of the sets have a Point. If k = m — 1 and

LUDWIG DANZER, BRANKO GRUNBAUM, AND VICTOR KLEE

124

fl' C = 0, Then each m — 1 of the sets have a common point in every closed set X such that X U U' C1 is convex. The second part of 4.8, describing the "hole" surrounded by the sets in question, may be compared with the description in 4.3+. The result 4.7 can also be approached by means of the Euler characteristic (Hadwiger [5], Klee [71).

For a family 5 of sets, let us define the Helly-number a(9)

to

be the

is a finite subfamily of 5 and smallest cardinal k such that whenever Helly's with card then 0 for all = n + 1 when is the family of all convex theorem asserts that The notion of Helly-number is especially appropriate for subsets of c families jr which are intersectional in the sense that E 9— for all

0 for each compact Xc E*, with 4(X) = 0 characterizing the convex sets and For example, GrUnbaum [10] defines a measure of noncon vex ity 4(X)

4(X) < oo if and only if X is the union of a finite family of pairwise disjoint compact convex sets. He proves 4.8.

For finite c

with 4(X)

e.

0, let

Then

e) is

denote the family of all compact sets Xc E" intersectional and has finite Helly-number.

Of course 0)) = n + 1. Griinbaum gives estimates for e)) and examples showing the impossibility of improvement in certain directions. The following result was proved by Motzkin [21: 4.9. Let d) denote the family of all varieties in (real) affine or projective n-space which are defined by one or more algebraic equations of degree Then ..4'(n, d) is intersectional and a(...,f(n, ii)) = (fl + d) In seeking Helly-type theorems for various non-intersectional families

one may encounter difficulties caused by the fact that the intersections of members of

can be structurally much more complicated than the individual

members. For such 5, it may happen that the Helly-number a(9) is infinite and that a more appropriate notion is the Helly-order a°(5), defined as the smallest cardinal k such that whenever is a finite subfamily of .5

ir.9' 5 for all .5' c .'? with card .9"
sectional.

Now let "2/(n, j) denote the family of all nonempty subsets of R' which are the union of j or fewer disjoint compact convex sets. Then a(/f(n, 1)) = 1)) = n + 1 by Helly's theorem. An example of Motzkin (in the correction of Santaló [7J; see also § 25 of Hadwiger-Debrunner [31) shows that 2)) = 4.10.

-

However, Griinbaum-Motzkin [1] establish

The family of all "twins" in E" has !-lelly-order2n + 2; i.e.,

2)) =

2n + 2.

of

The same result holds if "compact" is replaced by "open" in the definition but not if it is deleted. It is unknown even whether the family

HELLY'S THEOREM AND ITS RELATIVES

125

3) of all "triplets" in R2 is of finite Helly-order, but Griinbaum and = j(n + 1). Their paper Motzkin conjecture, more generally, that contains other intersection theorems in an abstract setting, with interesting is a family applications to convexity. In particular, they show that if

of subsets of R', and the intersection of each n members of .9 is the union of n or fewer pairwise disjoint closed convex sets, then n.9 is itself such a union. (This is a theorem of Helly-type in the sense described below.) A nonempty compact metric space is called a homology cell provided it is homologically trivial (acyclic) in all dimensions. In his second paper on intersection properties, Helly 12] employed Vietoris cycles and homology over the integers mod 2 to prove 4.11. In the family of all homology cells has Helly-order n + 1.

A combinatorial form of this result appears on p. 297 of Alexandroff-Hopf Helly actually proved a somewhat stronger formulation: If is a family of subsets of c ,.9 with card is a ho'nology cell for all c5 with n, and irs' * 0 for all = n+ 1, then irJ7 is a [1].

homology cell.

It is interesting to contrast the fate of Helly's topological Theorem 4.11 with that of its special case concerning the intersection of convex sets. For the geometric theorem, many generalizations, variations, and applications were found, but the topological theorem was nearly forgotten during more than twenty years. Except in 1-lelly's paper [2] and the book of Alexandroff-Hopf [1], we were unable to find any use of it prior to Molnár's paper [1] in 1956. In R2, Helly's topological theorem applies to simply connected compact sets. Helly [2] gave an elementary proof for this case, and Molnár [1; 2] established the following improvement: A family of at least three simply connected compact sets in R2 has nonempty simply connected intersection provided each two of its members have connected intersection and each three have nonempty intersection.

Now for a metric space X, let X denote the family of all homology cells in X which can be topologically embedded in R'. From Helly's result it follows easily that

n + 2, whence of course n + 1 n + 2 for every n-dimensional manifold M". As observed by Molnár [3], the 2-sphere S2 is the only 2-dimensional manifold in which the family of homology cells is of Helly-order 4. More generally, it is evident that = n + 2,

and another result of Helly [2] (also p. 296 of Alexandroff-Hopf [1]) implies that X contains a homology n-sphere whenever a°(, X) = n + 2. (See Soos [1] and Molnfir [4] for related material.) 2 Using reduced homology groups with coefficients in an arbitrary field, J. Berstein 111 has generalized Molnár's theorem as follows: If A1, -, Am are compact subsets of -

an n.manifold M 8uch that A1 U U Am * M and 8uch that for each k n + 1 and for every i,, -, the set A41 fl -• fl is (n — k-acyclic, then every union of intersection8 of the A4 is oo.acyclic. In particular, the set A1 fl ... fl Am is nonempty and "o-acyciic. (Here (—1)-acyclic means nonempty and 0-acyclic means connected.) See also .

Jussila 11].

126

LtJDWIG DANZER, BRANKO GRUNBAUM, AND VICTOR KLEE

There are many geometric problems in which the application of 1-lelly's original intersection theorem (if possible at all) may require considerable ingenuity, but to which Helty's topological theorem can be applied quite simply. Apparently this was first noticed by Grunbaum [11] in connection with common transversals (see § 5). Undoubtedly, many new fields of applicability of Helly's topological theorem remain to be discovered.

The topological intersection theorem and other results of Helly [2) are closely related to the fact that if a space be covered by a finite family of sets all of whose intersections are homologically trivial, and all members of are closed or all are open, then the space is homologically similar to the

nerve N5'. (For explicit formulation of one version of this result, see p. 138 of Leray [1] or §4 of Chapter IV of Borel [1]. A related result on the Euler characteristic is given by Borsuk (3] and there are similar results involving homotopy type (Borsuk [2), Weil [1]).) As an illustration, let us derive Helly's geometric theorem from these results and the fact that every subset of is homologically trivial in all dimensions > n. Let be a finite family of at least n + 1 compact convex sets in R", each n + 1 of which have a common point. Suppose 0 and let k be the smallest number such that some (k + 1)-membered subfamily of has empty intersection. Since each k members of have a common point, the nerve is isomorphic with the (k — 1)-skeleton Sk_1 of a k-simplex. Since is homologically similar to the union U of all members of it follows that the (k — 1)-dimensional homology of U is not trivial. But this is impossible, for U c and k — 1 n. *

*

*

*

*

*

*

*

*

Helly's geometric theorem gives some information about "intersection patterns" of convex sets in and additional information is supplied by the result of Hadwiger-Debrunner [2] (4.4 above). It seems natural to ask what is meant by "an intersection pattern of convex sets in Re" and, having settled on a definition, to attempt to determine all such patterns. We suggest two alternative but essentially equivalent definitions: An intersection pattern of convex sets in is the (nerve N.5T of a finite family of convex subsets of R array A(F1, . . -, of an ordered k-tuple > In the first case, is an abstract complex whose vertex-domain is .9,

with

NJr

(Le.,

is

the vertex set of a simplex in N5) if and only

* 0. The problem is to characterize intrinsically the abstract complexes which (up to isomorphism) can be obtained in this way. In the second case, .., is a function on (1, --, to {O, l} such that A(11, "-, 1 if and only if 0. The problem is to characterize intrinsically the FiG arrays which can be obtained in this way. Of course the two problems are equivalent, but the different formulations may suggest different approaches. It seems that the ideas of Hadwiger [2; 5] might be useful here. Notice that the desired characterizations must depend on the dimension of if

the space, for with unrestricted dimension every intersection pattern is possible. Helly's theorem implies that for families in the complex NJJr is determined

HELLY'S THEOREM AND ITS RELATIVES

is determined by by its n.dimensional skeleton and the function A(FI, •-, having at most n + 1 different entries. its restriction to k-tuples (i1, .. •, the analogously Alternatively, one may consider in place of A(F,, ., ,I+1 to {O, ij, and ., k} . which is a function on {1, •, defined array n + 1. Further Fe's for j expresses only the incidence of j-tuples of the restrictions are imposed by the result of Hadwiger-Debrunner [2). However, the problem of general characterization appears to be very difficult and is not trivial even for the case of R'.' The one-dimensional case was encountered by Seymour Benzer [1; 2] in connection with a problem in genetics. He was concerned with those k x k for some k-tuple of intervals on matrices which have the form M2(F1, - •-, the line. Alternatively, one may ask for an intrinsic description of those graphs G which are representable by intervals—that is, which can be realized of linear as the 1-skeleton of a nerve N.9r for some finite family intervals. In this form, the problem was recently solved by LekkerkerkerBoland [1], whose result is as follows: •

-

4.12. A finite graph G is representable by intervals if and only if it satisfies the following two conditions:

(a) G does not contain an irreducible cycle with more than three edges; among any three vertices of G, at least one is directly connected to each path joining the other two.

They also characterize the representable graphs as those which fail to have subgraphs of certain types, and they describe some practical methods for deciding whether a given graph is representable by intervals. We turn now to an abstract description of some of the types of problems which arise in connection with Helly's theorem and its relatives. Suppose X is a given space and is a hereditary property of families of subsets of X. implies That is, is a class of families of subsets of X, and .5- c e

For each cardinal K, define the

by agreeing that

and card 5-
feeling of Grlinbaum and Klee that the Helly-number

defined below

should turn out to be n + 1 rather than n + 2. If this requirement were abandoned, less awkward formulations would result from using "card 9
the individual members of 5 as well as on the structure of .5, and of course will depend on X, ¶13, and A. Essentially the same problem is often viewed the other way around: Given X, conditions on .5-, and [resp. A], what is the greatest A [resp. the smallest *] such that .5 £113., implies ..9 e $ It seems difficult even to determine all intersection patterns which are realized by parallelotopes in R" whose edges are parallel to the coordinate axes. Some special aspects of this problem have recently been treated by Danzer and Grünbaum.

128

LUDWIG DANZER, BRANKO GRONBAUM, AND VICTOR KLEE

A more general problem is said here to be of Gallal-type. conditions on 5 are given. Even though 9- e

Suppose X,

x, A,

may not imply 5 e it is clear that each 9 e can be split into subfamilies having the property The problem is to determine the smallest cardinal çl' such which satisfies the given conditions and has property that every family can be split into (or fewer) subfamilies each having property See 8.8+ for a still more general type of problem. Problems of these types have been considered in the literature mainly for convex subsets of linear spaces, and the remaining work is almost entirely for spherical spaces (see § 9).4 Nevertheless, we believe that of the many and

theorems reported here which can be formulated without linearity, most have

counterparts not only for the sphere but for any manifold. Of course the property of special interest is that of having nonempty intersection.' We dc note this by if and only if so that .9- e * 0. The literature deals only with problems in which x is given extremely small or A is given extremely large. In fact is the simplest interesting property and in the other direction (denoted henceforth by is of special interest since when 5 E a compactness argument can often be used to show that 5 Note that provided each two members of 5- have a common point and ..9 E

9-

5 has the finite intersection property. In a sense marks the borderline between combinatorial geometry and general topology. provided

Existing results on the above problems involve principally the numbers and which are defined below in terms of the property Note, however, that the definitions may be given for other properties as well,

and that some interesting problems can be formulated in this way. Let be a family of sets. Then: the Ath ifelly-number at.(g') is the smallest cardinal g such that (5 c and implies ..9 the xth Hanner-number

of card s'), the

and

implies

and

e

the smallest cardinal p such that each 5 c can be split into p (or fewer) subfamilies, each having property

GalIai-number

which is in

is the largest cardinal A with A + I (successor

such that (5 c is

The Ilelly. and Hanner-numbers are dual in the sense that if card A, x if and only if Ms') A. Note also that = where a(g') is the Ilelly-number defined earlier in this section, and that J 1ff = 1. Helly's theorem asserts that if is the family of all convex subsets of then n + 1 and § 7 discusses the Hanner- and Gallai-numbers for certain subclasses of and describes the problems of Hanner [1] and Gallai (in Fejes T6th [3J) which motivated this then aA(g')

terminology. • Prom an abstract combinatorial viewpoint, such problems have recently been studied by Danzer.

'Also of special interest is the property Q of being disjoint. That is, 3€

F, Il F2 = 0 whenever F4 €3 with

* F2.

Cf. 8.6-8.8.

if

HELLY'S THEOREM AND ITS RELATIVES

5. Common tranavereals. An rn-transversal of a family of sets in R" is an rn-dimensional flat which intersects each member of the family. Thus Helly's theorem deals with O-transversals and Horn's generalization 4.3 assures the

existence of m-transversals under certain conditions on the existence of 0transversals. The present section is devoted to the following problem, first formulated by Vincensini [11: For 1 m n — 1 and j 2, find conditions on a family 5 of sets in R" which assure that if each j-membered subfamily of .5- admits an rn-transversal, then so does .9. At present, only a little is known about the case m = n — I (see 5.1 and 5.2). All the other results are for m = 1, and some only for n = 2. The literature abounds in examples showing that even in R2, rather stringent conditions must be imposed (Frucht [1], Grunbaum [1; 6], Hadwiger [91, Hadwiger-Debrunner Li; 3], Harrop-Rado [1], Kijne ti), Kuiper 11], and Santal6 [1; 2)). Vincesini's problem involves the manifold of all rn-flats in R". While is of course identifiable with R', the manifolds F,,.,,, are not contractible

for I

n — 1, and herein lies part of the difficulty. It seems that any m comprehensive study of m-transversa)s, seeking new theorems under minimal hypotheses, must take account of known results on the structure of F,,,,,, and of the closely associated Grassmannian manifold of all (m + 1)-dimensional linear subspaces of For accounts of some such results and references to others, see Steenrod [1J and Milnor [1). The first significant results on common transversals were those of Santalô [ii, who proved

5.1.

If .9 is a family of parallelotopes in

with edges parallel to the

coordinate axes, and an (n — 1)-transversal is admitted by each subfamily with card + 1), then .9 itself admits an (n — 1)-transversal.

c.9

SantalO's proof is based on an analogue of Radon's theorem. By extending the methods of Hadwiger-Debrunner Li; 3] and GrUnbaum [5; 6; 11], GrUnbaum [201 has recently used Helly's theorem to obtain a generalization of 5.1. The

following terminology is convenient: A convex cone C is an associated cone of a polyhedron P provided the vertex v of C is one of the vertices of P, and C is the cone from v generated by P (i.e., C: = v + [0, oe[(x — v)). A polyhedron P' is related to P provided each associated cone of P' is an intersection of translates of associated cones of P. Then GrUnbaum's theorem is 5.2. Suppose P is a centrally symmetric polyhedron in with vertices, and is a family of polyhedra related to P. If an (n — 1)-transversal is admitted by each subfamily c .9 with card p(n + 1), then .9' itself admits an (n — 1)-transversal.

Increasing the number P(n + 1) leads to a similar result without the assumption of central symmetry. Even under additional restrictions on the family .9', the bound in 5.1 is the best possible for n = 2. (See Frucht [1], Santaló [2], and Griinbaum [6).) The best bounds in higher dimensions are unknown.

LUDWIG DANZER, BRANKO GRUNBAUM, AND VICTOR KLEE

130

For other results on transversal hyperplanes, see the paper of Valentine [91 in this volume.

The rest of the section will be devoted to 1-transversals. We say that a admits a 1-transversal, and family 5- of sets has property Z provided property

of at most j

provided a 1-transversal is admitted by every subfamily consisting members of 5-. The following result (Grunbaum [20)) is a

partial converse of 5.2: 5.3.

j

If K is a centrally symmetric convex body in Rt and if there exists

< cx, such

that

implies

alifamilies of homothets of K, then K is a polygon.

An early result of Santaló [1) is 5.4. axes,

For a family of parallelotopes in (2*-I)

with edges parallel to the coordinate

implies

It would be interesting to find an extension (analogous to 5.2) of 5.4 to more general classes of polyhedra. is said to be separated if there exists a finite of sets in A fainily II,.,) of parallel hyperplanes in R' and an enumeration = (H0, sequence such that each K, lies in the open set bounded by H1_1 (K1, . . ., K.) of and

As a direct application of

topological theorem, Grünbaum [11] proved

be the are as above, and for 1 i m let and x H,., such that [u, vi intersects K.. If the sets K. set of all points (u, v) H0 are all compact and convex and zf the intersection of any 3,4, ., 2n — 2 of implies Z. is a homology cell, then (for .2') the sets 5.5.

Suppose

From this he deduced 5.6. For a family of compact convex sets in R' whose members lie in distinct implies parallel hyperplanes,

such that the distance between any 5.7. For a family of Euclidean cells in two centers is greater than the sum of the corresponding diameters, implies

The result 5.6 is an easy consequence of Helly's geometric theorem, and was first proved for n = 2 by Santaló (21 (see also Dresher [11, RademacherSchoenberg [1)); 5.7 extends earlier results of Hadwiger [7J and Hadwiger-

Debrunner [3]. and With

is called f-simple provided each as above, and j 2,. From reasoning similar to that have connected intersection. j is of Brunn [21, it follows that every separated family of convex sets in 3-simple. With the aid of this remark, Grünbaum 111] derived the following from 5.5: of the sets

5.8.

For a 4-simple scparated family of compact convex sets in R3,

His examples show that none of these conditions can be dropped.

implies

It would

HELIX'S THEOREM AND ITS RELATIVES

be interesting to extend 5.8 to higher dimensions, and in particular to decide whether connectedness of sets of the form implies that they are homology cells. There are some theorems on common transversals for infinite families of sets which seem to have no counterpart for finite families. The following results from Hadwiger's proof [8] of a weaker statement: 5.9. For a family of compact convex sets in whose union is unbounded implies while the diameters of its members have finite upper bound, We turn finally to R', for which Vincensini's problem has been more thoroughly studied. Two approaches have led to interesting results. One permits the individual convex sets to be quite general, but places stringent

restrictions on their relative positions in the plane. Generalizing earlier results

of Vincensini (5), KIee [3], and GrUnbaum [1), Hadwiger [10] proved the following (where a disjoint family of sets is one whose members are pairwise disjoint):

5.10. A disjoint family .5w' of compact convex sets in R2 admits a 1-transcan be linearly ordered in such a way that each 3membered subfamily of admits a 1-transversal intersecting its members in

versal if and only if the specified order.

For j

2, a family

of sets in R* will be said to have property

provided each at most j-membered subfamily of a way that, for each i with 1 i j — 1, the

can

be ordered in such

convex hull of the first i members is disjoint from that of the remaining j — i members. The main result of Kuiper (1] and HarropRado [11 may be formulated as follows: 5.11. A family of compact convex sets in has property if it has and and also if it has and

The result was given in this form by Harrop-Rado. Kuiper's version was in the projective plane, which in a sense is the appropriate medium. Griinbaum (11) employed Helly's topological theorem to establish (in the projective plane) a stronger form of the second half of 5.11.

Another approach permits more freedom in the relative positions of the sets, but assumes that they are mutually disjoint and congruent. (The virtual

necessity of these assumptions is shown by various examples (HadwigerDebrunner [1; 3)).) 5.12. For a disjoint family of congruent circular discs in E1, For such families with at least six members, imftlies 5.13.

implies

For a disjoint family of translates of a parallelogram in

R2,

implies %. The first part of 5.12 is due to Danzer [1], solving a problem of HadwigerDebrunner [1) and Hadwiger [6]; 5.13 and the second part of 5.12 are due to GrUnbaum [6]. For additional results on common transversals in the plane, see Grunbaum [11] and especially HadwigerDebrunner [3).

LUDWIG DANZER, BRANKO GRONBAUM, AND VICTOR KLEE

332

implies % for disjoint families of congruent squares in E1, and for disjoint families of translates of an arbitrary convex body in R'. (2) Z implies % for disjoint families of congruent compact convex sets in E'. The, second conjecture is unsettled even The following are conjectures of Grünbaum [6]. (1)

for disjoint families of congruent segments. We end the section by stating a result of GaIlai type on common trans versals, due to Hadwiger-Debrunner [1; 3]: 5.14.

R',

For a family of Positive homothets of a compact convex set in imJ'lies that .5t can be split into 4 or fewer subfamilies, each having

property

This section is included because Heily's 6. Some covering problems. theorem is a valuable tool in connection with some of the problems, and because certain covering theorems are used in § 7 to treat intersection properties of special families. Discussing only the results most directly related to Helly's theorem would yield a distorted picture of an interesting and active area of research, so we include additional material to round out the picture. Our starting point is the theorem of Jung [1), whose popularity may be judged from its appearance in books by Bonnesen-Fenchel [1], Yaglom-Boltyanskil (1], Hadwiger [15], Eggleston [3], and in many research papers. (To those cited by Blumenthal-Wahlin [I] and Hadwiger [15], we add Straszewicz [11, Lagrange [1], Verblunsky [1], Gustin [i], Ehrhart (1] and Melzak El].)

Jung's theorem, proved in 2.6 with the aid of Helly's theorem, asserts lies in a spherical cell of radius that each subset of E of diameter where of course the terms are defined with respect to the [n/(2n +

Euclidean distance. An interesting problem arises when this is replaced by other distance functions. It is discussed directly below, and other generalizations or

relatives of the problem treated by Jung are discussed later in the section. Though the known results are rather scanty in each case, general formulations will suggest avenues for further study. *

*

*

*

*

*

*

*

*

£ is a set and p is a function on E x E to [0, oo[. For each point s): = (ye E: p(x, y) For x E and Ic E, the (p, x)-radius of Y is the number = sup{p(x,y):ye Y} = inf {e Yc and the p-radius of I is the number 1: = with Yc = The p-diameter of Y c F is the number Y= sup {p(y,y'):y,y'e Y) sup Y}. Now for two (possibly different) functions p and p' on E x E to [0, oo[, the problem is to determine the function J(p, p'; d) defined as follows for each Suppose

x E and each e > 0, let

d

0:

HELLY'S THEOREM AND ITS RELATIVES

J(p, p'; d) = sup (rad, Y: Y c E, diamw Y

133

d}.

Existing literature treats only very special cases of the problem, and in particular is restricted to the case p = p'. Accordingly, we define J(p, d): = J(p, p; d). In this notation, Jung's theorem asserts that if p. is the (Euclidean) distance Santaló [51 describes the function J(p, d) in E*, then J(p,, d) = [n/(2n + (See also the where p is geodesic distance on the unit sphere S" in discussion of spherical convexity in 9.1.) The above formulation subsumes various problems concerning covering by positive homothets of a set in a linear space E. In particular, suppose C is a convex body in E with 0 e mt C, and for x, y e E let O:y

= inf {t

—

xetC).

Then p satisfies the triangle inequality arid is symmetric when C = —C. (For metric spaces with nonsymmetric distance, see Zaustinsky 11].) The function J(pc, d) is positively homogeneous in d and hence is determined by the value when E is n-dimensional of JCoc, 1). It is known that J(po, 1) n(n + and C = —C (see 6.4-6.5 below).

In the above setting, it is natural to discuss centers as well as radii of = rad, I. For sets, where a p-center of I is a point xe E for which distances pa,, such centers always exist in the finite-dimensional case, and under the Euclidean metric the center is unique (for bounded I) and lies in the closed convex hull of Y. The last condition is almost characteristic of the Euclidean metric, for Klee [5) has proved the following: For a normed linear space E, the following three assertions are equivalent: a° E is an inner-product space or is two-dimensional;

6.1

b° ,f a subset I of E lies in a cell of radius <1, then Y lies in some cell of unit radius centered at a point of cony I; c° if a subset Z of E lies in a cell of radius <1, then Z is intersected by every cell of unit radius centered at a point of cony Z.

Results related to the equivalence a°

b° were also obtained by Kakutani

[1], Grtinbaum 171 and Comfort-Gordon [11, the first and last making use of

Helly's theorem. *

*

*

*

*

*

*

*

*

A basic relationship between covering and intersection properties was suggested by 2.1. In particular, the following simple corollary of Helly's theorem is very useful: 6.2.

If XcR" and each n+lorfewer points ofXcan be covered by some

translate of the convex body K c R", then X lies in some translate of K.

Closer inspection of the reasoning for 6.2 leads to the following lemma, trivial but fundamental.

LUDWIG DANZER, BRANKO GRUNBAUM, AND VICTOR KLEE

134

6.3. Suppose G is a grouft (written additively but not assumed to be abelian), W and Z are subsets of C, and e and 2 are cardinal numbers. Then the following two statements are equivalent: a° whenever Y c G with card Y < 1 +1, and every set X c Y with card X <e + 1 lies in some member of {q + W: g G}, then Y lies in some member

of {g +Z:g€G}; b° whenever U c C and the family { W + u : u U) is such thaI each of its subfamilies of cardinality <e + 1 has ,gonemfrty intersection, then each subfamily of {Z ÷ u : u U) of cardinality
Suppose a° holds and consider U as in b°. Let V c U with card (Z ÷ v) is nonempty, or, equivalently, + 1. We wish to show that But this follows from a°, for the hythat — V lies in some translate of Z. pothesis of b° insures that every subset of — V of cardinality <e + 1 lies in some translate of W. Thus a° implies b°, and it follows similarly that b° PROOF.

V<

A

implies a°.

It is also easy to prove a generalization of 6.3 which is related to 6.3 as 2.1 is to 6.2. From 6.3 it follows that with W and A fixed and Z = W, the Helly-number aA({ W + g: g G)) (defined in § 4) is exactly the smallest cardinal for which a° holds. If every a({ W + g : g GJ-) points of a finite subset Y

of G can be covered by some left-translate of W, then Y itself can be so covered; under suitable compactness conditions, this applies to infinite sets Y as well. The following is easily verified: For a convex body C in R* with 0€ mt C, let 10* [Jo) denote the smalles( 0 for which aC covers by translation every set Y such that y' e y + 2C for all y, y' e Y [each two pointed subset of Y lies in some translate of C]. Then always JJ = J(pc, 2), while Joe' Jo when C = —C. 6.4.

number è

The number Jo is called the Jung constant of C, and each translate of the set IcC is a Jung cell associated with C. Note that J deponds on the position of C relative to the origin, while Je is translation-invariant; also, diam = diam Y and hence Jo4' Jon i-c,. When C = —C, both the above conditions on V express the fact that diam P0 V 2. In general, however, the two conditions are not equivalent and it is convenient to define a C-distance which is invariant under translation of C. This is merely the distance generated by the gauge functional of the Minkowski symmetrization C4' of C, where C4': = (C + (—C))/2. Equivalently,

0: {x, j'J- lies in some translate of aC), and diamo V = II x — y Ho: = inf {2a diam,0.Y= sUp{Ilx—ylIc:x,y€ Y).

The following is known: 6.5.

If C isa convex body ink' with 0€intC, then

2n(n ÷ IY' (whence Jo 2n(n ÷ 1)' when C =

and

—C).

The inequality on 10* was proved by Bohnenblust [1), and later in simpler

}IELLY'S THEOREM AND ITS RELATIVES

fashion by Leichtweiss [1J and Eggleston [2J. The same result (in the language of analysis) is hidden in Volkov [1]. Leichtweiss also described the bodies for

which the maximum value of J is attained.. A body C c

will here be

callea a Leichiweiss body provided there is a simplex T such that T + (— T) C 2C c (n + 1)(— T) (which imples that the origin is the centroid of T). Leichtweiss [1) showed that for a convex body C c 2n(n + 1)_l if and only if C is a Leichtweiss body; the convex hull of the corresponding Y is necessarily

J=

(up to translation) such a T. With this result one proves easily that Jo fl with equality exactly when C is a simplex; the corresponding set cony Y is a translate of —C. The above results were obtained also by Grunbaum [2; 81, who employed the following definitions: with 0 jut C, the Jung constant Jo [expansion 6.6. For a convex body C c constant Ed is the smallest number o > 0 such that whenever fx + C: x C X} [{x + a8C: xe X}] is a family of pairwise intersecting translates [(positive) ho,nolhets] of C, then the expanded family {x + oC: x X)[.(x + ca1C: x C has nonem/.'ty intersection.

From 6.3 it is clear that this definition of Jc agrees with the earlier one. In particular, Jo is translation-invariant though E0 is not. Clearly 1 Jo E0, and from results of Nachbin [11, Sz.-Nagy [1J and Hanner [1], it follows that 1 or E0 = 1 exactly when C is a paralleLotope (cf. 7.1). Gri.inbaum [8J Jo proved

6.7. Fora convex body C in R* with C=—C, Even in the plane, Jo and E0 need not be equal.

However, they are equal when-

ever either is equal to 1 or to 2n/(n + 1), and also when C is an ellipsoid (a Euclidean cell).

For the close relationship between expansion constants and the extension of linear transformations, see 7.2. Now for a cardinal j 2, let Ic' and E0' be defined as in 6.6 with respect to families of cells having the property rather than merely Thus, for example, Jc' = Jc and JJ is defined by substituting "triply" for "pairwise" in 6.6. By HeLlys theorem Jo' Ec' = 1 for every convex body C C R* and j n + 1. The following is known: 6.8.

If BZ is the n-dimensional Euclidean cell, then EA' =

for

2

JA"

= [nj(n +

— 1)_h]h/2

n-fl.

The second equality is due to Danzer (unpublished manuscript; see also a comment in the proof of 2.6) and the first follows from reasoning employed by GrUnbaum (81 for the case j = It is interesting to note that

—C,C a convex body in R8)=3/2,

136

LUDWiG DANZER, BRANKO GRUNBAUM, AND ViCTOR KLEE

which is also the maximum of JJ. (If C = — C and JJ = 3/2, then of course = 3/2 and C is a Leichtweiss body. On the other hand, every Leichtweiss which is also a Hanner body (see 7.1 if.) has = 3/2, and the body D c regular octahedron is such a body.) See 7.5-7.6 for the relationship between Gallai's problem and the numbers

Another generalization of Jung's problem is the following: Given a class of subsets of a linear space E and a class C of transformations of E into a set U c £ is called a G-universal cover of provided for each Ye there exists g e G such that gU Y. For a set C which is starshaped from the origin in E, the problem is to evaluate the number inf {a 0: oC is a C-universal cover of In addition to the results on Jung constant, the following is known: 6.9. Suppose C is a centrally symmetric convex body in RN, T is a simplex such that diamc I 2 (i.e., each twocontaining C, and Y is a subset of pointed subset of V lies in some translate of C). translate of T or by a translate of — T.

Then

Y can be covered by a

PROOF. Let T1 be the smallest homothet (positive or negative) of T which covers I and let x + aC be the largest positive homothet of C which lies in 7',. We claim that Y is covered by the simplex T2: = (2 — a)a'(2x + (— 7',)). If this be so, then of course a 1 (for otherwise 7'2 is smaller than T,) and the proof is complete. To see that Y, note that 7' is a negative homothet of T1 such that

corresponding faces of dimension n — 1 lie in parallel hyperplanes at Cdistance

Then use the facts that diamo Y 2 and ci V intersects every (n — 1)dimensional face of T1. The following can be derived from 6.9 by standard continuity arguments. 2.

6.10. Suppose G is the grout of (orientation-preserving) motions in E, S is an n-simplex circumscribed to the Euclidean unit cell sisch that —Se GS, and

is the class of all sets Yc E of Euclidean diameter

Then S fl —s is

a G-universal cover of Another consequence of 6.9 is that if C is a centrally symmetric convex body in R' and T is a simplex of least volume among those which will cover (after suitable translation or point reflection) every set of C-diameter 2, then 7' is a translate of a simplex of least volume among those which are circumscribed to C. The special case of 6.10 where S is regular is due to Gale [2]. The above proof of 6.9 is essentially due to Viet El], a student of SUss. Using the same idea, Suss himself [1J gave an elegant proof of Gale's theorem and tried to employ this for a short proof of Jung's theorem. Unfortunately, his calculation to this end contained an irreparable error, for if S is regular and circumscribed to then for n 3 the set S fl —S will not fit into the Jung cell (Ja,s)B'. Theorem 6.9 remains true when the assumptions are replaced by: "Suppose C is a convex body in S is a simplex containing (C + (—C))f2...." It

HELLY'S THEOREM AND ITS RELATIVES

would be interesting to find a similar result under the original assumption on S but dropping the symmetry of C. *

*

*

*

*

*

*

*

*

Instead of asking how large a must be for oC to cover any set of C-diameter

2, one may ask, for fixed a, how many translates of aC are needed. In parttcular, ô(C) will denote the smallest integer k such that every set of Cdiameter 2 can be convered by k translates of C. Griinbaum [3] has proved 6.11.

If C is a centrally symmetrix convex body in three translates of C; hence ö(C) 3.

the Jung cell fcC can

be covered by

For related results on the Euclidean plane, see Gale [2), GrUnbaum [17], and

papers by H. Lenz listed by Grunbaum 117). Griinbaum conjectures that = n + 1, even though for n 3 it is clear that the Jung cell The best known result in the cannot be covered by n + 1 translates of direction of this conjecture is the following, due to Danzer (unpublished manuscript). 6.12.

The Jung cell i/2n(n +

/(n

can be covered by

+

translates of Be.' Close numerical investigations starting from the universal cover of GrUnbaum [4) and Heppes [1) seem to verify that 5(B') = 4. For the relationship between J(C) and Gallai's problem, see 7.5. Grunbaum's conjecture and many other interesting questions are closely related to the famous conjecture of Borsuk [1] that in every set of diameter 1 can be split into n + 1 sets of smaller diameter (cf. Hadwiger [1; 3]). For the literature on Borsuk's conjecture and related matters, see the extensive report by Griinbaum [17) in this volume. Another problem closely related to Borsuk's conjecture is due to Levi [3) (discussed also in two forms by Hadwiger [14; 161): Given a convex body C in R*, what is the smallest number e(C) of translates of mt c to cover C. Obviously = for an n-dimensional parallelotope and probably e(C) for every convex body C c On the other hand, considering the spherical image of C obtained from parallel supporting hyperplanes, we see

from the result of Borsuk [1) that *(C) n + 1, with equality when C is smooth since the (n — 1)-sphere can be covered by n + I open hemispheres. (This was the argument by which Hadwiger [1] proved Borsuk's conjecture for smooth bodies.) Levi [3] showed that E(C) 4 in R3, with equality characterizing the parallelograms (Levi [4]). See also Gohberg.Markus [1], and, for related problems, Boltyanskil [1] and Soltan [1]. 6 Perhaps this bound is already outdated by the method of Erdds-Rogers 11]. (Cf. the footnote to 7.10.)

138

LUDWIG DANZER, BRANKO GRUNBAUM, AND VICTOR KLEE

Replacing the word "translates" in the definition of s(C), by "images under affine transformations, each of determinant one," leads to another constant =2 in R' (as conjectured by Levi [3)), e'(C). Though easily seen that it is unknown whether €(C) assumes greater values in higher dimensions.

.

*

*

*

*

*

*

*

*

The covering theorems to be discussed next involve the group of isometries

(Euclidean motions) in E' and the group of affine transformations with determinant 1. Since both destroy any metrical concepts except those for the

Euclidean distance, there is no longer any connection with Minkowskian geometry. Further, there is no analogue to 6.3 replacing translates by more general images. Nevertheless, the results 6.14-6.15 complete our knowledge

of facts discussed earlier, and 6.17-6.18 are Helly-type theorems in the sense of §4. Considering circular discs D inscribed in a plane continuum C, Robinson [1)

showed that if C itself is not a circular disc, there exists a positive e such that any three points of the set F = (1 + t)D can be carried into C by a Euclidean motion. (He assumed superfluously that C was simply connected.) This proved

the circular discs are the only continua C with nonempty interior 6.13. In which have the following property:

If Fc Et and each triple of points of F can be covered by some isometric image of C, then so can F itself. Soon afterwards, Santalô [3] proved the polar counterpart: 6.14.

In E the circular discs are the only convex bodies C which have the

following property: Q83) If Fis a convex body in Ez and eve,y triangle containing F contains also an isometric image of C, then F itself contains such an image. That the circular discs have properties and is evident from Helly's theorem (cf. 2.1). Blumenthal [1), Santalô [3], and Kelly (1] ask whether there are analogous results for other classes of plane convex bodies. In particular, what happens when the number three (of points in F for %, of sides of the polygon containing F for is replaced by some k > 3? Of course the circular discs still have the corresponding properties and but it is undecided whether they are still characterised by these properties. For the convex hull of a circular disc and a nearby point may be a counterexample; for intersect the disc with a halfptane. Considering, for F, an insphere plus a thin "cap" of slightly larger radius (instead of a slightly expanded insphere), Danzer recently generalized 6.13 to E', assuming C to be a convex body. The corresponding polar argument is valid for 6.14. (Similar results have been reported by K. Böroczky in Buda-

pest.) Theorem 6.13 was inspired by the notion of the congruence order of a metric

space, due to Menger [1) and studied extensively by Blumenthal [1; 3] and

HELLY'S THEOREM AND ITS RELATIVES

A metric space M has congruence indices (n, k) with respect to a class .9' of metric spaces provided each member S of .9' with card S> n + k can be isometrically embedded in M whenever such an embedding is possible others.

for every XcS with cardX n. AndMhascongruenceordernwithrespect to .9' provided (n, 0) are congruence indices. (Note the close similarity of these ideas to those associated with Helly's theorem.) It is known that both E" (in the Euclidean distance) and the Euclidean ndimensional sphere S' (in the geodesic distance) have congruence order n + 3 with respect to the class of all metric spaces (see 38—39 of Blumenthal [3)). Among the plane convex bodies, the circular discs are characterized in 6.13 as having congruence order 3 with respect to the class of all plane sets. Kelly [ii notes that except for the circular ones, no elliptical disc has finite congruence order. It would be interesting to characterize the plane convex bodies of congruence order k for k > 3. A more algebraic problem is to determine the congruence indices of various curves in E'. Though solved in a few cases (see Kelly Li], Blumenthal 131), the problem is not settled even for all conics in Et. It is known that each conic has congruehce indices (6, 0) and (5, 1), but not whether it has (5, 0). This asks the following: Suppose Q is a conic in E2, X is a set of six points in E2, and for each x e X the set X {x) can be moved isometrically into Q. Must Q contain an isometric image of X? (The problem is trivial when Q is a parabola. For Q an ellipse, an affirmative solution was recently obtained by J. J. Seidel and J. van Vollenhoven in Eindhoven.) *

*

*

*

*

*

*

*

*

The affine transformations of determinant 1 carry ellipsoids in E* onto other

ellipsoids of equal volume. A basic fact is 6.15. For each convex body C in there is a unique circumscribed ellipsoid of least volume and a unique inscribed ellipsoid of greatest volume.

Existence of the ellipsoids is trivial, and uniqueness of the first became well known since K. Löwner used it in his lectures; uniqueness of the second was apparently discovered independently by Danzer and Zaguskin. See DanzerLaugwitz•Lenz [11 and Zaguskin [1] for proofs and applications.

Now let us say that a convex body C has property 2 if its LOwner-ellipsoid property has volume if cony X has property 2 for each Xc C with card X < k, property if if its inellipsoid has volume 1, and property each intersection of fewer than k halfspaces, each containing C, has property With these definitions, we have 6.16.

If C is a convex body in R, Ce 2

6.17.

If C is a convex 'body in R2, Ce

For n =

(*42) /2 implies

C €2.

implies Ce

2 these results are due to Behrend [1; 2], who also gave an exact description of how 3,4, or 5 points on a circle [resp. tangents to a circle] must be located in order for the disc to be the Löwner-ellipsoid [resp. the

LUDWIG DANZER, BRANKO GRUNBAUM, AND ViCTOR KLEE

140

inellipsoidi of the polygon determined by these points [reap. by the tangents through these pointsj. The general result 6.16 was found by John [1], whose

use of Lagrange's parameters should also be applicable to extend 6.17 to higher dimensions. *

*

*

*

*

*

*

*

*

We turn now to concepts and results which are in a sense polar to some of the material treated earlier. This polarity, which appeared earlier between

results of Robinson (6.13) and Santaló (6.14), involves a correspondence between points on the one hand and hyperplanes or halfspaces on the other. Though

the correspondence is often called a duality, we prefer the weaker term since the relationship is much less strict than the true duality of, e.g., projective geometry. There is no exact duality principle but only a strong duality feeling. This leads to interesting new results and problems, but there are many true statements whose polar statements are false, and often it is even unclear how to "dualize" a given problem. There seems to be no polar counterpart to 6.3, but the statement polar to 6.2 is the following consequence of 2.1: 6.18.

If a family

of closed halfspaces in r

has

bounded intersection,

and each 'intersection of n + 1 members contains some translate of the convex body C, then contains some translate of C.

Now consider a convex body C in and the associated C-distance ii Ito as defined in 6.4+. The C-distance between two parallel hyperplanes H and H' is inf{llx — X' tic : xE H, x' e H'), or equivalently,

0: some translate of aC fits between H and H') Further, the C-width of a bounded set F c is given by widthc F = inf {ll H, H' ho: F lies between H and H'}. ii

H, H' tic: = sup {2a

Now recalling the definition 6.4 of the Jung constant and replacing c by define the Blaschhe constant B0 of C by

we

> 0: every convex body I of C-width 2 contains some translate of aC). The polar counterpart of 6.5 is Bc: =

sup {a

6.19. If C is a convex liody in then B0 2(n + 1)_I when C is centrally symmetric, while B0 1/n in any case.

PROOF. By 6.18, the widest simplex T which can be circumscribed to C has width 2/B0. If C = —C, at least one of the hyperplanes bounding T will be no farther from the origin than the centroid of T. Thus width0 T n + 1

and B0 2(n + 1)". For the second half of 6.19, we employ an argument polar to one used by Grünbaum [8] for Jo. Since (with C* = (C + (—C))/2) ii tic. = Ii lie, it follows from the inequality already established and from

HELLY'S THEOREM AND ITS RELATIVES

the definition of Jo. that each convex body of C-width 2 contains a translate and consequently Bc 1/n. of 2(n + 1)'Jó.'C. But Jr7. 2n(n +

The first part of 6.19 was proved initially by Leichtweiss [1] and later also by Eggleston [21.. For the symmetric case' Leichtweiss showed that B0 = 2(n + if and only if there is a simplex T such that 2C c T + (— T) and (n + 1) T is circumscribed to 2C (implying that the origin is the centroid of T); this T may serve as Y and vice-versa (up to translation). Hence for general C, B0 = 1/n exactly when C is a simplex (with Y a translate of — C). Recall that attains its lower bound 1 exactly when C is a parallelotope. The polar situation is quite different, for Eggleston [2) proved 6.20. B0 = 2/3 for every centrally symmetric convex body C in

When n = 3, Leichtweiss's conditions are easily verified for parallelotopes, affinely regular octahedra, elliptic cylinders, and centrally synunetric double cones with convex base. For higher n, exact upper bounds are not known; so far as we know, they may even be attained for C = B" (6.21). For this Euclidean case the Blaschke constant is the width of the regular simplex circumscribed to B', where the parallel supporting hyperplanes realizing the width each contain about half the vertices. This case was treated by Blaschke [1] for n = 2 and in the general case by Steinhagen [1], whose calculations were later simplified by Gericke [1). The result is as follows: 6.21. If B" is the n-dimensional Euclidean cell, then

B=

forevenn, for oddn.

As Steinhagen remarked, for n 3 there are many subsets of a regular simplex T c E' having the same Euclidean width and the same insphere. Nevertheless, it should be possible to conclude from his or Gericke's calculies in a regular simplex Qf the same lations that every extremal set (for width and inradius. In his paper devoted to B1s, Blaschke [1) also noted the following consequence

of Jung's theorem: 6.22. 2—

Every set of constant width 2 in E' contains a sphere of radius

l/2n(n

+ 1)—1.

An apparently difficult problem is to determine the largest number such that every convex body of constant width 2 in E' contains a hemisphere of radius Clearly < 1 for n 3, but it seems probable that = 1. A partial result in this direction is given by Besicovitch [21. 7 In fact he does not assume symmetry. His "Minkowskische Dicke e" would be in our notation. In similarity to the distinction between 4' and Jo in 6.4 and 6.5, his result could be stated as follows: B0' 2(n + 1)—' for every "width

convex body C with 0€ mt C. Since Cij(—C) is symmetric and contains C, the general case of Leicbtweiss's theorem is immediate from the symmetric one.

LUDWIG DANZER, BRANKO GRONBAUM, AND VICTOR KLEE

142

Now we shall define the polar correspondents of the expansion constant E0 (6.6) and the constant J(C) (6.11). Let C be a convex body in E' with C = —C.

For a set XcE', let X—C: ={x:x+CcX}. Then in analogy with 6.6, we give another description of the Blaschke constant and simultaneously define the contractwn constant. 6.23. For a convex body C c R' with C = —C, the Blaschke constant [contraction constant S0J is the largest number a such that whenever is a family of closed halfspaces in R' with * 0 and each two me,nbers of { H-C: He He have a common point, then the contracted family {H—oC: He

[{H—aa11C: He

has

intersection.

We ask for lower bounds on and for the value of Se.. Finally, we denote by the smallest number Fe such that whenever Y is a convex body of C-width 2, then C can be covered by Fe translates of Y. We conjecture that J(B 5) = 3, consideration of a regular triangle showing that > 2. (For other related questions, see the report of Grunbaum [17] on Borsuk's problem.) Simple computation comparing the volume of B' with

that of a regular simplex of width 2 in E' shows tha( J(B') > n + 1 for all large n. 7.

IntersectIon theorems for special fallIes. For a set Xc R' and a group

G of transformations of I?' onto itself, GX will denote the family {gX: g e G).

Of special interest are the group T' of all translations 'in R' and the group H' of all positive homotheties; these will often be denoted simply by T and H. The present section summarizes known results concerning the Hannerand Gallai-numbers (defined at the end of § 4) of families TC and HC for various convex bodies C. Hanner [1) proved the following basic theorem: 7.1.

If C is a convex body in R', then is infinite when C is a

=3 whenCisntaparallelotopebntisa centrally =

p5(HC)

=2

symmetric polyhedron in which every Iwo disjoint maximal faces are Parallel; in all other cases.

Thus C is a parallelotope if and only if each family of pairwise intersecting translates of C has nonempty intersection. Hanner also proves some facts about the special polyhedra C for which p5(TC) =3 (called Hanner bodies in 6.8+) and describes a general construction for many of them. There are no Hanner bodies in Et and the only Hanner bodies in E5 are those which are affinely equivalent to the regular octahedron. That ft5(T'C) is finite (and hence for every non-parallelotope C was proved previously by Sz.—Nagy [1J. For j > 2, the literature contains no characterization of convex bodies C having ft,(TC) The work of Hanner [1] and Sz.-Nagy was inspired by a paper of Nacbbjn

HELLY'S THEOREM AND ITS RELATIVES

143

[1] concerning normed linear spaces E with the following extension proPerty:

whenever Y is a linear subspace of a normed linear space Z and

is a

continuous linear transformation of Y into E, then v can be extended to a One of continuous linear transformation C of Z into E with IICII = Nachbin's results was as follows: 7.2. For a normed linear space E with unit cell U, the following three assertions are equivalent: E has the extension property; every family of pairwise intersecting cells in E has nonem ply intersection (i.e.,

> card flU); E is equivalent to the space of all continuous real functions over an extremally disconnected compact Hausdorff space.

Part of Nachbin's proof assumed U to have an extreme point, but this assumption was removed by Kelley [1J. Similar results were later obtained by Aronszajn-Panitchpakdi [11 for more general metric spaces. See also Grünbaum [121, Nachbin [2].

The fact that a(TC) = 2 (equivalent to > card TC) if C is a parallelotope is generalized by the following, whose proof uses projections in an obvious way. 7.3.

If X, and

which span

are nonem ply sets in two linear subspaces RNI and

RNZ

then

+ X,)) = max

a(T*ZX2)}

+ X2)) = max {a(HNJX,),

1 + max {n, , n2) when the

are convex).

Nachbin [21 and others have asked whether a(WC) = n + 1 for every convex body C in RN which is not a cartesian sum. A counterexample (previously unpublished) was found by Danzer. Let K be a convex body in RM' and let f be a real-valued function on [0, 1) which is non-negative, antitone, and upper semicontinuous. Define

{(x, a): xef(a)K} c RN_I + R' = RN; C: = then a(H"C) = a(H*1K) + 1. In particular, a(H2P) = 3 when P is a pyramid

in E8 whose base is a square. Among all convex bodies in RN, the parallelotopes are characterized by an intersection property of their translates (7.1). The same is true of simplexes, as was proved by Rogers and Shephard [11: 7.4. A convex body C in RN is a simplex :f and only if the intersection of each pair of translates of C is empty, onepointed, or a (positive) homothet of C. An equivalent condition is that C, fl C2 E HC whenever C, e HC, C2 e HC, and C, n C2 includes more than one point. This was used by Choquet [1; 21 to define infinite-dimensional simplexes. (See Bauer [2) and Kendall [1) for additional details.)

144

LUDWIG DANZER, BRANKO GRUNBAUM, AND VICTOR KLEE

Before discussing the problem of Gallai, we should mention once more the problem of Hadwiger-Debrunner [2), described earlier in 4.4 if. In addition

to the number Jfr, s, n) defined there with respect to the family v"'

of all

convex sets in R", one may consider J5,(r, s, n) defined in the same way for a subfamily of '2"'. Hadwiger-Debrunner [2) observe that if .9" is the family of all parallelotopes in R" with edges parallel to the coordinate axes, then s, n) — s, n) r—s+1 when ns (n — 1)r+n. 'i); further,

It would be interesting to study the numbers J,.(r, s, n) for families .9 the form T"C or H"C. *

*

*

*

*

*

*

*

of

*

The oldest question of Gallai type is that of T. Gallai, apparently first published in the book of Fejes Tóth (3) (cf. Hadwiger (13)): What is the

smallest number of needles required to pierce all members of any family ..9: = (B,: eel) of pairwise intersecting circular discs in Et? In our notation, this asks for the value of r0(FIB2). From Figure 7, P. Ungár and G. Szekeres

FIGURE 7

concluded that rz(HB2) 7. In fact, seven points are enough even when, instead of only BIn B0 0 for all eel is assumed, where B0 is a

smallest disc in 9. Under this weaker assumption, 7 is the best number. Later A. Heppes proved ri(HB') 6. It was reduced to 5 by L. Stachó (L. Szentmártony), and by Danzer to 4, thus settling the question as examples show. All proofs are still unpublished; one example is given by Griinbaum

a related remark by Schopp [1). For guidance in generalizing the original Gallal problem, we review the

[9],

following facts:

It seems natural to consider only families of convex sets, since otherwise the Gallai-numbers may be infinite even for families of translates (a)

(Danzer [3)). (b) Clearly

= I when 5 aC.9), and in particular when 5 consists of convex sets in R' and I n + 1. (c) If .9" is the family of all parallelotopes with edges parallel to the coordinate axes in R", then = 1. The condition r:(TC) = 1 characterizes the parallelotopes among all convex bodies C in R", and r2(JIC) = 1 characterizes the parallelotopes among compact sets C. (Probably r2(TC) = I

HELLY'S THEOREM AND ITS RELATIVES

145

characterizes the parallelotopes among connected compact sets.) (d)

For families of the form CC, where C is a group of linear transfor-

mations in with G D T", it seems clear that finite Gallai numbers cannot or C is very special. be expected unless G In view of the above facts, we restrict our attention to the numbers r,(T"C)

where C is a convex body in R" (but not a parallelotope) and families of translates will be treated first (7.5-7.8) and then j those of homothets. Some related questions on covering and on families of sets adjacent to a given set are discussed at the end of the section. For the application of 6.3 to Gallai's problem, some special notation is convenient. Suppose r > 0, j is a natural number, Y is a bounded set in and C is a convex body in Then we define = mm {k: Y can be covered by k translates of C),

and

2

n.

The

called the covering number of Y with respect to C;

each j +1 points of Ycan

Y: = infI2a

covered by a translate of aC

I.

J

called the jth C-diameter of Y; and

ö'(C,r): =max{[Y/C1: YCR* with We shall write ö'(C) for o'(C, 1), whence ot(C) = J(C) as defined in § 6. From 6.2 it follows that for all j n, diam Y = diam Y, the circumdiameter of Y with respect to C. Although diam Y Y when j 1, this is not true in general when j > 1; but of course Y= (- Y) and thus o'(C, r) = o'(—C, r) for all j, C and r. In conjunction with 6.3, the above definitions lead at once to the following results: 7.5. {x

If C is a convex body in R", F c R", and

+ C:x€F), then

if and only if

7.6. For a convex body C in all translates of C is given by

is the family of translates 2.

the jth Gallai-number of the family TC of =

=

hence is not greater than the number of translates of C needed to cover a translate expanded by the jth Jung constant of C; i.e.,

r'(TC)

[(J01)C/C].

Jung constants JJ were defined in 6.7+.) Thus the Gallai problem for translates is equivalent to a covering problem, evaluation of the numbers o't(C). For general j, there are significant results (The

only for the Euclidean case C = results 6.8:

and these are derived from Danzer's

________________

LUDWIG DANZER, BRANKO GRUNBAUM, AND VICTOR KLEE

146

7.7. (See

1))

6.12 for the case j =

2;

Bn/Br](

(n

+

+.V#_:_. r—1

3

Ill

R

) )

for the methods used and for values of j close

to n + 1, see 7.14 if.) For more general C, the main results are due to Grunbaum and are restricted

to the case j =

1:

For a convex body C, 2 when C is an affinely regular hexagon in R'; o'(C) 3 when C = —C c Rt; n + I when C — C c R" and C is strictly convex. ot(C) 7.8.

o'(C)

the others in Grunbaum [91 for n 2. The second result is in Grunbaum Beyond this, almost nothing is known. On the one hand, we know of no for general C and n and even n we have only very rough upper bounds on the numbers

for large values of j no explicit numerical bounds on r,(T'C) better than those stated below for Very little is known about the relative magnitudes of T,(TC) and r( TC*), where

C (C*: = (C + (—C))/2). Consider

a set F and the corresponding families of translates, 9 = (x + C: xe F) and = {x + : xe F). An elementary calculation shows when j = 2, 5 £ if and only if but for any higher j neither of these statements implies the other. Hence no inequality between o''(C) and is trivial, although we do not know of any C for which > o''(C). Note that for a triangle S in R8, o'(S) 3>2= We turn now to Gallai's problem for families of homothets, where even less is known. The geometrical situation is so complicated that it seems impossible to find an equivalent covering problem. However, two different approaches by means of covering do lead to rough upper bounds. One possibility is to generalize the approach of Ungár and Szekeres mentioned earlier. For a convex body C in R', let ?(C) be the smallest number j which

is such that whenever Xc 1 and (x + a,C) fl C * 0 for each x€X, = {C) U {x + aRC: xe X), then .9 admits a f-partition (i.e., some f-pointed set intersects all members of $r). Then clearly Tz(HC) ?(C); and 2? C if 11± is the group of all homotheties in Further, with .9and

as described, a1 e (a1 + C) 11 C and yx: = a'(x — Zg) + zg, it is easily verified that + Ccx + a2C and + C) fl C * 0. Thus ?(C) may be defined alter.

natively in terms of families of translates of C. From this it follows, by reasoning analogous to that in 6.3, that ?(C) is equal to the covering number [C + (—C)/CJ. Thus we state 7.9.

r2(HC)

' Perhaps

?(C) = [C + (—C)ICJ.

this bound is already outdated by the method of Erdös-Rogers (11. (Cf. the

footnote to 7.10.)

HELLY'S THEOREM AND ITS RELATIVES

147

The principal, known results on the function 7 may be stated as follows (Danzer [3; 4], Grunbaum [9]): 7(C) 5' when C is centrally symmetric, (4n + 1)' in any case. 7.11. If C is a convex body in R2, then 7(C) 7 when C is strictly convex, while 7(C) 7 if an affinely regular hexagon A can be inscribed in C in such a way that C admits parallel supporting lines at opposite vertices of A.

7.10. For a convex body C in

while 7(C)

To show that 7(C) 5' when C = — C, one obtains a covering of 2C by first packing into 5C/2 as many translates of C12 as possible, and then expanding each of them by a factor 2. For general C, assume the centroid to be at the origin and let K: = C A (—C). Then C c nK by the result of Minkowski [1] and Radon [1) (or use 2.7) and hence C + (—C) c 2nK. Covering 2nK by translates of K leads to the inequality 7(C) (4n + 1)'. The first inequality in 7.11 follows from the possibility of arranging six translates of C so that they are all adjacent to C and are cyclically adjacent among themselves. For detailed proofs ot 7.10—7.11, as well as for relevant examples and values of rt(HC) for a few special C c R, see Danzer [2; 4]; for the case C = —C, see also Grtlnbaum [9]. Danzer [41 gives conditions on a family .9 in R' which imply the existence of a 7-pointed set intersecting the interior of each member

of .9.

For a Euclidean cell K, the idea of studying by means of 7(K) was extended by Danzer [3] to r,(K). With the aid of a general theorem on intersections of metric cells (stated in 9.9 below), he proved the following crucial lemma: 7.12.

Suppose 2 j

n + 1 and

is a family of Euclidean cells in E'

Let C0 be an intersection of j —

1 members of which has smallest (Euclidean) diameter among all such intersections, and suppose

with

KflC0*Ø (for all Then E' contains a flat F of dimension n + 2 — j such that F fl C0 is a Euclidean cell; diam C0 = diam (F fl C0) diam (F fl K) for all K€ (Ffl K) fl C0 0 for all K€ and (*) holds, then intersection with a suitable In other words, if flat wilt reduce the dimension by j — 2 and yield again a family of Euclidean ceLls in which each meets the smallest one. Consequently,

7.13.

• As pointed out by C.A. Rogers in a letter to the authors, these bounds may be replaced by 7(C) 3"O,, in the symmetric case and by in general, + where o,, : = n log n + n log log is+ 5n. Here 0,, is the density always obtainable by by translates of a convex body (Rogers (1]), while covering + 1)-' iS an estimate for the number vol ((1/3)C + (213)( — C))/vol (C) obtained by consideration of the (n + 1)-dimensional convex body associated with C (Rogers.Shephard 121).

148

LUDWIG DANZER, BRANKO GRONBAUM, AND VICTOR KLEE

It would be interesting to know whether, for 1 e family JY of Euclidean cells in E' with

i j — 2 and for every there exists a fiat F of

= (Ff1 K: Ke .2"I deficiency i in such that the family of intersections intersects and some smallest member of has property -' (or at least all members of In particular, what about the case i = j — 3? A second and simpler approach to the Gallai problem for families I-IC was recently developed by Danzer (as yet unpublished). Consider a family .9: = 1, and {x + : xE X} of homothets of C with Ce..9' and with always

For each k,.. can be split into k subfamilies, i k) e): = {x + aRC: xe X, (1 + < (1 + (1 then and a remainder Jr'(k, €) consisting of rather large cells. If .5 each of the k families €) can be pierced by ö't(C, 1 + e) or fewer points, while 9'(k, €) can be pierced by a rather small number p of points. (If C is smooth and (1 + is large enough, p n + 1.) It follows that Tj(HC) ko't(C, 1 + €) + p k[(1 + c)(J'o)C/Cl + p. suppose

0.

For large n this should yield much better results than the inequalities mentioned above, except when C = and I is close to n. In particular, this method

leads to the inequality k[(1 + €)(

—

+

1))'B"/B"] + RI + (1 +

where the best numerical results are obtained for (1 + e)i/n)(n + 1)'(j — 1 + (1 + Of course all the upper bounds for which have been mentioned here

are very rough. We know of no example which contradicts T,(HB*) n + 4—f and also of no C for which r,(TC) > n + 3—j. As to lower bounds for nothing is known for j > 2. *

*

*

*

*

*

*

*

*

In attempting to find values or upper bounds for the Gallai numbers and we were led to various covering problems. For general C and

n, difficulty is caused by the fact that the bodies to be covered by translates of C are themselves not much larger than C. For the special case of B", there is more hope of improvement. Recall from = o' t(C) 7.6 that whence (7.7) + 1XJ —

And by 7.9, 32(HC) ?(C) = [C + (—C)/CJ, whence

r,(HB")

[2B"/B"]

Thus we are interested in finding "good" coverings of rB" by translates of

especially for 1
and for r 2. As Danzer observed, the task is (almost trivially) equivalent to a similar one on the sphere

HELLY'S THEOREM AND ITS RELATIVES

the minimal number of spherical caps of (spherical) radius in E', then a that will cover the unit sphere 7.14.

If

is

[(cosec

a,

for 2r/2> a

C_(a)

1

+1

for

ff14

>

a

ir/6.

For small n and specific valifes of a, one may use particular coverings to

get upper bounds. For example, Danzer [3] shows whence f(B') 21 and 7.15. Ct(ff/6)=6, C2(ir/6)_20, and (See 7.9 — for definition of ?.) For larger n and arbitrary a one

71.

may instead estimate /ackings of radius

a12 by use of Blichfeldt's method (see Rankin [1]). This yields (Danzer [3]): 7.16. <

— 2)(rt — 1)1/2 +

1r,2)112rra —

+

By a theorem of Rogers [2], the density of a packing of congruent cells in

cannot exceed the density in a regular simplex with the vertices being would lead to a slight centers of cells. The same result for packings in improvement of 7.16. (This result was proved by Fejes Tóth [2; 3] for n = 3, but is unknown in general.) are especially interesting for r = 2, since we Lower bounds for know so little about rt(HB'). Dividing the (n — 1)-measure of SRI by that of a spherical cap of radius ff16, Danzer [3] showed that for n 3, > By Coxeter-Few-Rogers [1], the density of a covering of E" by

congruent cells cannot be less than the corresponding density in a regular this inequality could simplex. If the same were known for coverings of be improved by a factor of about n12. (For n = 3 the result was proved by Fejes Toth [1; 3], though it is uncertain in general. It follows in particular that Csfr/6) 19.) Hadwiger [11] asked how many translates C4(c e I) of a convex body C in E*

can be arranged so that each intersects C but no two of them have

C*0 interior points. (The similar problem with C1 fl was treated by Danzer-Grunbaum [11.) From (*) in 7.8-i- it follows that the maximum number is the same for C as for and then a simple argument shows that the maximum is attained only if C itself is among the C1. Thus Hadwiger's number is equal to ?(C) + 1, where ?(C) is the maximum number of translates of C which can be adjacent to C while having no interior points in common with each other. (Two sets are adjacent provided their closures

meet but not their interiors.) When C is smooth, ?(C) serves as a trivial lower bound for f(C). The following is known: '° Perhaps this bound is already outdated by the method of Erdos-Rogers footnote to 7.10.)

Ill.

(Cf. the

LUDWIG DANZER, BRANKO GRUNBAUM, AND VICTOR KLEE

150

7.17.

For every convex body C in RW, nz + n

C is a parallelotoPe if and only if 7(C) =

3*

7(C) =

3W

— 1.

Further,

— 1.

The inequality n2 + n 7(C) was proved by GrUnbaum [151, who observed that it is always possible to have n + 1 translates + C /.,airwise adjacent to each other, and then the n2 + n sets x, — x + C (i * j) are in the desired position with respect to C. (See also Swinnerton-Dyer [1).) The inequality 3W — 1, given by Hlawka [1], Hadwiger [11], and Groemer [I], follows 7(C) at once from the observation that 1 + 7(C) = 1 +

(vol 3C*)f(vol C*).

The characterization of parallelotopes is due to Griinbaum [15] for n = 2 and to Groemer [11 for general n. (Groemer [21 corrects an erroneous remark of Griinbaum [151.) There seems to be no general characterization of those C for which 7(C) = n2 + n, though Grunbaum [15] shows that 7(C) = 6 for every C in R' which is not a parallelogram. Griinbaum [15] conjectures that 7(C) is always an even number. It would be interesting to study the numbers 7(C) for sets C in R' assumed This is done for n = 2 by merely to be homeomorphic with Levin-Straus [1], who show that even without the assumption of Convexity, 7(C)

6.

The problem of finding ?(B*) is famous and of long standing. For a discussion of it, see Fejes Tóth [3] for n = 3, and the report of Coxeter (1) in this volume for general n. At present the best estimates for 7(B") are obtained by the methods described above for ?(B'). Substituting r = (4/3)h/* (corresponding to a/2 = ir/6) in the inequality 7.16 to obtain an upper bound, and computing a quotient of volumes for a lower bound, we obtain / / 4 \(ii—l)II n+ 4 2 1 11 v(2fl± 7.18.

8.

Other intersection theorems. The material summarized in this section

makes little contact with Helly's theorem, but it does treat intersection One group of results owes its initial stimulus properties of convex sets in to coloring problems (see also 8.6-8.8). Two convex bodies in are called neighbors if their intersection is of dimension n — 1, and a family of convex bodies is neighborly provided each two of its members are neighbors. For an obvious reason, the neighborly families require special attention in connection with coloring problems. A neighborly family of convex (or much more

general) bodies in Ez can have at most four members. On the other hand, Tietze [1] answered a question of Stäckel by constructing in E3 an infinite neighborly family of convex polyhedra. (See Tietze [2] for a survey of this and related questions.) M. Crum posed the problem independently about forty years after its solution by Tietze, and a second construction was given by Besicovitch [1]. A more detailed study of intersection properties of convex

polyhedra in R" was made by Rado [3) and Eggleston [1], who proved the

HELLY'S THEOREM AND ITS RELATIVES

151

following (the first statement in each case being that of Rado, the second that of Eggleston). + 1)12, R' contains an infinite family 8.1. When the integer m is of convex polyhedra such that for all j with 1 j m, each j members of ..9 have an (n j + 1)-dimensional intersection. When m > (n + 1)12, such a family does not exist in contains a family of n + 2 convex polyhedra, each j having (n — j + 1)8.2. cannot n. Such a family in dimensional intersection for all j with 1 have n ÷ 3 members. These results on neighborly families are closely connected with the neighborly polyhedra of Gale [3; 51 (see also Carathéoclory 121, Motzkin [3J, and 3.9 above). For example, let Pé be a convex polyhedron in R' which has k vertices such that each segment joining two vertices is an edge of Pk. Then the dual polyhedron Qb has k 3-faces, each two of which intersect in a 2-face of Let K be a 3-face of z a point exterior to K but very close to a (relatively)

j

interior point of K. and from z as center, project all the other faces of Qe into K. The resulting configuration consists of k—I 3-dimensional polyhedra

whose union is the convex polyhedron K, and each two of these polyhedra intersect in a common 2-face. In the constructions of Tietze, Besicovitch, and Rado there was no restriction

on the numbers of vertices or faces of the polyhedra in question. However, Bagemihl [1) asked for the maximum number k of letrahedra in a neighborly family in he proved that 8 k 17 and conjectured that k = 8. Baston [11 has recently proved that k 9, but it remains undecided whether 8 or 9 is the actual value. More generally, the following problem is nearly untouched: For j n + 1, determine the maximum number N(n,j) of convex polyhedra having j vertices each which can appear in a neighborly family in

R'. An extension of the reasoning of Bagemihl [1] shows that always N(n,j) < One might ask the same question with the restriction that the polyhedra in question be affinely or combinatorially equivalent to a given one,

but even here the literature is silent except for the results of Bagemihi and Baston. (For a related result involving translative equivalence of tetrahedra, see Swierczkowski [11.) *

*

*

*

*

*

*

*

*

Another group of results owes its basic notion to the theory of probability. of subsets of a set E is said to be independent in E if for every

A family

subfamily 5 c

of sets as the least upper bound of the cardinalities of independent subfamilies of Then the results of Let us define the rank of a family ..9

Rényi-Rényi-Suranyi [11 may be stated as follows. 8.3. The family of all open coordinate axes, is of rank 2n.

parallel to the

152

LUDWIG DANZER, BRANKO GRIJNBAUM, AND VICTOR KLEE

The family of all (n — 1)-dimensional Euclidean spheres in E" is of rank n ÷ 1. 8.5. If rj is the rank of the family of all open convex polygonal domains of then limj.... r,/logj = 1/log 2. at most sides in The proof of 8.3 is straightforward, while that of 8.4 depends on an estimate can be divided by k of the maximum numbers of parts rn into which spheres. The exact value of B'i is tabulated for 1 n 10, 1 k 10, and several problems are suggested. 8.4.

*

*

*

*

*

*

*

*

*

For the problems of Gallai 7) and of Hadwiger-Debrunner [2] (4.4 above), which assure that one seeks conditions on a family .9- of convex sets in .9- can be partitioned into m subfamilies, each with nonempty intersection. On the other hand, Bielecki [1], Rado [4], and Asplund-Grünbaum [11 have sought conditions under which .9- is mcolorable, this meaning that .9- can be partitioned into m families of pairwise disjoint sets. For a family .9- of denote the smallest natural sets and for natural numbers p and q, let .9- and each p-membered number r which has the following property: if = 00 when is r-colorable subfamily of is q-colorable, then no such r exists). The result of Rado [4] and Bielecki [1] is = k. 8.6. If .5 is the family of all intervals in an ordered set,

Asplund-Griinbaum (1] prove 8.7. If is the family of all plane rectangles with edges. parallel to the coOrdinate axes, = 6.

Further, a graph-coloring theorem of Grotzsch [1] 8.8. For each convex body K in R', all (positive) homothets of K).

3 (where HKis the family of

The paper of Asplund and Grunbauni contains other results on colorability problems. The following geneial problem seems worthy of mention: For what families .9- of convex < oo? How sets in R" and for what p and q is the coloring number < co when .9- is the does it depend on p, q and n? In particular, is family of all convex bodies in R2, or when is the family of all parallelepipeds in R' with sides parallel to the coordinate axes? (Some partial answers have recently been obtained Danzer and GrUnbaum.) These "coloring problems" suggest the following extension of the Gallai-type problem formulated near the end of § 4: 2, , and conditions on .9- are given. Determine the Suppose X, smallest cardinal 4' such that for every family .9- which satisfies the given c .9- and card conditions, and for which that can <x + 1 be split into p families each with property it is possible to split .9- into 4' subfamilies each having property

of various families, and raises some

HELIX'S THEOREM AND ITS RELATIVES

153

is the property "the £ =3, p = 2, and 8.6 asserts that if Xis an ordered set, .5- = = 6;

In the above terms, the result 8.7 asserts that if

members of the family are pairwise disjoint," X = R',

.9 =

then 4' and A = = k + 1, p = k, and A = successor of card X, then çb = k.

An unsolved problem related to Gallai's original geometric problem arises when X = R', = (nonempty intersection), .5- = TK for a convex body When K is strictly, convex and centrally K c R2, K = 2k, p = k, and A = symmetric, a result of Grlinbaum [9] implies that 4' 3n. It is easy to verify that 4' 8 when K is a circle and k = 2; Grlinbaum conjectures that 4' 6 in this case. *

*

*

*

*

*

If çø is a transformation of a metric space (X,

*

* p)

*

into a metric space (Y, a),

then is called Lipschitzian with constdnt A provided o4ox, 'ox') Ap(x, x') for all x, x' e X; and so is a contraction provided always a(ç'x, tx') p(x, x'). Several

authors have studied the problem of extending (to all of X, with-preservation of constant) a Lipschitzian transformation which is initially 4eflned only on a subset of X. Two early papers on the subject were those 1of McShane [1] and Kirszbraun El]. The method of McShane (and, later, of Banach [11 and Czipszer-Gehér [1]) provided explicit formulae for the extension in certain cases, while the "point-by-point" method of Kirszbraun, Valentine [1; 2; 31, and Mickle [1] used an interesting intersection property. We shall discuss here only contractions, for the general Lipschitzian extension problem can be reduced to that for contractions when Y is a normed linear space. and are families of metric cells in (X, p) and (Y, a) respectively, When we shall write if the families can be indexed simultaneously in such )a way that corresponding cells have equal radii and the distance between any two centers from .5- is at least that between the corresponding centers from Thus with forall

these circumstances, it is natural to ask whether *c9 * 0 implies * 0. That is, if the cells in .5-have a common point, should they not again have a common point if we displace their centers so as not to be farther apart than they were before? In fact, the situation is as follows (Valentine Under

[3)):

For metric spaces X and Y, the following two statements are equivalent: whenever .5- and are families of metric cells in X and Y respectively, then ,r.9 * 0 implies with 5>* 0; whenever W c X, each contraction of W into Y can be extended to a contraction of X into Y. 8.9.

From 8.9 it is clear that results of all the authors mentioned above may be interpreted as intersection theorems, though they were not always stated as such. There seems to be no complete picture even of what pairs of Minkowski

LUDWIG DANZER, BRANKO GRONBAUM, AND VICTOR KLEE

154

spaces X and I are related as in 8.9, though Zorn observed that the conditions may fail for large classes of such pairs (see Valentine [3)). In any case, the following is known: 8.10. For each of the following cases, the conditions of 8.9 are satisfied:

a° X an arbitrary metric space, Y = b'

X=Y=

(spherical space).

8.11. When X and V are the same two-dimensional Minkowski space with unit sphere S, the conditions of 8.9 are satisfied if and only if S is an ellipse or a parallelogram.

The result 8.11 is due to Griinbaum [7). The proof of 8.IOa is given by McShane [1), Valentine [1), Banach [1), Mickle [1], and [1],

with various analytical generalizations (e.g., to functions satisfiying a LipschjtzIFolder condition). From 8.lOa it follows that the conditions of 8.9 are also

satisfied for arbitrary X when Y is the space of all continuous real functions on an extremally disconnected compact Hausdorif space, and thus in particular

when I is a Minkowski space whose unit cell is a parallelotope (cf. 7.1 and 7.2).

Case b' is discussed by Kirszbraun [11, Valentine [31, Mickle [1), Schoen-

berg [1), and Grunbaum [19], and (as noted by Valentine) extends easily to the case X = Y = complete inner-product space. The case C' is treated by Valentine [3), who also 12) establishes the result for X = V = n-dimensional hyperbolic space. Helly's theorem is employed by Kirszbraun [1] and Valentine

(1; 2; 31; in particular, it is clear that for 8.7b it suffices to consider families consisting of n ÷ 1 cells. All the proofs employ analytical notions (inner. products, etc.) even for X V E2, but it would seem worthwhile to seek a more geometrical proof. The for 8.lOb are those of Schoenberg [11 and of Grunbaum [19], who proves a generalization (8.12 below) which a and includes (by suitable choice of , both Kirszbraun's theorem and

a recent theorem of Minty [1]. Suppose fri, - •, x,j C the inner product (x, — {Yi, - . , Yiui} C are real numbers not both zero. Then — y) 0 for all. i, j, and a and there exists z E E such that + az, + $z) 0 for all i. 8.12:

There are some other interesting results and problems which involve conBy considering the case of cells all of the same radius, we tractions in conclude from Kirszbraun's theorem that if a set X c admits a contraction onto a set I, then the circumradius of I is not greater than that of X. While

there exist X and I in E such that X can be contracted onto I even though I has greater width, this cannot happen when X and I are both convex bodies and the contraction is biunique (Gale [4]). Nothing seems to be known about the behavior of inro4ii in this case. There is an unsolved problem of Helly type concerning continuous contractions For ordered k-tuples (x1, .. ., of points in a metric and (Yz, space (M, p), a continuous contraction of the first onto the second is an ordered of continuous mappings of 10, 1] into M such that for all k-tuple -, in

-

-

-

HELLY'S THEOREM AND ITS RELATIVES

155

= çoj(a)) whenever 0 = and a 1. The problem is to evaluate the number rn,, defined as the smallest are k-tuples from E' for j such that whenever (xj, . -, and (yi, , which each j-tuple of the xi's admits a continuous contraction onto the cor-

i and j,

•

responding j-tuple of the ye's, then (x1, onto(y1,

•,

=

admits a Continuous contraction Itiseasytosee that rn1 =2.

co.)

A. H. Cayford has remarked that from consideration of the positive and negative unit vectors along the n axes in together with another point far removed from this set, it follows that rn% 2n + 1 when n 2. A striking problem is that of Thue Poulsen [1) and M. Kneser (1]: When 5 and are families of unit cells in does 9 >- imply vol x5

(where iz indicates union) and vol ir9 vol irs'? An affirmative solution would be useful in connection with Minkowski's theory of surface area. Kirszbraun's theorem gives a bit of information on the problem, and 3fl vol Kneser shows that vol (For other details see M. Kneser (I) and Hadwiger [9].) It seems clear that in non•Euclidean Minkowski spaces, the answer to the above question is negative, in the two-dimensional case,

vol

this follows from 8.11. For a Minkowski space X with unit cell C, it would be of interest to study the number defined as the smallest u > 0 such that whenever 9- and are families of homothets of C with 5 >and * 0, then a common point is obtained upon expanding all the members of about their centers by the factor c. The number is defined similarly for families of translates of C. Clearly 1 and 1 E0' are where Jo and the Jung constant and the expansion constant of 6. Is it possible to have when C is a Leichtweiss body (cf. 6.5+)? = E0? What is 1 9. Generalized convexity. The applicability and the intuitive appeal of convexity have led to a wide range of notions of "generalized convexity." For several of them, theorems related to Helly's were either a motive or a

by-product of the investigation. For others, it seems probable that no attempt has been made to find analogues of Helly's theorem. In order to encourage such attempts and to facilitate comparison of the various notions which have been considered, our discussion will include several generalizations of convexity which seem at present unrelated to Helly's theorem.

The usual procedure in defining a generalized convexity is to select a property of convex sets in R or E • which is either characteristic of convexity or essential in the proof of some important theorem about convex sets, and to

formulate that property or a suitable variant in other settings. Many properties of convex sets are useful for this purpose. We shall first describe in general terms the most important procedures which have been adopted, and then review' briefly some of the results obtained in specific cases. The usual definition of convexity in I?' can be generalized according to the following scheme. In a set X, a family of sets is given together with a function 11 which assigns to each Fs a family of subsets of X. A set K c X is called 'i-convex provided K contains at least one member of ij(F)

156

LUDWIG DANZER, BRANKO GRUNBAUM, AND VICTOR KLEE

whenever,F c K and Fe .F. This is the most common form of generalization, and appears in most of the cases discussed below. Usually .9 is the family. of all two-pointed subsets of Xand Prenowitz [1), for example, studies convexity

in abstract geometries based on the notion of "join." For other possibilities see 9.7 below and Valentine [4), where one such variant is discussed and references to others are given. is an Another approach derives from the fact that every convex set in intersection of semispaces (Motzkin [11, Hammer 11; 2), Klee [41), and every

closed (resp. open] convex set is an intersection of closed [resp. open] halfof subsets of a set X is given, and a set Kc X is spaces. A family called

-convex provided

K is the intersection of a subfamily of

of this appear in 9.1, 9.3, and 9.9 below. See also Ghika [3).) The '-convexity can also be expressed in terms of "separation": a definitions of -convex if and only if for every x E X K there exists He set K c X is such that K c H and x e H. (For a generalization in this direction, see Ellis [1].) The family " may be defined in terms of a family of functions, or more directly we may be given a family 0 of real-valued functions on X and say that a set K c X is 0-convex provided for each XE X — K there exists 'p €0 such that ço(x) < inf çoK. This procedure is followed by Fan [1] in his generalization of the Krein-Milman theorem, and enters also in various notions of "regular convexity" (see Berge [1) and his references). is convex if and Another approach is based on the fact that a subset of (Instances

only if its intersection with each straight line is connected. Working in R' and using in place of the lines a two-parameter family of curves, Drandell [lj obtained interesting results under very weak assumptions. His paper also gives references for previously known results in this direction. (See Skornyakov [1] for a relevant result on curve families in R, and Valentine [3], Kuhn [I), Fáry [I], Kosiflski [1), and their references for some characterizations of convexity in terms of k-dimensional sections.) Of course one may generalize, in various ways, the underlying algebraic

structure in terms of which convexity is formulated. Many of the combinatorial results of this paper are valid in a linear space over an arbitrary ordered field. Monna [1] discusses convexity in spaces over nonarchimedean ordered fields. Other algebraic formulations are those of Rado 121 and Ghika [1; 2], the former being described briefly in 9.4 below.

One may ignore the surrounding linear space and seek a more intrinsic notion of barycentric calculus or convex

space.

This was done by Stone (1; 21,

H. Kneser [1], and Nef [I]. Their theories turn out to be embeddable in linear spaces, but might be of independent interest if reformulated to introduce topological as well as algebraic structure. For an open subset D of a metric space (M, p), let ö, and denote the inner and outer distance functions of D. These functions are defined on D and M respectively, with O,,(x): = p({x}, M-'- D) and vn(x): = p D is equivalent to that of the function — log ÔD and also to that of the function This suggests that for a family 0 of functions on subsets of a metric space (M, p), one could

HELLY'S THEOREM AND ITS RELATIVES

157

say that an open set K c M is #-convex provided —log Jj eG, or perhaps provided Vg

The notion of convexity in spaces of several oomplex vari-

ables may be approached in this way (see 9.11). However, a more useful idea suggested by these considerations is that one should try to generalize the sort of "convex structure" which is formed by the convex subsets of R' together with the convex or concave functions defined over them. This leads in one direction to the theory of complex convexity (9.11) and in another to the abstract minimum principle of Bauer [11, generalizing the Krein-Milman theorem.

Finally, we mention the fact that each new characterization of convexity in R' may lead to new generalized convexities, and that, conversely, the search for a suitable notion of convexity in a given setting may lead to discovery of new and useful properties of sets which are convex in the classical sense. For an interesting example, see R&dström's work ([1; 21 and 9.10 below)

in topological groups. We proceed now to discuss some specific "convexities," commencing with

those which appear to be most closely related to Helly's theorem. 9.1. Spherical convexity. Convexity on the n-dimensional sphere S" has been considered from various points of view, and is certainly the most significant of the generalized convexities so far as Helly's theorem is concerned. Several different definitions of spherical convexity have been studied, though not always with due regard to the limitations which they impose. For the sake of simplicity, we shall discuss only closed sets, though in most cases this restriction can be avoided. (S) A set Kc is strongly convex if it does not contain antipodal points and it contains, with each pair of its points, the small arc of the great circle determined by them. Equivalently, K (being closed) is strongly convex if it does not contain antipodal points and is an intersection of closed hemispheres. strongly convex K is an intersection of open hemispheres, and vice

versa.

(W) A set Kc S' is weakly convex if it contains, with each pair of its points, the small arc or a semicircular arc of a great circle determined by them. Equivalently, K is connected and is an intersection of closed hemispheres.

(R) A set K c S' is Robinson-convex if it contains, with each of its non-

antipodal points, the small arc of the great circle determined by them. Equivalently, K is an intersection of closed hemispheres. (H) A set K c S* is Horn-convex if it contains, with each pair of its nonantipodal points, at least one of the great circle arcs determined by them. Obviously, every strongly convex set is weakly convex; weak convexity implies Robinson-convexity, and that implies Horn-convexity. The strongly convex sets and the Robinson-convex sets form intersectional families. All strongly convex sets are contractible, as are all the Robinson- or weakly-convex sets except for the great rn-spheres c S". According to purpose, one or another definition of spherical convexity may be more appropriate. Horn-

158

LUDWIG DANZER, BRANKO GRUNBAUM, AND VICTOR KLEE

convexity was introduced by Horn [1] in proving his generalization of Helly's theorem. Robinson-convexity was used by Robinson [2) in studying congruenceindices of spherical caps (see also Blumenthal [2]). Weakly convex sets were considered by Santaló [4] among others, though the definition at the beginning of his paper is different and includes only a subclass of the strongly convex sets.

An exhaustive bibliography of papers dealing with spherical convexity would be very extensive. Almost every notion and result on convexity in

can be extended to Sn, and in many cases the extension has been at least partly accomplished in connection with needs arising from other problems. As an example, we mention a little-known paper of Vigodsky [1] which proves (Vigodsky's main theorem was an analogue of Carathéodory's theorem for proved earlier by Fenchel [2).) It seems that Carathéodory's theorem and its

variants have not been generalized in full to the spherical case, though such an extension is surely possible. For certain types of problems, the treatment in S' is more satisfactory than that in R', due to the possibility of dualization in Sn. For example, the of Jung's theorem [11 on circumspheres and Steinhagen's [1] analogues in on inspheres are dual aspects of the same result (Santaló [51). On the other hand, some results in have no Euclidean analogue. An example of this kind is furnished by the difference between the following two relatives of Jung's theorem, in which connectedness plays an essential role though it is irrelevant in Rn: If a compact subset of has diameter less than arc cos (n + 1)', it lies in a closed hemisphere; a compact subset whose diameter is equal to arc cos (n + 1)-' need not lie in any hemisphere (MolnIr [4]). If a compact connected subset of sn has diameter arc cos n1, it lies in a closed hemisphere (Grunbaum [14]).

Due to the close and obvious relationship between spherical convexity in

many results can be interpreted in both the and convex cones in spherical and the Euclidean setting. This fact has been used repeatedly (e.g.,

s"

Motzkin [1), Robinson [11, Horn [1]).

For spherical analogues of Helly's theorem, the simplest approach seems to be through Helly's topological theorem. From this it follows that the family of all homology cells in n + 2. Alternatively, the result has

may be stated as follows: A family .9 of homology cells in

has non-

empty intersection provided each uniofi of n + 2 members of ..9 is different from sn and each intersection of n + 1 or fewer members of is a homology cell. This applies, in particular, to the case of strongly sets considered by Molnár [4]. Similar corollaries may be derived for the other types of spherical convexity, but their formulation is somewhat more complicated due to the absence of intersectionality, or contractibility, or both (see Horn [11, Karlin-Shapley [1], Griinbaum [14]).

Other results of Helly-type for s" may easily be derived from results on convex sets in For example, the case j = 1 of Theorem 4.1 (essentially contained in Steinitz's result 3.2) may be reformulated as the following fact, first stated by Robinson [2]: If is a family of at least 2n +2 Robinson-

HELLY'S THEOREM AND ITS RELATIVES

159

and each 2n +2 members of .9- have a common point, then r.9- * 0. This and related results were employed by Blumenthal [21 in connection with linear inequalities. For some intersection and covering theorems a question involving hemispheres, see Hadwiger [4] and Blumenthal [2; raised by Hadwiger is answered by GrUnbaum [9]. Horn-convex sets have been studied by Horn [1J and Vincensini [3; 4; 5]. Horn shows that if 1 k n + 1 and 9- is a family of at least k Hornsuch that each k members of ..9 have a common point, convex sets in lies in a great (n — k + 1)-sphere which then every great (n — k)-sphere in convex sets in

intersects each member of .9 (cf.

4.3).

For k =

n + 1,

this asserts the

existence of a point z such that every member of F includes z or its antipode. 9.2. Projectlve convexity. A set K in the n-dimensional (real) pro jective space P' is called convex provided it contains, with each pair of its points, exactly one of the two segments determined by these points. Such sets have

been considered repeatedly since their introduction by Steinitz [1] and VeblenYoung [1], though most authors have discussed only the equivalence of various definitions. (See de Groot-de Vries [11 and other papers listed by them.) The

projectively convex sets are contractible but do not form an intersectional family. The intersection of k + 1 convex sets in may have up to components, each of which is projectively convex (Motzkin [1]). On the other

hand, if each two of the sets have convex intersection, then so has the entire family. Thus the following is an immediate consequence of Helly's topological is such theorem: If a family 9 of at least n + 1 closed convex sets in that

of .9- have convex intersection and each n + 1 members intersection, then x.9 * 0 (Griinbaum [11]). let which are the k) denote the class of all subsets of

each two members

have

For k

1,

union of k or fewer pairwise disjoint closed convex sets. It seems probable that k) is of finite Helly-order, but apparently this has not been established and surely the exact value even of 2)) is unknown (cf. 4.11). For other results related to projective convexity, including some on common transversals, see Fenchel [3], Kuiper [1], de Groot-de Vries [1], Schweppe [1J, Gaddum [1], Marchaud [1], Grunbaum [11] and Hare-Gaddum [1]. 9.3. LevI's convexity. Like that of Rado discussed in 9.4, this notion was motivated primarily by Helly's theorem. Levi [1] considers a family of subsets of a set X, and assumes: (I)

for all

For each set Y lying in some member of the is defined as the intersection of all members of 'if' which contain Y. The second axiom is (ha) Every (n + 2)-pointed subset of a member of contains two disjoint sets whose have a common point. Generalizing Radon's proof [21, Levi deduces from the above axioms that if

.9- is a finite family of at least n + 1 members of and each n + 1 members of 9- have a common point, then * 0. As corollaries he obtains Helly's theorem on convex sets in R", its extension to n-dimensional

160

LUDWIG DANZER, BRANKO GRUNBAUM, AND VICTOR KLEE

geometries satisfying Hubert's axioms of incidence and order (Hubert [11), and an intersection theorem in free abelian groups with n generators. (In is the family of all subgroups, the neutral element being the last case, omitted from each.)

It would be of interest to study, in a system assumed to satisfy (I) and perhaps other simple "axioms of convexity," the inter-relationships of Radon's property as expressed by Helly's property as stated above, and Carathéodory's property expressed as follows: (ilL1) Whenever a point p lies in the

of a set Y, then p is in

the

so,neai-nwst-(n + 1)-pointed

subset of Y. 9.4. Rado'. convexity. Rado [21 considers an abelian group A under the action of a commutative ring of operators, and sets forth conditions on A and which assure the validity of Hefty's theorem (with a proper

interpretation of "convex set"). His reasoning yields not only Helly's theorem on convex sets in R*, but also a generalization of a theorem of Stieltjes Eli on arithmetic progressions. With respect to a given coordinatization of a lattice is the set of all points of the form x0 + where the xj's are n + 1 given points with integral coordinates and the ai's are-arbitrary integers. Rado proves that if ..9 is a finite family of at least n + 1 lattices in such that each n + 1 of them have a common point, then * 0. 9.5. Ryperconvexity. If K is a compact convex set in R*, a set A c R' is K-convex (or hyperconvex with respect to K) provided A contains, with each pair of its points, the intersection of all translates of K which contain those points. K-convex sets form an intersectional family and the connected Kconvex sets are intersections of translates of K. K-convex sets need not be connected, but attention is usually restricted to their connected components. The study of hyperconvexity was initiated by Mayer [1], and a complete set of references can be found through Pasqualini [1), Blanc (I], and Santaló [6). Most of the papers treat only the planar case, and under restrictions on K. They deal with support properties, equivalent definitions, hyperconvex hulls, extremal problems, etc. 9.6. Quasiconvexlty. Let A be a subset of [0, 1). A set Kc is A-convex (or quasiconvex with rvspect to 4) provided Ax + (1 — 2)y

K whenever A e 4

and x,y e K. This notion was studied by Green and Gustin El), who give references for the previously considered special cases (such as A = (1/2)). A detailed study of even more general concepts is the paper of Motzkin [4J. 9.7. Three-poInt convexity. A set K in R" is 3-Point convex if it contains,

with each three of its points, at least one of the three segments determined by these points. Valentine [4; 5] has made an interesting study of this property, and a more general notion has been formulated by Allen [1]. Hare and Gaddum [1] discovered a connection between 3-point convexity and projective convexity. It seems likely that the family of all 3-point convex subsets of is of finite Hefty-order, but this has not been determined. (See also Valentine [6; 7].)

HELLY'S THEOREM AND ITS RELATIVES 9.8.

161

A set K in a partially ordered space is order-convex

K whenever x, z K with x
For results on extreme points and convex hulls with respect to order-convexity, see Franklin [1). provided y

Apparently the literature contains no results of Helly-type about order-convexity

in general partially ordered linear spaces, though in any linearly ordered space the family of all order-convex sets is intersectional and has Hellynumber 2. 9.9. MetrIc eseveilty. The usual definition of metric convexity is due to Menger [1): A set K in a metric space (M, p) is metrically convex if for each pair of distinct points x, z 6 K there exists a point y K, different from x and z, such that p(x, z) p(x, y) + p(y, z). When K is not only convex but also metrically complete, then the points x and z of K must lie in a subset of K which is isometric with a linear interval of length p(x, z). A closed subset strictly convex Banach space) is metrically convex if and of E' (or of only if it is convex in the usual sense. For geodesic distance on the sphere Sn, a closed set is metrically convex if and only if it is weakly convex in the sense of 9_i. For various geometric developments involving Menger's and other closely related notions of metric convexity, see Blumenthal [3], Busemann [2), Pauc [lJ, Aronszajn-Panitchpakdi (1], and references listed by these authors. A theorem of Whitehead 11] asserts that for any point p of a Riemannian manifold with positive definite metric, all sufficiently small "spherical" neighborhoods of p are metrically convex in a rather strong sense. This result was

sharpened by NIjenhuis [11.

Several authors have studied the relationship between the topological structure of a metrizable space and the existence of a convex metric compatible with the given topology. Menger (i] asked whether every continuous curve admits a convex metric, and after several partial results the problem was finally settled affirmatively by Bing [1]. See Bing [1; 2; 3], Plunkett (ii, and Lelek-Nitka [1] for this and related results and references. In order to extend some of his selection theorems, Michael [1] formulated the notion of a convex structure on a metric space, describing a situation in which, as in a Riernannian manifold, it is meaningful to form "convex combinations" of certain (but not necessarily all) k-tuples of points in the space. For another metric notion related to convexity, see Michael (2). Danzer [2; 3; 4) formulates and applies yet another concept of convexity in metric spaces. For each pair x and z of distinct points of M, let H(z, z) = eM: p(x,y) < p(y, z)}. Such a set is called a halfspace, and a set is called convex provided it is an intersection of halfspaces. This is closely related to Leibniz's idea (see Busemann [1]) of defining a plane as the set of all points equidistant from two given points. From results of Busemann [11 it follows that when (M, p) is a normed linear space with metric generated by the norm, then the closed Danzer-convex sets in M are all metrically convex (in the sense of Menger) only when (M, p) is an inner-product space. Sets of the form {y eM: p(x, y) = p(y, z)} are studied also by Kalisch and Straus (1].

162

LUDWIG DANZER. BRANKO GRUNBAUM, AND VICTOR KLEE

Defining the convex hull of a set as the intersection of all halfspaces containing it, Danzer [5] proves that in a metric space, any family of spherical cells having none,npty intersection covers the convex hull of the set of its centers. This was the principal tool in his proof of 7.12 above. 9.10. Convexity in topological group.. Though another approach to convexity in topological groups has been formulated by Ghika [4; 5], we consider here only the approach of [1; 21. Let G be a topological group and 20 the class of all nonempty subsets of C. A one-parameter semigroup of subsets of G is defined by [ 21 to be a biunique mapping A of ]O, oo[ into where the righthand 2° such that A(o1 + o1) = A(o1) + for all €10, [2] proves the following, indicates the usual addition of sets. "+"

suggested by the observation that in E', "high powers of small sets are in Hausdorff linear space, the some sense almost convex": In a locally of compact sets are exactly those of the form A(ö) = one-Parameter f(ö) + oK, where I is a one-parameter semigroup of points and K is a compact convex set. In

earlier paper [1], a slightly different definition of one-param-

the one-parameter semieter semigroup leads to the conclusion that in groups of sets are exactly those of the form A(J) = 8K for compact convex K. This suggests that in an arbitrary topological group G, one might define convex sets in terms of the one-parameter semigroups, or at least that these semigroups may play a role similar to that played by convex sets in R. El) shows that if G is a Lie group, then every one-parameter Semigroup of subsets of G has, in a sense, an infinitesimal generator which is a convex set. 9.11. Complex convexity. We turn finally to the generalized convexity which plays such an important role in the theory of functions of several

complex variables. This generalizes simultaneously the notions of convex set and convex function, with striking parallelism between the real and complex theories. (Of course they differ in details ot proof, the complex theory being much more difficult in some respects.) See Bremermann 11; 2; 81 for an excellent description of this parallelism and for references to other work in the field. In particular, Bremermann [1) supplies a "real-complex dictionary" with the following translation of terms: Real

linear function of one real variable convex function of one real variable convex function of n real variables convex region in R Complex

harmonic function of one complex variable subharmonic function of one complex variable plurisubharmonic function of n complex variables (the space of n complex variables). region of holomorphy in

HELLY'S THEOREM AND ITS RELATIVES

is plurisubharmonic provided A real-valued function V on a region D c is upper semicontinuous on D, and for every analytic plane V< V it is a subharmonic function of A on + Aa: 2 e C'} (where z, a e P(z, a) = {z is holothe set D fl P(z, a). A complex-valued function on a region D c if it is single-valued and is holomorphic in each of the n comp'ex variables separately. The region D is a region of holomorphy if there exists a function holomorphic in D and not extendable to a function holomorphic in —

a larger region.

Bremermann [1] justifies the above translations by an

impressive list of parallel definitions and theorems, of which the following are samples:

a convex [plurisubharmonic] function assumes its maximum only at the boundary of a domain unless it is constant in the domain; [D c a domain Dc is convex [a region of holomorphy] if and only if — log ö,, is a convex [plurisubharmonic] function in D. (Here is the inner distance function of D, so that is the radius of the largest Euclidean sphere which is centered at p and contained in D.) As might be guessed from the above parallelism, the plurisubharmonic functions and the regions of holomorphy are also called pseudoconvex functions and pseudoconvex regipns. (The pseudoconvex regions were originally defined in

a different way which turned out to be equivalent to regions of holomorphy.). It would be most interesting to describe axiomatically a structure consisting

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W. W. Comfort and H. Gordon 1. Inner product spaces and the tn -spherical intersection property, Proc. Amer. Math Soc. 12 (1961), 327-329. H. S. M. Coxeter 1. An upper bound for the number of equal nonovenlapping spheres that can touch another of the same size, these Proceedings, pp. 53-71. H. S. M. Coxeter, L. Few, and C. A. Rogers 1. Covering the space wit/i equal spheres, Mathematika 6 (1959), 147-157. J. Czipszer and L. Gehér 1. Extension of functions satisfying a Lipschitz condition, Acta Math. Acad. Sd. Hungar. 6 (1955), 213-220. L. Danzer 1. Ober em Problem aus den kombinatonischen Geo,netnie, Arch. Math. 8 (1957), 347-351.

2.

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3.

(Jber Durchschnittseigenschaften n-dimensionaler Kugelfamiien, .1. Reine Angew. Math. 208 (1961), 181-203. 4. Uber Durchschnittseigenschaften konvezer hoinothetischer Bereiche in den Ebene,

J. Reine Angew. Math. (to appear). L. Danzer and B. Grünbaum 1. Uber zwei Problems bezüglich konvexer .Korper von P. Erdos und von V. L. Klee, Math. Z. 79 (1962), 95-99. L. Danzer, D. Laugwitz and H. Lenz 1.

Uber dos Löwnersche Ellipsoid und sein Analogon unter den einens Eikorper

einbeschniebenen EUipsoiden, Arch. Math. 8 (1957), 214-219. R. De Santis 1. A gendro.iization of Heliy's theorem, Proc. Amer. Math. Soc. S (1957), 336-340.

L. L. Dines and N. H. McCoy 1. On linear inequalities, Trans. Roy. Soc. Canada Sect. III 27 (1933), 37-70. M. Drandell 1. Generalized convex sets in the plans, Duke Math. J. 19 (1952), 537-547. M. Dresher 1. Problem 4361, Amer. Math. Monthly 56 (1949), 557; Solution, ibid. 58 (1951), 272-273.

1. G. Dukor 1.

On a theorem of Holly on collections of convex bodies with common points, Uspehi

Mat. Nauk 10

(1944), 60-61. (Russian) A. Dvoretzky 1. A converse of Holly's theorem on convex sets, Pacific J. Math. 5 (1965), 345-350. M. Edeistein 1. Contributions to geometry in topological groups, Thesis, Technion. Haifa, 1958. (Hebrew. English summary) H. G. Eggleston 1. On Rado's extension of Cram's problem, J. London Math. Soc. 28 (1953), 467-471 2. Notes on Minkowsici geometry. I, Relations between the circumro4ius, diameter

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Convexity, Cambridge Univ. Press, Cambridge, 1958. E. Ehrhart 1. Le plus petit couvercie circulaire de n points, Enseignement Math. (2) 6 (1960), 47-50.

J. W. Ellis 1. A general set-separation theorem, Duke Math. J. 19 (1952), 417-421. P. Erdös and C. A. Rogers 1. Covering space with convex bodies, Acta Arith. 7 (1962), 281-285. P. Erdös and G. Szekeres 1. A combinatorial probiem in geometry, Compositio Math. 2 (1935), 463-470. 2. On some extremum problems in elementary geometry, Ann. Univ. Sci. Budapest Eotvos. Sect. Math. 3-4 (1960/61), 53-62. K. Fan 1. On the Krein-Milnsan theorem, these Proceedings, pp. 211-219. I. Fáry 1. A characterization of convex bodies, Amer. Math. Monthly 69 (1962), 25-31. L. Fejes Tóth 1. Uber die tfberdeckung einer Kugelftilche durch kongruente KugeikoAotten, Mat. Fix. Lapok 50 (1943), 40-46. (Hungarian. German summary) 2. Uber die Abschätzung des kürzesten Abstandes zweier Punkte eines ouf Kugelftäche liegenden Punktsyste,ns, Jber Deutsch. Math. Verein. 53 (1943), 66-68.

'3. Lagerungen in der Ebene, auf der Kugel und im Raum, Springer, Berlin, 1953. W. Fenchel 1. tIbet Krilmmung und Windung geschlossener Ranmnkurven, Math. Ann. 101 (1929), 238-252.

2. Guchloasene Raumkurven mit vorgeschriebenen Tangentenbild, Jber. Deutsch. Math. Verein. 39 (1930), 183-185.

3. A remark on convex sets and polarity, Medd. Lunds Univ. Mat. Sem. Tome Supplémentaire, 1952, 82-89.

4. On the convex hull of point sets with symmetry properties, Rend. Mat. e Appi. (5) 14 (1955), 355-358.

S. P. Franklin 1. Som. results on order convexity, Amer. Math. Monthly 69 (1962), 357-359. R. Frucht de segmentos, Rev. Un. Mat. Argentina 7(1941), 134-135. 1. Sobre un cierto J. W. Gaddum 1. Projective convexity in Euclidean space, Abstract 542-155, Notices Amer. Math. Soc. 6 (1958), 309. D. Gale 1.

Linear combinations of vectors with non-negative coefficients, Amer. Math. Monthly 59 (1962), 46-47.

2. On inscribing n-dimensional sets in a regular n-simplex, Proc. Amer. Math. Soc. 4 (1953), 222-225.

Neighboring vertices on a convex polyhedron, pp. 255-263, Ann. of Math. Studies, No. 38, Princeton Univ. Press, Princeton, N. J., 1956. 4. On Lipschitzian mappings of convex bodies, these Proceedings, pp. 221-223. 5. Neighborly and cyclic polytopes, these Proceedings, pp. 225-232. D. Gale and V. Klee 1. Continuous convex sets, Math. Scand. 7 (1959), 379-397. 3.

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H. Gericke 1. (iber die groaste Kugel in siner konvexen Punktinenge, Math. Z. 40 (1936), 317-320. A. Ghika (all rn Romanian; Russian and French summaries) Corn. Acad. R. P. Romine 2 (1952), 1. Ensembles A-converse dana lea 669-671.

2. Propriétes de liz convexité dana certains modules, Corn. Acad. R. P. Romine 3 (1953), 355-360.

3.

Separation des en8enlbles convexes dana lea espaces hgnés non vectoriels, Acad. Mat. Fiz. 7 (1955). 287-296. Sect. R. P. Rornine Bul.

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Ensembles entiers, converse, 8errC8 et absorbants, dana tee grou pea a radicaux, Corn. Acad. R. P. Rornine 5 (1955), 1229-1233.

5. Grou pea topologiques loca,lement paraconvexes, Corn. Acad. R. P. Rornine 5

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1235-1240.

A. Ghouda-Houri 1.

Sur l'etude combinatoire des families de convexes, C. R. Acad. Sd. Paris 252 (1961), 494-496.

1. C. Gohberg and A. S. Markus 1. A problem on covering a convex figure by similar ones, lzv. Mold. Fit. Akad. Nauk SSSR 10 (76) (1960), 87-95. (Russian) J. W. Green and W. Gustin 1. Quasiconvex sets, Canad. J. Math. 2 (1950), 489-507. H. Groemer 1. AbscMtzungen für die Anzahl der konvexen karper die einen konvexen Kôrper berühren, Monatsh. Math. 65 (1961), 74-81. 2. fiber die dichteste gitterformige Lagerung kongruenter Tetraeder, Monatsh. Math. 66 (1962), 12-15. J. de Groot and H. de Vries 1. Convex sets in projective space, Cornpositio Math. 13 (1957), 113—118. H. Grötzsch 1. Zur Theoria der diskreten Gebilde. VII.Kin Dreifarbenaatz für dreiecksfreie Netze auf der Kuget, Wiss. Z. Martin-Luther Univ. Halle-Wittenberg Math.-Nat. Reihe 8 (1958159), 109-119.

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2. A theorem on convex sets, Bull. Res. Council Israel Sect. A 5

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Sect. F 7 (1957), 25-30. 4. A simple proof of Borsuk's conjecture in three dimensions, Proc. Carnbridge Philos. Soc. 53 (1957), 776-778. 5. On common secants for families of polygons, Riveon Lematernatika 12 (1958), 37-40. (Hebrew)

6. On common transversals, Arch. Math. 9 (1958), 465-469. 7. On a theorem of Kirszbraun, Bull. Res. Council Israel Sect. F 7 (1958), 129-132. 8. On some covering and intersection properties in Minkow8ki spaces, Pacific 3. Math. 9 (1959), 487-494. On intersections of similar sets, Portugal. Math. 18 (1959), 155-164. A ,.nriant of !feUy'8 theorem, Proc. Amer. Math. Soc. 11 (1960), 517-522. Common tranaversals for families of sets, 3. London Math. Soc. 35 (1960), 408416. 12. Some application8 of expansion constants. Pacific J. Math. 10 (1960), 193-201.

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14. 15. 16. 17. 18. 19.

Math. 10 (1960), 1257-1261. Subsets of S" contained in a hemisphere, An. Acad. Brash Ci. 32 (1960), 323-328. On a conjecture of Hadwiger, Pacific J. Math. 11 (1961), 215-219. Measures of symmetry for convex sets, these Proceedings, pp. 233-270. Borsuk's problem and related questions, these Proceedings, pp. 271-284. The dimension of interections of convex sets, Pacific J. Math. 12 (1961), 197-202. A generalization of theorems of Kirazbraun and Minty, Proc. Amer. Math. Soc. (to appear).

20. Common secants for families of polyhedra (to appear). B. Grunbaum and T. S. Motzkin 1. On components in some families of sets, Proc. Amer. Math. Soc. 12(1961), 607613.

2. On polyhedral graphs, these Proceedings, pp. 285-290. W. Gustin 1.

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2. Review of Verblunsky Ill, MR 14, 495. H. Hadwiger 1. Mitteilung bet reff end nieine Note: (Jberdeckung einer Menge durch Mengen kleineren Durchmessers, Comment Math. HeIv. 19 (1946/47), 72—73. 2. Uber sine symbolisch-topologi.sche Formel, Elem. Math. 2 (1947), 35-41. (Trans-

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5.

(1955), 45-48. Eulers Charakteristik und kombinatorieche Geometrie, J. Reine Angew. Math. 194 (1955), 101-110.

6. Ungelöste Problems, No. 7, Elem. Math. 10 (1955), 111. 7. Problem 107, Nieuw Arch. Wiskunde (3) 4 (1956), 57; SoLution, Wisk. Opgaven

20 8. 9. 10. 11.

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Ober einen Satz Hellyscher Art, Arch. Math. 7 (1956), 377-379. Ungelöste Problems, No. 11, Elem. Math. 11 (1956), 60-61. Uber Eihereiche mit gemeinsamer Treffgeraden, Portugal. Math.

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tlber Treffanzahlen bei translationsgleichen Eikörpern, Arch. Math. 8 (1957), 212-213.

Problems, No. 15; Elem. Math. 12 (1957), 10-11; Nachi rag, ibid., p. 62. Problems, No. 19. Elem. Math. 12 (1957), 109-110. Problems, No. 20, Elem. Math. 12 (1957), 121. über Inhalt, Oberfiilche und Isoperimetrie, Springer, Berlin, 1957. 16. Ungeteste Problems, No. 38, Elem. Math. 15 (1960), 130-131. H. Hadwiger and H. Debrunner 1. Einzelprobteme der kombinatorischen Geometrie in der Ebene. Enseignement Math. 1 (1955), 56-89. (Translated into French, ibid. 3 (1957), 35-70.) 2. (Jber sine Variants zum Hellyschen Satz, Arch. Math. 8 (1957), 309-313. Kombinatorieche Geometrie in der Ebene, Monographies de l'Enseignement Math., No. 2, Geneva, 1960 (revised and expanded edition of [1]). H. Hadwiger, P. Erdos, L. Fejes Tóth and V. Klee 12. 13. 14.

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P. C. Hammer 1. MaximaL convex sets, Duke Math. J. 22 (1955), 103-106. 2. Semispaces and the topology of convexity, these Proceedings, pp. 305-316. 0. Hanner 1. Intersections of translates of convex bodies, Math. Scand. 4 (1956), 65-87. 0. Hanner and H. 1.

A generalization of a theorem of F'enchel, Proc. Amer. Math. Soc. 2 (1951), 589-593.

W. R. Hare, Jr. and I. W. Gaddum 1. Projective convexity in topological linear spaces and a certain connection with property P3. Abstract 576-228, Notices Amer. Math. Soc. 7 (1960), 984-985. R. Harrop and R. Rado 1. Common transversais of plane sets, 3. London Math. Soc. 33 (3958), 85-95. E. HelLy 1. (Jber Mengen konvexer Körper mit gemeinechaftlichen Punicten, Jber. Deutsch.

Math. Verein. 32 (1923), 175-176. 2. Uber Systems abgeschiossener Mengen mit gemeinschaftlichen Punkten, Monatsh. Math. 37 (1930), 281-302. A. Heppes 1.

On the splitting of point sets in three•space into the union of sets of sinaUer

diameter, Magyar Tud. Akad. Mat. Fiz. Oszt. KözJ. 7(1957), 413-416. (Hungarian) D. Hilbert 1. Grundlagen tier Geometrie, 8th ed., Teubner, Stuttgart, 1956. J. Hjelmslev 1.

Contribution a Ia géométrie infinitesimale tie la courbe resiZe, Overs. Danske Vidensk. SeLsk. Forh. 1911, 433-494.

E. Hlawka 1. Ansfi4Liung und tiberdeckung konvexer Kbrper durch konvexe Kbrper, Monatsh. Math. 53 (1949), 81-131. E. I4opf 1. Losung tier Aufgabe 299 von H. Kneser und W. Silas (51 (1941), 3), Jber. Deutach. Math. Verein. 53 (1943), 40. A. Horn 1. Some generalizations of JieUy's theorem on convex sets, Bull. Amer. Math. Soc. 55 (1949), 923-929.

A. Horn and F. A. Valentine 1. Some properties of Lists in the plane, Duke Math. J. 16 (1949), 131-140. F. John 1. Estremum problems with inequalities as subsidiary conditions, Studies and essays presented to R. Courant, pp. 187-204, lnterscience, New York, 1948. H. W. E. Jung 1. Ober die kleinste Kugel, die sine rdumliche Figur einschliesst 5. Reine Angew. Math. 123 (1901), 241-257.

0. K. Jussila 1. On a theorem of Floyd, Abstracts Ins. Cong. Math. Stockholm 1962 133. S. Kakutani 1. Some characterizations of Euclidean space, Jap. J. Math. 16 (1939), 93-97.

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G. K. Kalisch and E. G. Straus 1. On the determination of points in a Banach space by their distances from the points of a given set, An. Acad. Brasil Ci. 29 (1958), 501-519. S. Karlin Mather,w.ticai methods and theorij in games, programming, and economics, Addison-Wesley, Reading, Mass. 1959.

S. Karlin and L. S. Shapley 1. Some applications of a theorem on convex functions, Ann. of Math. (2)52 (1950), 148-153.

J. L. Kelley 1.

Banach spaces with the extension property, Trans. Amer. Math. Soc. 72 (1952), 323-326.

L. M. Kelly 1. Covering problems, Math. Mag. 19 (1944-45), 123-130.

D. G. Kendall 1.

Siinplexes and vector lattices, J. London Math. Soc. 37 (1962), 366-371.

D. Kijne 1.

On coilinearities of two-dimensional convex bodies, Nieuw Arch. Wiskunde. (3) 5 (1957), 81-83.-

P. Kirchberger 1. Uber Tschefnjachefsche Annâherungsrnethoden, Math. Ann. 57 (1903), 509-540. M. D. Kirszbraun 1. Ober die zusammenziehenden und Lipschitzschen Transformationen, Fund. Math. 22 (1934), 77-108. V. Klee 1. On certain intersection properties of convex sets, Canad. J. Math. 3(1951), 272-275. 2. The critical set of a convex body, Amer. J. Math. 75 (1953), 178-188. 3. Common secants for plane convex sets, Proc. Amer. Math. Soc. 5(1954), 639-641.

4. The structure of semispace, Math Scand. 4 (1956), 54-64. 5. Circumspheres and inner products, Math. Scand S (1960), 363-370.' 6. Infinite-dimensional intersection theorems, these Proceedings, pp. 349-360. 7. The Euler characteristic in combinatorial geometry, Amer. Math. Monthly (to appear).

8. The generation of affine hulls, Acta Sci. Math. (Szeged) (to appear). B. Knaster, C. Kuratowski and S. Mazurkiewicz 1. Em Beweis des Fixpunktsatzes. für n-dimensionale Simplexe, Fund. Math. 14 (1929), 132-137.

H. Kneser 1. Konvexe Räume, Arth. Math. 3 (1952), 198-206. M. Kneser 1. Einige Besnerkungen iiber das Minkowskische Flächen,nass, Arch. Math. 6 (1955), 382-390.

D. Konig 1.

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M. A. Krasnosselsky 1. Stir tin critère pour qu'un domain soil etoilé, Mat. Sb. (N. S.) 19(61) 309—310. (Russian. French summary)

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R. Lagrange 1. Sur 1. tJtraldre it our Ia sphlre minimum continual un ensemble de points, Bull. Sd. Math. (2) 67 (1943), 108-115. F. Lannér 1. On convex bodies with at least one point in common, Kungl. Fysiogr. Sällsk. I Lund Förh. 13 (1943), 41-50. K. Leichtweiss 1. Zwei Eztremalprobleme dir Minkowaki-Geometrie, Math. C. G. Lekkerkerker and J. C. Boland

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Representation of a finite graph by a set of intervals on the real line, Fund. Math. 51 (1962), 45-64.

A. Lelek and W. Nitka 1. On convex metric spaces. 1, Fund. Math. 49 (196!), 183-204. J. Leray 1. Sur to forms des eapaces topologiqn.s it ear les points fixes des representations, J. Math. Pures App!. 24 (1945), 95-167. F. W. Levi 1. On Kelly's theorem and the axioms of convexity, J. Indian Math. Soc. (N. S.) Part A 15 (1951), 65-76. 2. Em. Erganzung sum HdUy'schen Salve, Arch. Math. 4 (1953), 3. Em geometrieches tfberdeckungsproblem, Arch. Math. 5 (1964), 476-478. 4. Oberdeckung eisiss Eibereiches durck Paralleiverachiebungen seine. offenen Kern., Arch. Math. 6 (1955), 369-370. J. C. Mairhuber 1. On Ifaar's theorem concerning Chebyshev approximation problem, having unique solutions, Proc. Amer. Math. Soc. 7 (1966), 609-615. A. Marchaud 1. Convexite it connexite Zineaire, C. R. Acad. Sci. Paris 249 (1959), 2141-2143. A. E. Mayer 1. Eine Uberkonvexitdt, Math. Z. 39 (1935), 511-531. E. J. McShane 1. Extension of range of function., Bull. Amer. Math. Soc. 40 (1934), 837-842.

Z. A. Melzak 1. A note on sets of constant width, Proc. Amer. Math. Soc. 11 (1960), 493-497. K. Menger 1. Untersuchungen über allgemein. Metrik, Math. Ann. 100 (1928), 75-163. E. Michael 1. Convex structures and continuous selection., Canad. J. Math. 11 (1959), 556-575. 2. Paraconvex sets, Math. Scand. 7 (1959), 372-376.

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On the simultaneous solution of a certain system of linear inequalities, Proc.

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Acta Math. Uber ems Ubertragung des Heliyachen Satzes in sphärische Acad. Sci. Hungar. 8 (1957), 315-318. 5. Ober Sternpolygone, Pub!. Math. Debrecen 5(1958), 241-245. A. F. Monna 1. Ensembles convexes dane lea espaces vectoriels eur un corps value, Nederl. Akad. Weiensch. Indag. Math. 20 (1958), 528-539. T. S. Motzkin 1. Linear inequalities, Mimeographed lecture notes, Univ. of California, Los Angeles, 4.

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R. L. Plunkett 1. Concerning two types of convexity for metrics, Arch. Math. 10(1959), 42-45. B. Thue Poulsen 1. ProbLem 10, Math. Scand. 2 (1954), 346. W. Prenowitz 1. A contemporary approach to cLassical geometry, H. B. Slaught Memorial Paper No. 9, Amer. Math. Monthly 88 (1961). I. V. Proskuryakov affine apace connected with Heily's theorem, Uspehi 1. A property of Mat. Nauk 14 (1959), no. 1 (85), 219-222. (Russian) V. Pták

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poiists donné et sur is problem. de l'app insatton minimum, Tray. Inst. Math. Acad. Sd. RSS Ukraine Kiev 1 (1938), 115-130. (Ukrainian. French summary) A. Rényi, C. Rényi and J. Surányi 1. Sn, des domains. simpiss dana L'espacs euclidien I n diewnstons, CoIloq. Math. 2 (1951), 130-135. C. V. Robinson

1. A characterization of the disc, Bull. Amer. Math. Soc. 47 (1941), 818-819. 2. Spherical theorems of HeUy type and congruence indices of spherical caps, Amer. J. Math. 64 (1942), 260-272. C. A. Rogers 1. A note on coverings, Matematika 4 (1957), 1-6. 2. The packing of equal spheres, Proc. London Math. Soc. (3) 8 (1958), 609-620. C. A. Rogers and G. C. Shephard 1. The difference body of a convex body, Arch. Math. 8 (1957), 220-233. 2. Convex bodies Gsaociated with a given convex body, J. London Math. Soc. 33 (1958), 270—281.

L. Sandgren 1. On convex cones, Math. Scand. 2 (1954), 19-28; Correction, ibid. 3 (1955), 170. L. A. Santaló 1. Un teorema söbre conjuntos do paralelepipedos do arteta. paralelas, Pubi. Inst. Mat. Univ. Nac. Litoral 2 (1940), 49-60. 2. Corn ple,n.nto a La ucLa: Un teore,na .öbre conjnnics do paralelepipedos do arictas paraleias, Pubi. Inst. Mat Univ. Nac. Litoral S (1942), 202-210. 3. Una propiedad caracteristica deL circulo, Math. Notae 3 (1943), 142-147. 4. Prcpiedades de Las figures convenes sôbre La esfera, Math. Notac 4(1944), 11-40. 5. Convex regions on the n.dimeissional spherical surface, Math. Ann. 47 (1946), 448-459.

6. Sôbre figures pIanos kiperconvexa., Summa Brasil. Math. 1 (1946), 221-239. 7. Sfibre pares de figures convexas, Gas. Mat. (Lisboa) 12(1951), no. 50, 7-10; rection, ibid. 14 (1953), no. 54, 6. 1. j. Schoenberg

1. On a theorem of Kirszbraun and Valentine, Amer. Math. Monthly 60 (1953), 620-622.

J. Schopp 1.

Ob., einen Kreisiiberdeckungssatz und seine Aquivalenz mit dem Gallaischen Krei.abstechens problem, 2nd Congr. Math. Hunger., Budapest, 1960, Vol.1, pp. 57-59.

E. 3. Schweppe

1. A projective generalization of convexity, Abstract 305, Bull. Amer. Math. Soc. 63 (1957), 137. M. Shimrat 1. Simple proof of a theorem of P. Kirchberger, Pacific 3. Math. 5 (1955), 361-362. L. A. Skornyakov 1.. Systems of curves on a plane, Dokl. Akad. Nauk SSSR 98(1954), 25-26. (Russian)

L. 0. Snirel'man 1. On uniform approximations, lzv. Akad. Nauk SSSR Ser. Mat. 2 (1938), 53-59. (Russian. French summary) P. S. Sottan 1. On illumination of the boundary of a convex body from within, Mat. Sb. (N.S.) 57 (99) (1962), 443-448. (Russian)

LUDWIG DANZER, BRANKO GRONBAUM, AND VICTOR KLEE

176

G. Soos

des Hellyschen Satzes auf den Fall vollstdndiger konvexer Fldchen, Pub!. Math. Debrecen 2 (1952), 244-247. N. Steenrod 1.

Ausdehnung

The topology of fibre bundles, Princeton Univ. Press, Princeton, N. J., 1951. P. Steinhagen 1. tJber die grosst,e Kugel in einer konvexen Punkemenge, Abh. Math. Sem. Univ. Hamburg 1 (1921), 15-26. E. Steinitz 1. Bedingt konvergente Reihen und konveze Systeme. I-H-Ill, J. Reine Angew. Math. 143 (1913), 128-175; 144 (1914), 1-40; 146 (1916), 1-52.

T.-J. Stieltjes 1. Sur La théorie des nombres, Ann. Fac. Sd. Univ. Toulouse 4 T. 6. D. Stoelinga

(1890). 1-103.

1. Convexe Puntverzamelingen, Dissertation, Univ. of Groningen, Groningen, 1932. M. H. Stone 1. Convexity, Mimeographed lecture notes (prepared by Harley Flanders), Univ. of

Chicago, Chicago, III., 1946. Postulates for the bar ycentric calculus, Ann. Mat. Pura App!. (4) 29 (1949), 25-30. S. Straszewicz 2.

1.

Beifrfige zur Theorie der konvexen Punktmengen, Thesis, Zurich, 191".

W. Süss 1.

(flier die kteinste Kugel, die jede Figur gegebenen Durchmessers einschliesst,

Abh. Math. Scm. Univ. Hamburg 20 (1955), 115-116. S. Swierczkowski 1. On chains of regular tetrahedra, Colloq. Math. 7 (1959), 9-10. H. P. F. Swinnerton.Dyer 1.

Extremat Lattices of convex bodies, Proc. Cambridge Philos. Soc. 49 (1953), 161-162.

B. Szbkefatvi.Nagy

1.

Satz über Paro2letverschiebungen konvexer KOrper, Acta Sd. Math. (Szeged) 15 (1954), 169-177.

Em

H. Tietze 1. (Iber das Problem der Nachbargebiete On Raura, Monatsh. Math. 16(1905), 211-216. *2. Getoste und ungeloste inathematigche Problem. one alter und nener Zeit, Bieder-

stein, Munich, 1949. F. A. Valentine 1. On the extension of a vector function so as to preserve a Lipschitz condition, Bull. Amer. Math. Soc. 49 (1943), 100-108. 2. Contractions in non-Euclidean spaces, Bull. Amer. Math. Soc. 50(1944), 710-713. 3. A Lipschitz condition preserving extension for a vector function, Amer. Math.

67

4.

5. 6. 7.

S.

(1945), 83-93.

properties determined by conditions on linear sections, Bull. Amer. Math. Soc. 10 (1946), 925-931. Three point arcwise convexity, Proc. Amer. Math. Soc. 6 (1955), 671-674. A three point convexity property, Pacific J. Math. 79 (1957), 1227-1235. The intersection of two convex surfaces and property Ps, Proc. Amer. Math. Soc. 9 (1958), 47-54. Some topics in the theory of convex sets, Hectographed lecture notes, Univ. of Set

California, Los Angeles, Calif., 1959.

HELLY'S THEOREM AND ITS RELATIVES

177

9. The dual cone and Helly type theorems, these Proceedings, pp. 473-493. 0. Veblen and J. W. Young *1. Projective. geometry, Vol. 2, Ginn, Boston, 1918. S. Verblunsky 1. On the circumradius of the bounded set, J. London Math. Soc. 27(1952), 505-507. U. Viet 1. Uber die ebenen Eibereichen umbeschriebenen Dreiecke, Math.—Phys. Semesterber. 4 (1954), 57-58. M. Vigodsky 1.

Sur lea courbes fer,nées a indicatrice des tangents donnée, Mat. Sb. (N. S.) 16

(58) (1945), 73-80. P. Vincensini 1. Figures convexes et variétts linéaires de Z'espace euclidien a n dimensions, Bull. Sci. Math. 59 (1935), 163-174. 2. Sur une extension d'un theorems de M. J. Radon sur lea ensembles do corps convexea, Bull. Soc. Math. France 67 (1939), 115-119. 3. Stir lea ensembles convenes sties involutions aigebr-iques de directions du plan, C. R. Acad. Sci. Paris 232 (1951), 2075-2076. 4. Stir certains ensembles déduits des ensembles d'arcs de cercie ou Sc calottee epheriques, Bull. Sci. Math. 77 (1953), 1-9. 5. Lea ensembles d'arcs d'un méme cercie dons leurs relations avec lee ensemble8 Sc corps convenes du plan euciidien, Atti IV Congr. Un. Mat. Ital. 2 (1953), 456-464. V. I. Volkov

On the least radius of a sphere containing a given set of n continuous functions, Zap. Kalininsk Red. Inst. 16 (1953), 3-6; RZ Mat. 1955, No. 1, 91. (Russian) A. Well 1. Sur lee théorèpnes de do Rham, Comment. Math. HeIv. 26 (1952), 119-145. J. H. C. Whitehead 1. Convex regions in the geometry of paths, Quart. J. Math. Oxford Ser. 3 (1932), 1.

33-42.

I. M. Yaglom and V. G. Boltyanskii 1. Convex figures, GITTL, Moscow, 1951 (Russian); English transi., Holt, Rinehart and Winston, New York, 1961; German transl., VEB Deutscher Verlag der Wissenschaften, Berlin, 1956. V. L. Zaguskin 1. On circumscribed and inscribed eUipsoids of extrensal volume, Uspehi Mat. Nauk 13 (1958), no. 6 (84), 89-93, (Russian) E. M. Zaustinsky 1. Spaces with non-symmetric distance function, Memoirs Amer. Math. Soc. No.34 91 pp.

_______

________

LUDWIG DANZER, BRANKO GRUNBAUM, AND VICTOR KLEE

178

INDEX OF NOTATION

The general notation is described on pages 105-106. Much of the special notation is used only on the page where it is defined. Important exceptions are listed below, with numbers indicating where the definitions appear.

aCt) Belly-number a°(5) Belly-order

124 124

Helly-numbers Hanner-numbers Gallai-numbers property of having nonempty

intersection

properties Jung constant

lie C-distance

134

135

140, 142 145 145

I jth Cdiameter o(C), J'(C)

128 130

%, %j transversal

Jo, J

128 128 128

II

B0 expansion constant Be Blaschke constant (Y/CJ covering number

137, 145 146 149

?(C)

134-135

INDEX OF TERMS

adjacent sets 147, 149-151 algebraic topology

109, 125-126, 129

varieties 124 approximation theory 110-115 axioms of convexity 109, 155-160 Blaschke constant Ba 140-142 Borsuk's conjecture 137 Carathéodory's theorem 103, 115-117, 158, 160

applications

110, 112—113, 115—116

cell 105-106, 162 Euclidean B" 112-113, 130-133, 135-139, 141-142, 144-150, 154-155 characterization 133, 138-139 homology 125-126, 131, 158 Jung 112, 132, 140, 145

center p.

133

of symmetry 123, 130, 134—137, 140-142, 147, 153 chord 113 circle, circular disc (see cell) circumradius 132, 154 circumscribed (see Jung) cell 112, 134, 137 ellipsoid 139-140 simplex 136, 140 colorable 153 combinatorial topology 109, 125-126, 129

complex convexity 157, 162-163 cone 107, 129, 158 congruence indices and order 139 congruent families of sets 123, 131—132, 138-139

conics 139 connected sets 117, 125, 138, 157-158

constant width

141

Constants Blaschke 140-142 contraction 142 expansion 135, 155 Jung 112, 133-136

contraction constant contractions 153-155

142

Con vex

body 105 combinations 103, 115-119 cone 107, 129, 158 connectedness 117 function 123 bull 103, 115-119, 159-160, 162 independence 119 space 156

structure 157, 161 convexity axioms of 109, 155-160 complex 157, 162-163 generalized 155-163

HELI.Y'S THEOREM AND ITS RELATWES

partitionable 122-123

Convexity (cont.) Born- 157-159 hyper- 160 LeVi's 109, 159—160

metric 161—162 order 161 projective 159 pseudo- 162-163 quasi- 160 Rado's 109, 160 Robinson- 157-159

spherical 157-159

strict 105, 123, 146-147 strong 157-158 three-point 160 topological group 162 weak

157-158

covering number I nd

145-149

by congruent sets 138-139 by Euclidean cells 112, 133, 135-138, 150

by balfspaces 111, 113, 140 by homothets 132-136 by translates 133-134, 136-137, 145-147 deficiency of a flat (=codimension) 119-121, 147-148

diameter

diam, Y 145-146 Y 132-133 disjoint families of sees 124-125, 130, 132, C-

152-153

rank of 151-152 separated 130 simple 130 translates 131—137, 144—150, 154-155 finite intersection property 106 flat 105, 118—121, 129, 147—148 GaIIai 128, 144449 •numbers rg -type problems 128, 152-153 original Gallai p'roblem 144 generalizations of convexity 155-163

of Hefty's theorem 119-128 genetics 127 graphs 118, 127, 152 Grasamannian manifold 129 halfapace 111, 113, 140-142, 156, 161-162 Hanner 128, 142 -numbers body 136, 142 Helly

Eduard 101-102

-number a(s) -numbers

-order a°($)

-type problems 127-128 Belly's theorem 101-104, 106-109 applications 110-115 converse 109-1 10 generalizations 119-128 topological generalization 125—126, 130,

distance Ux—yHc 134-135 duality 108, 140, 158 Dvoretzky's theorem 109-110 C-

ellipse, ellipsoid characterization 133, 154 circumscribed and inscribed 139-140 Euler characteristic 109, 124, 126 expansion constants E0, 135, 155 extension of contractions 153-155 property of normed linear spaces 143 families of sets congruent 131—132, 138-139 disjoint 124-125, 130, 132, 152-153 homothets 132-136, 146-149, 155 independent 151-152 intersectional 124-125 neighborly 150-152

124 128 124-125

158

hemispheres 137, 141, 157-159

hereditary property 127-128 homology cell 125-126, 131, 158 homothets 105-106

families of 132-136, 146-149 homothety 105-106, 146 hyperconvexity 160 incidence array 126-127 independent families 151-152 inscribed ellipsoid 139-140 inner-product space 133, 154-155 interior 105 5-

intjZ

of convex hull intersection 105

116 116

5-dimensional 120-121

pattern 126-127 property 104, 106, 150

_______ LUDWIG DANZER. BRANKO GRUNBAUM, AND VICTOR KLEE

180

radius

intersectional families 124-125 isometries (EUClidean motions) 138-139 Jung cell 112, 132, 134-137, 140, 145, 112, 133-136, constants

lattice

141, 154 132, 154

circum

Y 132-133 rank of a family of sets 151-152 semispaces 156

155

theorem

in

separated families

112-113, 131-133

130

separation by hyperplanes 106-107, 109,

160

Leichtweiss body

111, 130

135-136

Lipschitzian transformations (see contrac-

simple families

tions) Lowner ellipsoid 139-140 mass-distribution 113-115

simplex

measure of nonconvexity metric convexity 161-162

130

112-113, 123, 136-138, 140-141, 143, 151

characterization 135, 141, 143 regular 136

124

smooth 105, 149 sphere 105, 152 spherical caps 149-150

spaces 132-133, 139, 153-154 Minkowski

measure of surface area 155 measure of symmetry 113, 115

convexity 157-159 spaces 133, 139, 149-150, 154, 157-159 starshaped sets 111 Steinitz's theorem 115-116

spaces 133-137, 153-155 neighborly

families of sets 150-151

strict convexity 105, 123, 146-147

polyhedra 118, 151

supporting hyperplanes 109, 116, 136 symmetry

nerve of a complex 126-121 nonconvexity, measure of 124 order convexity 161

center of 123, 130, 134—137, 140-142, 147 measure of 113, 115

packings 147, 149-150 parallelograms, parallelotopes characterizations 137, 142-145, 150, 154 intersection properties 127, 129-131, 142-

topological groups, convexity in

translates, families of

162 131—137, 144—150,

154-155

(see also Gallai) polarity 108, 140-142, 158

transversals 110—111, 114, 121, 129—132 triangle, tetrahedron (see simplex) universal cover 136 varieties, algebraic 124 vertices 118—119, 151 visible points 111

polygons, polyhedra 106, 118-119, 129, 150-

width 154

144, 150-152, 154

partition of a family of sets

122-123

152

projective convexity 159 Radon's theorem 103, 107-109, 117-118, 159

constant C-

141 140

AN EXTREMAL PROBLEM FOR PLANE CONVEX CURVES' By

CHANDLER DAVIS 1. Peter Ungar conjectured an extremalperimeter property of the rectangle among convex curves (called the 'first problem' below). This and the corresponding extremal-area problem (the 'second problem') have been studied by

several mathematicians in the last few years, but as far as I know nothing has been published about them. The main purpose of this note is to give the solution of the second problem. In § 2 the problems are stated. The theorem of § 3 was told me years ago

as a result of Ungar; I give it here for the convenience of the reader, not to claim credit. Its effect is to reduce both problems to the consideration of quadrilaterals. The main theorem follows in § 4. 2. A 'cross' is the union of two perpendicular closed segments in the plane (the 'members') which have a point of intersection interior to both. The situation to be considered throughout the paper is this: a convex curve is 'around' a cross, i.e., passes through all four of its endpoints. The interior of the curve is then divided by the cross into four convex regions, and the curve itself into four convex arcs. The first problem is to prove Ungar's conjecture: Among all convex curves which are divided by a given cross into four equal arcs, none is longer than the rectangle through the endpoints with sides parallel to the members (and even this length is attained only by that rectangle—hence only in case the members bisect each other).

The second problem is to prove the corresponding statement for areas: Among all convex regions which are divided by a given cross into four equal

areas, none has greater area than the rectangle through the endpoints with sides parallel to the members (and even this area is attained only by. that rectangle—hence only in case the members bisect each other). 3. Hereafter assume without loss of generality that the cross lies on the coordinate axes, with endpoints (a1 .0), (0, a3), (—a3, 0), (0, —a4). The quadrants

are numbered as usual, and all numbers in such contexts are modulo 4; thus

it will be clear what is meant by such statements as that the ith quadrant lies between the ith and (i + 1)st arms of the cross. The symbol C will denote a convex region whose boundary OC is around the cross. By

will be meant the length of the portion of 8C in the ith quadrant,

if the first problem is in question; the area of the portion of C in the ith Research supported in part by the Federal Prison System. Opinions expreseed in this paper are the author's and are not necessarily those of the Bureau of Prisons. 181

CHANDLER DAVIS

182

quadrant, if the second problem is in question. In the statement of the problem only C for which mi(C) = ,n2(C) = m3(C) = m4(C) are admitted, but this restriction will not be imposed for the moment. Indeed it will be convenient sometimes not to exclude unbounded convex sets from the discussion. One may distinguish, for a given real value of those C 'consistent' with it, by which I mean those C having a supporting line at inclination through the ith endpoint of the cross. This is not to imply that fiC has a tangent

there; if it fails to, then C is consistent with an interval of values of

4,,.

Take normalization

(1) (except that a few expressions will have to be reduced modulo 2,r in a selfevident way).

Say C is 'filled for' a pair of values (4,,, in case that portion of C in the ith quadrant is the union of all convex subsets of the quadrant having supporting lines at the specified inclinations 4,., through the respective endpoints. LEMMA 1. —

Among those C consistent with a pair of values 4,, ,

4,, <2t, m4(C) is maximized exactly by those C which

such

that

are filled for (4,,,

For the second problem this is obvious. For the first, ii. is a non-elementary but well-known fact. Evidently for m4(C) is unbounded. By — admitting +00 as a value of the conclusion of the lemma can still Lx asserted, except that for — ç5, > it there may be C not filled for = 00• yet giving In order for a quadruple , #2, #4) satisfying (1) to be consistent with any C at all, it must satisfy other inequalities: indeed, if the line through the ith endpoint at inclination has any other point in common with the cross, it must be the preceding or succeeding endpoint. The set of quadruples satisfying all the inequalities forms a convex interval in 4-space. It may be identified (though not i—i on the boundary) with a certain set of regions C. Namely, to the quadruple , #2, associate that C

which is filled for each pair

(i =

1,

2, 3, 4).

It is

either a triangle or a convex quadrilateral, if bounded; it has at most one vertex in the interior of each quadrant. The bounded members of constitute a subset LEMMA 2.

,

is a decreasing function of 4,' and an in-

creasing function of when it is finite and when the variables have nonextreme values, this decrease and increase are strict and the function is continuously differentiable.

This is obvious. THEOREM 1. If ,n4(C°), then there

there exists some C such thai exists C'

such

m2(C°)

that m,(C') = •-- = ,n4(C')

m2(C°)

AN EXTREMAL PROBLEM FOR PLANE CONVEX CURVES

183

This reduces the problem to the case of Ce , PROOF. Consider, for varying , , equal, the finite-valued continuous function all

not restricted to having

= mm1 #4)). I Evidently its maximum is attained on the compact convex polyhedral set will show that, under the hypotheses, it must be attained for an are equal. such that all First, if a maximum point is in the interior of it corresponds to a F= F(9S1 ,

quadrilateral with a vertex in each quadrant. But in this case if not all were equal the smaller one(s) could be increased a little at the expense of the larger, by Lemma 2, which would increase F. There remains only the case that the maximum is attained on the boundary of therefore for an extreme value of some Say it is attained for minimal, therefore for ni1 equal to the smallest value attainable with the given cross. Here = 0, so by Lemma 2 suitable small increases in all the could properly increase all the m1 unless this was prevented by some other being maximal; therefore suppose this. There are three cases. First case: minimal, maximal; m1 = m2 = F, otherwise m3 = either increase of or decrease of would increase F. But no C around this cross has a higher value of m2, clearly—or a lower value of m2 or m3. So if m1 = m2 = = rn can be satisfied by any C° around this cross it can be satisfied by

#2, #8, #4), with #2 minimal,

maximal, and F maximal.

(In reaching the conclusion the fact is required that in this case clearly and oo by proper #2, #3, #4)) can be given any value between choice of

(or #4).)

case: F is maximized for #2 minimal, #4 maximal; m1 Pt, m4 = = m3 = F, otherwise by increase of #2 and decrease of #4 (adjusting #3 if needed) F could be increased. If any C around this cross has a higher value Second

of m2, clearly, it has a higher value of #3 and a lower value of m3, and m2> m3. That is, subject to m2 = m2 has been maximized. So if m8 = = m3 = can be satisfied by any C° around this cross, it is satisfied for the unique

, #2, #3 ,#4) maximizing F.

Third case: F is maximized for #2 minimal,

iiiaximal m2 = The reasoning applied to m2, m3 in the preceding case applies mutatis mutandis

in this case. In these extreme cases, the conclusion of the theorem has actually been proved. Return to the "general" case. Given the C° assumed, choose consistent with it. For each to

i, m1(C°)

Therefore m4(C°) = F(C°)

S max F(C);

since the max is attained for a C' with all m4 equal, this proves the conclusion. 4.

In this section the second problem stated in § 2 is solved. By § 3, it is

CHANDLER DAWS

184

enough to deal with Ce In order that the computations below should not be confusing, Let me arrange the notation judiciously. The lengths of the members of the cross can be taken each equal to 2, because, areas being in question, the scale in each coordinate direction can be adjusted separately. (Indeed the second problem is an affine one, so the proof to follow will establish the seemingly more general theorem in which the arms of the cross may be skew.) Call the endpoints of the cross, accordingly, (1 + 0,0), (0, 1 + r), (—1 + c,O), (0, —1 + r);

a and r are less than I in absolute value. C is a quadrilateral or triangle with at most one vertex in each quadrant and with the endpoints of the cross lying on its sides. Its area in each quadrant has the same value, call it A. C is otherwise arbitrary. The object is to show that A 1, and to determine the condition for equality. Denote the vertices of C by

(

/

2A

1

1

\

2A \

2,

1+t' 2A

1—r'

(

i+c)

2A

--22

2A \

1

1+22 1+r 1+22 1—a)' 2A 1 2A \ 2A / 24 i—a)' i—t — 1+24 1+a

—

The ith of these expressions represents, whatever the value of (0, + 00), a point which determines, with the adjacent arms of the cross, a quadrilateral of area A. The further conditions binding the A are the equations expressing that successive vertices of C are collinear with the appropriate endpoint of the cross. E.g., the first two are collinear with (0, 1 + r), so

[

2A

—p1+

Li+2, 1+a

'

_[

1

1.

1

2A

1+2, 1+r

2A

(1+r'1• 'J

2!

2A

1+2k 1+r I spare the printer the trouble of displaying the other three formulas in raw form. After obvious reductions one gets

11+A! 1—a

+

1+r

i

=

(A,

+

+ 2),

+

r +

=

(22+24+2)

=

(24 +

+ 2).

Now we are free to suppose A 1. Combined with this, the equations (2) imply four inequalities which may be written as

i=l,2,3,4; here the definitions are

AN EXTREMAL PROBLEM FOR PLANE CONVEX CURVES P1

(1+cXl+r)

1,

(1—cXl+r)

—1, (1—uXl—r) Adding the inequalities (3) gives P3

(1+QX1—v) +

185

_'

8,

4. Now it is easy to finish the argument; the following be the quadratic Maclaurin approximay be the most convenient way. Let as a function of a and r. Equivalently, mation for + 1XPs + 1)' = (1 — a2Xl — v2) One computes at once that P. Q1 for a and re(—1, 1), with equality exactly when (a, t-) (0, 0). But one also computes = 4. Combining these con-

which implies

clusions,

4=

Q.

4,

giving a contradiction to the supposition A 1 unless a and r are both 0. When a = 0, r = 0, that is, when the arms bisect each other, equality can must be 1 for all hold exactly if it holds in each of (3). = 1 for all 1. 1; we have a rectangle, as alleged. This finishes the proof, except for filling one gap: it has not been excluded that for an extreme value = 0 or co (for some i), A might be 1. Say = 0. The first three of equations (2) still hold as before. From (23), supposing A

1, and using

P

4.

0,

follows 23 2/P3. Substituting in (23) yields similarly

Now I will derive a contradiction from the assumption that in the first quadrant 0C simply joins the two endpoints of the cross. For this triangle to have such a large area as 1, we must have (1 + aX1 + r) 2, (1 — aX! — r) 1/2, P3 3, P > 4, a contradiction. For the first problem to be solved in an analogous way would presumably require much messier manipulations. MATHEMATICAL REVIEWS

In discussion, several analogues of Ungar's problem were proposed. H. S. M.

Coxeter suggested, for example, the extremal.perimeter problem for curves divided equally by three fixed points instead of four. A. Dvoretzky asked what the extremal quadrilaterals are like for a fixed cross whose arms do not bisect each other. Davis noted that they are determined by the property that rtrsrsT7 = r2T4r6r8, where the Ts are, in order, the segments into which OC is divided by the endpoints of the cross and the vertices.

NOTIONS GENERALIZING CONVEXITY FOR FUNCI'IONS DEFINED ON SPACES OF MATRICES BY

CHANDLER DAVIS 1. IntroductIon. Most of what I say will concern real functions f of real variables, subjected to certain conditions implying the usual "functional inequality" of convexity. That is, the function classes discussed (of which there are several) are most of them properly contained in classes of functions convex in the usual sense. The narrowing is the result not of adjoining captiously some supplementary conditions, but of the following quite natural consider-

ations.

A real function I of a real variable may be considered in the standard way (see § 2) as giving a function (for each n = 1,2, ...) whose domain and range are n x a real symmetric matrices. Indeed, it can be seen that a function f having such domain and range (for each n) is obtained from some real function in the way described if and only if it satisfies

f(V*AV)= Vf(A)V for every suitable A and every partial isometry V whose range reduces A. (The behavior of I for different a is related by (1.1), because V need not be a square matrix.) Now consider the convexity condition

+ 2B) (1— 2)1(A) + 2f(B) where domain and range of I are a x a real symmetric and where the order f((1 — A)A

is that in which positive (semi-) definite matrices are Among the functions from this n(n + 1)12-dimensional partially ordered vector space to itself, (1.2) determines the subclass of those convex in the usual sense. However,

for f

to

satisfy (1.2) and in addition to be obtained from a function of a

single real variable—this requires I to belong to a quite narrow class of onevariable functions, the 'matrixconvex' functions. Charles Loewner first recognized the possibility of describing this class; actually he began with the related class of 'matrix-monotone' functions determined by requiring His remarkably original and successful treatment of this problem [14; 131 has been followed by a number of papers by others (most important for present purposes are L12; 2; 23]) giving alternate proofs, added information, and applications.

Here I will begin by giving an exposition of the main facts of the Loewner 187

CHANDLER DAVIS

188

theory 4, following preliminaries in 2-3). This exposition's brevity should not be taken to imply that the theory can be developed much more economically now than it was at its discovery. In the first place, I skip over a great deal;

in the second place, some of the new methods turn out if examined closely not to be as different from Loewner's as they look; finally, I make stronger smoothness hypotheses here than are required. The next part of this paper discusses some other conditions resembling, and related to, matrix-convexity: in § 5, a sketch of a still unpublished theory of Korfinyit [11J; in § 6, a rather full account, in part new, of a condition proposed by Sherman; in § 7, the statement of a proposed subject for future research. The unifying idea is the question: what conditions for functions of several real variables are natural analogues of matrix-monotonicity and matrixconvexity?

The last section, §8, though labeled "appendix," may be the most useful

part. It is a collection of simple special cases with independent proofs. For

example, the reader may turn there at once if he wants to be quickly persuaded that the class of matrix-convex functions does not include all convex functions (see Example 5) but does have nonlinear members (see Example 6). 2. Some conventions. As already indicated, linear transformations of real n-space will be the basic objects of study. The n-space will be denoted by H,, or H, its elements by lower-case italic letters x, y, and its inner product by ( , ). The space of all linear operators on H will be denoted by or its elements by italic capitals. Important subsets are or the hermitian (symmetric) members, and [0, 11, the order subinterval of consisting of those A such that 0 A 1. Occasionally the replacement of this H by more general hubert spaces will be discussed. The space Hk consists of the k-tensors over H; for example, x1 ® — y2 e H2. The space of skew k-tensors will be denoted by thus xØy — y ® x = i/Tx A y H(23. ffL*J is void if k> n. The space of k-tensors over

is identical with the space of all linear operators on Hk. Let mean the space of all linear operators on Htk1. The notations are now self-explanatory. Among the elements of .-ifk, which will be

considered are power and adjugate matrices (A)" and [15, VII]. For A (A)" = A e ®A is the restriction of this operator

to its invariant subspace other main subject of study will be real-valued functions f, g, of (one or more) real variables s, t, - --. If is defined and measurable on an interval of R, say [0, 1J, then if A e is in [0, 11, The

/

A =

f(A) will mean 51(A) dE(A). In the form stated, this definition makes sense for H an arbitrary hilbert space. If I uniformly by a polynomial: 1

Added

is

continuous, it can be approximated

in proof. His paper has since been published, see [11].

NOTIONS GENERALIZING CONVEXITY

189

converge in the uniform sense to f(A). If H is again the operators and f(A) = n-space, the spectral representation of A is simply A =

The situation is unchanged 1ff is a real function of k real variables defined are commuting members of Rk, and A1, •••, on some interval the Cartesian product of whose spectra is contained in I": one can write in when

the matrix case, for instance, A = f(A1,

A novel convention will now be introduced which is essential both to the

method used in this paper for treating the one-variable case, and to the statement of the extended problems considered later 7). When 1 is a function of several variables, defined on an interval, and A has suitably restricted spectrum, f(A) will be defined. It will be clear enough to show only the case that I is given as a function on [a, b]5. Then for whichmakes f(A)willbe A

it sense according to the previous definition. Again letting A = The functions to which this A,,)P. 0 P, 0 follows that 1(A) = definition is applied in most of this paper have the further property of being symmetric in the variables. As a consequence, fiL$] is invariant under 1(A), which may be regarded as a member of Ac)Qs,c, the 1(A) = last symbol denoting the projection on the subspace of z A A xc, where = is The relation corresponding to (1.1) for these extended functions to

f(

V(CJ

V) =

for any partial isometry V whose range reduces A.

In the one-variable case, the central notions of matrix-monotonicity and matrix-convexity have already been explained. It should be observed that either condition (1.2) or (1.3) should be imposed only for matrix arguments with spectrum in some interval, the domain of definition of 1. More important, one might impose (1.2) or (1.3) only for n x n matrices, for some fixed n; it is then implied for all smaller n. Matrix-monotonicity for all orders n implies that (1.3) holds also when H is separable real hilbert space, as is easily seen [2], so one may speak of 'operator-monotonicity.' Similarly for convexity. In the next section and its applications, divided differences enter extensively;

accordingly I conclude these preliminaries by setting forth notations and simple facts about them.

Let I be again a real function on [a, bJ.

means 1.

[a, b]2, except the diagonal, by JLIJ(31)

f(s)

=

f(t)

is defined on

CHANDLER DAVIS

it is symmetric.

is defined by repeating the process with respect to one

variable: I, u) =

t)

:

fLzJ(s, u)

is defined recursively in the same

it is symmetric in all three variables. fashion. Alternatively, f(t0)

•.

/

.

I

k—i k

k—i tO

tki

... .

, . . . ,

J

1

•..

1/

. .

1

...

1

may be considered which makes the symmetry evident. The domain of to be extended by continuity to allow equal values of some or all arguments, at just those values where I is sufficiently often continuously differentiable. I, ... t)= One obtains then, for example, 3. The Taylor formula. When F(t) is a sufficiently often strongly differentiable function whose values are commuting elements of a Banach algebra, there is little difficulty in finding the proper generalization of the usual

Taylor formula to handle it. The situation is much lees clear when the

values fail to commute, and noncommutativity is essential to the occurrence of the phenomena this paper studies. (For instance, matrix-monotonicity of f involves the condition f(A + H) — f(A) 0 for symmetric A and positive definite H; if H was restricted to commute with A, this would force upon I nothing but ordinary monotonicity.) The Taylor formula required here is due to Daleckil [3]. For a function of one variable, k times continuously differentiable, it may be written in the form (different from his) (H)k (3.1) — H + •. + f(A + rH) f(A) + The notation has been explained in § 2 except for the bilinear operation It will be defined taking pairs (T, U), Te to T— Ue Ue for those elements of the tensor spaces which are simply tensor products of elements of ...4: Then the requirement of bilinearity extends the definition to all

x

It must further be specified in what sense the equality is asserted in (3.1). It is only an asymptotic expression for small r: each side is a k times differentiable function of r with values in and it is asserted that all these derivatives are equal at r = 0. It should be emphasized that the reason for the occurrence of divided differences in (3.1) is entirely different from situations in analysis where one wants to avoid computing a derivative, or asserting its existence. The derivatives' existence is assumed here, they may occur in the terms on the right, and it

NOTIONS GENERALIZING CONVEXITY

may even happen (in case H commutes with A) that all the divided differences

on the right become derivatives. The formula is nevertheless impossible to write in terms solely of derivatives. This is because, when H fails to commute with A, the effect of perturbation of A by rH involves different eigenspaces of A in an irreducible way. 4.

Conditions for matrix-monotonjelty and matrix-convexity.

Daleckil [3)

credits M. G. Kreln with the observation that his Taylor formula may be used to derive easily the quadratic form condition for matrixmonotonicity or matrix-convexity. This is so, provided one assumes sufficient differentiability. THEOREK 1 (LOEwNER). For a continuously differentiable function I defined it is necessary and sufficient that, for any t1, ••., t1, E (a, b), the matrix

on (a, b) to be matrix-monotone on (a, b) c ,

,

t1)

t1)

-

, I,,)

•..

be Positive definite.

(From this several other criteria follow more or less readily; see [21 and references there.) Indeed, assuming the differentiability, matrix-monotonicity is equivalent to the positive definiteness of df(A + rH)/dr for all A e (a, b) and H 0. By the Taylor formula, this derivative is — H. This expression being linear in H, it is enough to consider only H of the form Hij = xix,,

x€ H. The condition is then that for arbitrary ye H, H)y, y) =

0

Nothing

-

has restricted the choice of coordinate system yet, so choose to

diagonalize A, whose eigenvalues are the numbers (arbitrary in (a, b)) The condition becomes 0 where

t')yixixjyj =

.

.., t,,.

t.)z1zj

the z = xjy, are arbitrary reals. This is just the condition of positive

definiteness. U Tunoani, 2.

For a twice continuously differentiable function f defined on

(a, b) to be matrix-convex on (a, b) ç it is necessary and sufficient that, for s, t e (a, b), fW should be, as a function of t, matrix-monotone.

Bendat and Sherman [2] give this with a different proof. I give this version

because it seems the most natural form in which to reap the fruit of the Daleckjl-Kreln method.

Again it is clear, given the differentiability assumptions, that matrixconvexity is equivalent to the positive definiteness of dtf(A + rH)/dr2 I,-..., but here H is arbitrary in again the expression for the derivative is

CHANDLER DAVIS

192

taken from (2.1): 1 ttkA) (H)'. Making the same convenient choIce of coordinate system as before, the condition becomes (4.2)

, t,,

OS 43$

To establish the theorem it is, in virtue of Theorem 1, enough to show (4.2)

equivalent to

0 5 Zf11kt3, t,,

(4.3)

Now (4.2) bounds a sum over j of terms each of which is a sum of the form which enters in (4.3); this gives the implication (4.3) (4.2). For the other implication, consider the following special choice of y H and HE = while for 1> 1, = z,; H,1 u, while for i,j> 1, H1, = = 1, = 0. The terms .1 = 1 in (4.2) become then exactly (4.3); but these must indeed be 0 for (4.2) to hold identically, because all the terms j> 1 go to zero as I The condition for matrix-convexity in terms of quadratic forms Is already found, viz. (4.3), and other conditions follow from the corresponding ones for matrix-monotonicity by application of Theorem 2. THEOREM 3

monotone on

For a function I as in Theorem 1 to be matrixfor all n, it is necessa,y and sufficient that I be analytic on

(a, b) and be analytically extendable to the upper half plane, having non-negative imaginary part there. THEOREM 4 (BENDAT AND SHaaMAN).

Another necessary and sufficient condition

for f defined on (—1,1) to be ,natrix-monolone for all a is that / be representable by a Stieltjes integral (4.4)

1(1) = /(0) +

dm2).

1

p being (as a measure) finite and non-negative.

The equivalence of the criteria of these two theorems is fairly easy to

prove [9]. Here, I will content myself with sketching a proof, due to Korányi

f1O,23], of Theorem 4 from Theorem 1. The idea is to apply the positivity of the quadratic forms (4.1) to construct a functional hUbert space, in the manner familiar from the work of Aronszajn [1] and others. Namely, in the set of functions representable by finite sums = (ti (—1, 1), Ed complex), introduce the inner product E;ft13(s, ii, ( , ), where if it(s) u,),

= One

=

readily verifies that a prehilbert space results; complete it.

Next one defines an operator A on of the above form by Ar(s) = L Edf"ks, 0); note that the right side is again of the same form. The

NOTIONS GENERALIZING CONVEXITY

domain of A must not include ço0 (coo(s) fhh](s, 0)) at first, since 1" is not assumed to exist. However, A is proved uniquely extendable to a seif-adjoint on its operator. Korányi shows somewhat indirectly that A has norm original domain, hence everywhere. By direct computation, f(t) = f(0) + (1(1 — IA)1ço,, coo), coo being as above. ((1 — IAY1 need not be bounded for t = ±1, but ço, is in its domain.) From 4 dEA (the integral is over [—1, 1] because the spectral resolution A = one obtains

f(t)=f(0)+ç

d(EAçO0,ç'o),

converse is straightforward: substituting (4.4) into The gives an expression which can be written as the integral with respect to p of a formally non-negative function. i which is of the required form (4.4).

-

extension. As already mdireport is to suggest conditions extending those of matrix-monotonicity and matrix-convexity to functions of several real variables. 5. Functions of several variables: cated, one of the main objects of this

If I is an arbitrary function of two real variables, and A, B are real symmetric matrices, f(A, B) is not readily defined unless A B, in which case one proceeds as before: any XE H which is an eigenvector of A, B (belonging to eigenvalues A, p, respectively) is an eigenvector of f(A, B) belonging to eigenvalue f(2, p). Now if one is to compare the matrix f(A, B) with f(A + H, B + K), where each of H and K ranges over a reasonably general set, one is obliged, in order to preserve commutativity of the arguments of 1, to restrict them to lie in commuting subalgebras of But then there is no loss of generality in replacing H by Hz, and using as the domain of the extension of f, pairs A ® 1, 1 ® B; indeed, I will use (in the present section only) the notation f(A, B) for what would otherwise be written

f(AØ1,1ØB).

The Taylor formula now takes the following form: (5.1)

f(A + rH, B + rK) = f(A, B) + rf[10)(A, B) — H + rft0 + rtfu"(A,B)—H®K+

B)

K

Even with the proviso already made that f(A, B) this notation is not self-explanatory, indeed I will perhaps not be forgiven such an abuse of 'anguage. The divided differences occurring are s1 ,

t) = f(s0, t) —

,

t)

So —

s1 ,

t0, t1) =

ft*.0(s

s1

—

fri

,

,

to — ti

—

f(so, t0) —f(so, t1) —f(s1 ,t0)

—

(s0 —

— t1)

t1)

, e

c.

CHANDLER DAVIS

194

B) e

=

(I (1.9](A, B)

411,Hhi

etc., so that the interpretation of the symbol now requires reference to the superscript on the I preceding. The notation gets so little use in this paper that it is not worth amending it to correct this defect. Now Korányi found that the 2-variable problem to which the 1-variable methods extend without essential modification is the following: Comparef(A, B)

withf(A +H,B+K), where

In order toget a condition on! which genuinely involves both variables, one may look at the first term in (5.1) in which both H and K appear, and this suggests that the proper condition is (5.2) f(A + Jf,B+ K)—f(A + In fact, by application of the Taylor formula just as in the proof of Theorem 1 above, one finds a quadratic form requirement on to be equivalent to (5.2).

Korányi till obtains this result instead by an extension of Loewner's method. He then proceeds to prove analogues of Theorems 3 and 4 above. 6. The Sherman condition. As preparation for the discussion of the next section, I give some results on matrix-convex functions of one variable. The notation 1€ will mean that the extension of I to is convex. This property will be related to what I will call the Sherman property. (The main result of this section, that K. = S.., was obtained some years ago [4] following a suggestion by S. Sherman.) The notation 1€ S, will mean / has the Sherman property when extended to that for any projection F,

Pf(PAP)P Pf(A)P. If we agree to consider f(PAP) only on the range of F, or if we assume f(O) =0, this may be simplified to

f(PAP)

Pf(A)P;

then generalized to

f(

(6.2)

V)

Vs 1(A) V,

for any partially isometric V. In this form it is interesting to compare with the condition (1.1) satisfied (for special V) by arbitrary matrix extensions of scalar functions. THEOREM 5.

Se,, c

S.

This will be proved here in terms of an alternative form of the Sherman The condition that

condition [6; 17). (6.3)

whenever the F, are orthogonal projections adding to i, is clearly equivalent

NOTIONS GENERALIZING CONVEXITY

to the Sherman condition; as is the special case m =2, where I will write the by the "partly projections as P, P. The operation of replacing all Be will be called a pinching. diagonal" 'it'B = and projection P, it is evident how to Given 1€ K,,, and given A

express PAP + PAP, the pinched matrix, as a convex combination involving A:

PAP+ PAP= 4(P+ P)A(P+P)+f (P—P)A(P_ P) = But X = P — P =

4A I-4XAX.

X' is unitary, sof(XAX) = Xf(A)X by (1.1).

Hence

41A) + 4 XJ(A)X

f(PAP + PAP) =i(f A + 4 XAX)

= Pf(A)P + Pf(A)P. This proves fe Sn. For the other half, given 1€ S,,, and given A, Be A/2 + B/2. Equivalently one may consider, in

is to consider the operation

one

B) = (A/2 + B/2) (A/2 + B/2). The condiany A $ B by tion to be proved will thereby again be expressed in the form One advantage of this formal maneuver is that the following proof may then

replacing

be summed up as saying that this sort of averaging, though distinguished from pinchings for purposes of the structure theorem [6], is obtainable by pinching. Indeed, let X be the canonical isomorphism between the coordinate subspaces in (1

B)X. X being a symmetry, P = so that B A = X(A + X)/2 is a projection. Now for any Ce

cfc=4c +j-XCX=PCP+15CP by just the same computation as above. Hence

f(4A + B) 4 (A + B)) = f(P(A Pf(A =

B)P + P(A

B)P + Pf(A +

+ j-f(B)) = (4 1(A)

B)P

(4 1(A) + 41(B))..

There is no essential added difficulty in writing an arbitrary convex combination of A and B in the last half of this proof, but no essential added information either. There may be some interest to the following further extension, however. Define the class R,, by setting Ic R in case (6.4)

RiA;R?)

R,f(A1)R'

CHANDLER DAVIS

196

e whenever the matrices can be constants) and are the following relations.

and ZRjR' = 1. Evidently c KN (the (the R1 can be projections). More surprising

THEOREM 6. fE R,, if and only if the Sherman condition (6.1) holds whenever the range of P is n-dimensional. The proof differs from that of Theorem S only in the slightly greater bother of allowing for "matrix coefficients" in the convex combinations.

of P; B acts on a

Let f€ RN, and consider f(PBP) on the range H =

where B1 B1E1, E1 an n-dimensional space K H. Decompose B = projection (without loss of generality suppose dim K a multiple of n). Let be a partial isometry such that V1 V," = P, V," = E1. Let R1 = PV1 and

let A, = V7B, V1.

In this way PBP can be written entirely in terms of

operators on H: PAP =

(the identity in .-//N), on EK. Therefore

L R1A1R'. and

Furthermore,

=

P

A acting on H is a unitary image of B acting

f(PBP)

R1f(A1)R,'

=

= Pf(B)P = P(E In the other direction the proof is similar. Given the R,, one obtains the E1 in the larger space K, such that RIR' = PEIP (P the projection on the original space Ha), by Naimark's theorem 1221. Given the A,, the B1 are not determined by the condition that A, = for some partially isometric mappings of the appropriate spaces: the particular partial isometry to use is determined by the "phase" in the polar decomposition of Now = f(PBP) Pf(B)P = R,f(AJR,. THEOREM 7. fe R,. if and only if the defining condition (6.4) holds for m =2. This is trivial; I indicate the passage from m = + R21?,* + I, let S = (1 — Then

RLRI*

2

to m = 3. Given

+ exists as a projection Q (i.e., QH is the range of S). Let T1 = 1,2).

(1 =

Then = SBS +

B = TIAITI* + T2A2T2",

and two applications of the condition for m

2 (one on H and one on QH)

give the desired conclusion. COROLLARY.

For 1€ COROLLARY.

S2,,

RN.

must satisfy (6.4) for m = K..

S... = R...

2,

by the construction of Theorem 6.p

NOTIONS GENERALIZING CONVEXITY

(Here K.. denotes fl, K,, etc.) This is immediate.

I do not known which of the inclusions proved are strict for finite n; Namely, in = R1 = except for n = 1, where it is easy to see that St 1-space any pinching is trivial, so (6.3) does not restrict 1; the definitions of K1 and R1 are the same; and finally (the only nontrivial projections in 14 being 1-dimensional) (6.1) takes the form f((Ax, x)) (f(A)x, x) for xe H, which by the spectral decomposition of A is seen to hold for any convex f. 7. FunctIon, of several variables: a second type of extension. As in § 5, let 1 be a real function of several variables—for instance, of 3. Now, howthe ever, study the associated function of one matrix, not three. For A e

covention 1(A) = f(A 0101,10 A 01,1010 A), introduced in §2, extends f

to a function from ' to Since f will be assumed symmetric, 1(A) may be regarded instead as an operator in (hi) or is The condition that H 0 imply f(A + H) f(A) (whether in quickly disposed of as giving nothing new: it is equivalent to matrix-monotonicity in the usual sense of I as a function of one of its variables. The condition of convexity (1.2) makes sense, but is satisfied by almost no functions; for example, f(s, t) = s + t or st fails to be either operator-convex or operator-concave. However, the Sherman condition extends in a natural way to this situation. mean that the analogue of (6.3) holds in Let f€ (7.1)

PAP)

Q,f(A)Q,;

here e ranges over m-tuples of integers 0, e = (i1, = k); and Q1 means the projection onto the subspace of spanned by elements of the form x1 A x2 A ... A Xk with x, E P1H for 1 P j1, x, PmH for

CHANDLER DAVIS

198

replacing pure states by mixed ones. are familiar in matrix theory already. Next I point out that the classes Remember that for the construction of non-zero skew htensors over H,,, the is 1, smallest possible value of n is n = k. Then the dimeusionality of and may be identified with the real numbers. All pinchings on being therefore trivial, (7.1) becomes simply f(Z PAPj) f(A). (Here I has become a numerical-valued function of the k eigenvalues of its matrix argument; the invariance condition (2.1) has become f(U*AU) = 1(A) for unitary U, no other V in (2.1) being possible.) Therefore is exactly the class of Schur-convex functions [20; 19}. There is as yet no general description of the class

this seems a difficult and interesting open question. The one further remark that will be made here is that there are a lot of functions in the class: indeed, it is a convex cone, it includes the product function f(t1, - -' = - - - h and its negative, and g(11) + -- + g(tk). ..., it includes the symmetrized form of any g S.., The first two of these statements are direct consequences of the definition; the third, though almost as straightforward, will now be proved, for k = 2. 1(A) = g(A 01) + g(1 0 A). Now by the Sherman condition for g, = g((PAP + f(PAP + 01) + g(1 0 (PAP +

+(1®P)g(1®AX1®P)+(1ØP)g(1®AX1®P). This is to be compared with L Q,f(A)Q, as an operator on actually they are equal on H'. For instance, consider the first term in the above expression.

(P® l)g(A 0 1XPO 1) = (P® P)g(A 0 1XP® F) + (PØP)g(A 0 1)(P® P)

+(P®P)g(A®1XPØP)+(PØP)g(A ®1XPOP); on the right, the first term occurs in Q,.,f(A)Q,,,, the next two vanish, and the last occurs in Q1.1f(A)Q,.1. And so forth. Appendix: special cues. In this section 1 collect some special cases of

8.

the theory which are so much simpler to prove directly that invoking the general theorems seems clumsy. Some of these proofs are new; some are more or less known elementary folk theorems which nevertheless are often overlooked. Examples 1 and 6 are standard theorems. EXAMPLE 1.

The inverse is matrix-monotone on (0, oo); that is, 0

and A invertible implies B'

A'.

A

B

Paoop. For any xe H, II A'1xfl' = (Ax, x) (Bx, x) = Clearly B" is invertible; let T = A"B1" and y = B'1tx. Then the above inequality says I I y I I', for all ye H, so II T 1. This implies II T* II Ty 1, and since = this means that lix II'. Setting this time z= one obtains z) = = (A'z, z). • I

The square function is not matrix-monotone. The following is a direct proof that it is not matrix-monotone on any interEXAMPLE 2.

val.

Let

NOTIONS GENERALIZING CONVEXITY

r\

0\ 0

r

non-zero.

P1'

Then

e±v) and

where E = nor

= 2(a + r)r; this has negative determinant, so is neither

I

EXAMPLE 3.

The square-root function is matrix-monotone on (0, co); that is,

if Psoor. (I extracted the following from Kato's proof [8] of a more general theorem. It is much simpler than other known proofs.) Let A be any eigenvalue of the hermitian B — A, and x the corresponding eigenvector; it will be proved that 0. Now Bx — Ax = Ax; taking inner product with Bx, IIBxII'—A(Bx,x)=(Bx,Ax). Use the Schwarz inequality: IIIBxII'—A(Bx,x)I IIBxIIIIAxII = (B'x, x)"(A'x, (B'x,x)= UBxII', since A' B'. Hence A(Bx, x) 0, hence A 0. I Ex.&in'i.s 4. The function

I defined by f(t) = I v 0 is .not matrix-monotone on (—2,2) (evidently, then, not on any interval including 0). Paoop.

Let

A—'° 1\

0

0)'

0

Then 1(A) is the projection

Iz

1

2

22 Any matrix with only one eigenvalue >0, if it is to be must have PH as the corresponding eigenspace. But f(A + H) has only one eigenvalue >0 and fails to satisfy this condition. EZAurIz 5. The absolute value function v is not matrix-convex on (—2,2) (evidently, then, not on any interval including 0). Paoor. Let

A=('

r>0.

Then

v (f A + ÷ B) = 4

(r

4- v(A) +

÷ v(B) =

but for small i the latter expression has a smaller entry in the lower right

corner than does the former. i By positive homogeneity, this is the same as saying the "triangle inequality" V(A) + v(B) v(A + B) does not hold; indeed, the same example refutes any "weakened triangle inequality" of the

200

CHANDLER DAVIS

form v(A) + v(B) pv(A + B) 121]. EXAMPLE 6. The square function is operator-convex. This is a familiar computation: ((1 — A)A + (1 — 2)A' + AB2 is equiva-

lent to 0

(1 — A)A(A — B)2. EXAMPLE 7. The square function

P is a projection, II P11

satisfies the Sherman condition. 1, so ((PAP)2x, x) =

(PA'Px, x).

The inverse function satisfies the Sherman condition on (0, 00);

EXAMPLE 8.

that is, for invertible A

0 and projection F, (PAP)-' PA1P, the left side

being defined on PH. PROOF. Clearly PAP is invertible on PH. For any xe PH, define H, z = (PAP)'x PH. Now the object is to prove that, of the two positive numbers ((PAPY'x, x) = (z, x) anr (PA'Px; x) = (y, x), the second is the first.

But 0

so

(A(y

—

z), y — z)

= (Ay, y) — (Ay, z) — (z, Ay) ÷ (Az, Pz) = (x, y) — (x, z) — (z, x) + (x, z).

1. N. Aronszajn, La théorie des noyaux riproduiaanta et sea applications. 1, Proc. Cambridge Philos. Soc. 39 (1943), 133-153. 2. J. Bendat and S. Sherman, Monotone and convex operator functions, Trans. Amer. Math. Soc. 79 (1955), 58-71. 3. Yu. L. Daleckil, Integrirovanie i differencirovanie ftnkcit èrmitovyh operatorov, of parametra, Uspehi Mat. Nauk 12 (1957), no. 1(73), 182-186. Translation; Amer. Math. Soc. Transl. (2) 16 (1960), 396-400. 4. C. Davis, A Schwarz inequality for convex operator functions, Proc. Amer. Math. Soc. 8 (1957), 42-44. 5. , AU convex invariant functions of hermitian matrices, Arch. Math. 8 (1957), 276—278.

6.

•

Various averaging operations onto subaigebras, Uhinois J. Math. 3(1959),

538-553.

7. , Operator-valued entropy of a quantum mechanical measurement, Proc. Japan Acad. 37 (1961), 533-538. 8. T. Kato, Notes on seine inequalities for Linear operators, Math. Ann. 125 (1952), 208-212.

9. A. Korányi, Note on the theory of monotone operator functions, Acta Sci. Math. Szeged 16 (1955), 241-245. 10. , On a theorem of and its connections with read vents of esjfad-

joint transforinattons, Acta Sd. Math. Szeged 17 (1956), 63-70. 11. , On some classes of anal ytic funot ions of several carialijes, Trans. Amer. Math. Soc. 101 (1961), 521-554. 12. F. Kraus, ()ber konveze Matrixfunktionen, Math. Z. 41 (1936), 18-42. 13. C. Loewner, Advanced matrix theory, Lecture Notes, Stanford Univ., 1957. 14. K. Löwner, tfber monotone Matrixfuhktionen, Math. Z. 38 (1934), 177-216. 15. C. C. MacDuffee, The theory of matrices, Springer, Berlin, 1933.

16. M. Marcus, Convex functions of quadratic form., Duke Math. 3. 24 (1957),

321-326.

NOTIONS GENERALIZING CONVEXITY

201

17. M. Nakamura, M. Takesaki and H. Umegaki, A remark on the expectations of operator algebra., Ködai Math. Scm. Rep. 12 (1960), 82-90. 18. M. Nakamura and H. Umegaki, A iscte on the entropy for operator algebras, Proc. Japan Acad. 37 (1961), 149-154. 19. A. Ostrowski, Sur queique. applications dee functions convenes et concaves au eons de I. Schur, J. Math. Pures AppI. (9) 31 (1952), 253-292. 20. 1. Schur, Obey cuss Kiasse ven Mittetbildungen mit Anwendungen auf die minantentheori., S.-B. Berlin. Math. Ges. 22 (1923), 9-20. 21. W. F. Stinespring, Problem 4863. Amer. Math. Monthly 66 (1959), 728. 22. B. Sz.-Nagy, Proiongements des transformations de i'eapace de Hubert, qui sortent de cet espace, Appendice au livre "Lecons d'analyse fonctionelle" par F. Riesz et B. Sz.Nagy, Budapest, 1955. 23. B. Sz.-Nagy and A. Korányi, Operatortheoretiache Behandlung und VeraUgeineinerung cix.. Probiemkrei.ea in dcv hemplexen Funktionentheorie, Acta Math. 100 (1958), 171-202. MATHEMATICAL REVIEWS

SOME NEAR-SPHERICITY RESULTS BY

ARYEH DVORETZKY' 1. The sphere occupies a very special position in geometry and especially in the theory of convex bodies. The relevant results are too numerous to quote, yet the subject is far from exhausted and new aspects continue to be

The purpose of this communication is to discuss one such recently discovered new facet. The results discussed here will center around the theorems reproduced in the next section. Since their proof is rather lengthy, it will not be reproduced and the reader is referred to [4]. Some other results discovered.

from [4] will also be quoted. On the other hand, new corollaries of these results will be drawn. In § 5 we will prove a new result on the projections of a cube, which is a counterpart to a result given in [4] on the sections of a cube but is more precise and, also, easier to prove. We shall close with a discussion of some open questions and also state a new result on the convergence of series in Fréchet spaces, which is derived from near-orthogonality—

a rather poor relation of near-sphericity, but which is more widespread. Except for the last section we shall deal exclusively with Euclidean spaces (see [3J for further applications to functional analysis). Throughout the paper E* will denote n-dimensional Euclidean space, its points will be denoted by x = (x1, ..., X,,) in orthonormal coordinates. BW will 1} of E'. C' will represent the stand for the unit ball {x; + - -. + unit cube {x; —1 1 (i = 1, -, ,s)} while will denote the unit octahedron Cx; I xjI + -. + 1) of E'. By a subspace Ek of E' (1 k n) we shall always mean a subapace through the origin. The term symmetric will always signify symmetry with respect to the origin, thus a set C c E0 is symmetric if and only if C = —C. A set C is said to be a convex body in it it is a compact convex set with nonempty interior. I

To facilitate the statement of our result we give the following definition. DEFnnnoN. A convex set C in a Euclidean space is called spherical to within s (0 < e 1) if there exist in the flat space generated by C two concentric balls B1 and B, of radii (1 — e)r and r, respectively, such that B1 c C c B,. The g. 1. b. of the t having the above Property is called the asphericity of C 2.

and denoted by a(C).

If C is a compact set then there exist concentric balls of radii (1 — and 1

a(C))r

r, the one contained in and the other containing C. If P is the common The

research reported in this paper has been sponsored in part by the Air Force

Office of Scientific of the Air Research and Development Command USAF through its European Office. 203

ARYEH DVORETZKY

204

center of these balls then 1 — a(C) is simply the ratio of the distances from P to the nearest and farthest points on the boundary of C. in case C has a center of symmetry, the point P must coincide with it. Obviously a(C) 1 for a compact set C if and only if C is a ball. r(C) is a continuous functional of C in the usual Blaschke topology of compact sets in E". The main result of [41 is THEOREM 1. Given e (0 < e < 1) and a positive integer k, there exists an integer N = N(k; e) such that if C is any convex body in E", n N, then there is a subs pace E" for which a(C

< c.

A suitable N(k, e) is given by exp

log k).

The crucial fact is that N(k, e) is a function of k and e alone, thus giving a result of a uniform nature for all convex symmetric bodies. Since a is a continuous functional it is obvious that if (1) holds for a certain E" it will hold also for all E" sufficiently "near" it. It is natural to ask for some measure-theoretic statement about the Ek satisfying (1). The obvious measure to use is one which is preserved under orthogonal transformations. According to A. Weil's results on Haar measures in homogeneous spaces [9], there exists a unique (up to a numerical factor) regular measure of this kind. We shall denote this measure, normalized so that it becomes a probability

measure, i.e., so that the measure of the set of all E" c E" is 1, by jg,,,i,. Explicit expressions for this measure may be found in works on integral geometry; these, however, need not concern us here. It is very easy to see that2 inf .a,,,k{Ek; a(C n < e} = 0 for every n, £ > 0 and k> 1 when the jnf is taken with respect to all symmetric convex C c E'. To see this it is sufficient to consider ellipsoids with all but one principal axes equal and the remaining one very large. Thus to obtain significant results of this nature some suitable restrictions must be imposed on C. A result of the type urn inf p,,,tc{E"; a(C n Ek) < s} = 1 (2) would be interesting already when

denotes the class, say, of symmetric

i.e., C which lie between two concentric convex bodies C with a(C) < A < balls with fixed ratio of radii. However, much more is true; not only (2) holds for a much wider class of convex bodies, but the result can be uniformized. Indeed, it was proved in [41 that the following theorem holds. THEOREM 2. Given e (0 < s < 1), 8 (0 < 6 < 1), an integer k> 1 and a function g(n) defined on the positive integers and satisfying (3)

g(n)

1 (n =

1,

2, ...),

urn

there exists an integer N = N(k; €, 6; g) such that 2

(Ek; . . .} denotes the set of Ek possessing the property

SOME NEAR-SPHERICITY RESULTS

a(C

(4)

holds

for all n

fl

Ek) < e} >

1

205

ö

N and all symmetric convex bodies C in E" satisfying

C cg(n)C". (We recall that B" and C" are the unit ball and unit cube of E", respectively.) Though we do not reproduce the proof of these theorems, it should perhaps be remarked that no direct proof of Theorem 1 is known and that in [3] and [4] it Is derived from a weak version of Theorem 2 (with g(n) = const). We would like also to note that the proof proceeds in two main stages: first it is shown that for C satisfying (5) there are "many" directions for which the of C are nearly equal, then integral distances from the origin to the geometric methods are used to derive (4). 3. So far we have discussed the near sphericity of sections, let us now turn to that of orthogonal projections. Let us denote by Cl Ek the orthogonal projection of a set C in E" on (1 k n). If C is a closed convex set containing the origin and e is a unit B"

vector then the projection on the ray determined by e is the segment [0, H(e)e],

where H(e) is the support function associated with C (for this and all other assertions about the general theory of convex bodies used here see [1]). Since if C is a symmetric convex body, the union of all such segments is again such a body, it follows immediately from Theorem I and the fact that if H(e) r that we have for all ee E& then rB" c Cl THEOREM 3.

Given

(0 <

e

< 1)

symmetric body in E" with3 n

and a Positive integer k, then if C is any N(k;

there exists a subsfrace E" of E" for

which

a(C I Ek) < e.

However, something more can be done just as easily, and a statement about

the measure of the set of E" satisfying (6) can be derived. The support functional H is defined, of course, throughout E" (it is simply the positive homogeneous extension of H(e) given above for unit vectors e in terms of projections). The set C" = {x; H(x) < 1) is the polar reciprocal of C, and is again a symmetric convex body. Since the distance from the origin to the in the direction e is 1/H(e), it follows at once from the definition of asphericity that we have a(C* fl Ek) = a(C I Ek) for every Ek. Now C* satisfies B" c cg(n)C" if and only if C satisfies boundary of

—f-- 0" c C C

g(n)

B",

where, we recall, 0" is the unit octahedron in E". Thus, we have from Theorem 2: THEOREM 4. For every (0< e <1),ö(0 <8<1), integerk> 1,and function g(n) defined on the positive integers and satisfying (3), we have 3

The same N(k,c) as in Theorem 1.

ARYEH DVORETZKY

206

a(CIEk) < e}> 1—8 for all' n N(k; e, 8; g) and all convex symmetric bodies C in E* satisfying (7). On combining the results expressed by (4) and (8) we have

k and g be as in Theorems 2 and 4. Then we have a(CJEk)< e} > 1—8 ,1,I,k{E;a(C n Ek) < N(k; s, 8; g) and all convex symmetric bodies C in satisfying

THEOREM 5.

for all' n a(C)

Let

8,

g(n).

Thus for symmetric convex bodies in E" whose asphericity increases less rapidly than (log n)1", most k-dimensional sections as well as projections tend to become spherical.

One could give many further results of a similar nature, we shall content ourselves with one more. Let V(C1, . ., , i) denote the mixed volume of the convex, sets C1, . . ., and the segment [0, xl in E0. Then V(C1, ..., C,,-1, x) satisfies, as a function of x, the conditions imposed on the support function of a symmetric convex set. Hence we have

Let a (0 < a < 1) and a positive integer k be given, then for all' N(k; a) and any choice of convex bodies C1, '•', in E" there exists Ek c for which mm V(C1, .' ', C,,-1, e) (1 — a) max V(C1, . .., C,,-1, e) the mm and max being both relative to all unit vectors e e Ek. A special case of this theorem is obtained on taking C1 = C, = •.. = C,,_1 = C, then the• mixed volume in V(C1, . . •, C,,-1, e) is proportional to the (n — 1)dimensional volume of the projection Cl E%.I of C on the orthogonal to e. Similar results hold for the (n — P).dimensional volumes of projections THEOREM 6.

n

Cl 4. So far we have dealt with convex bodies possessing a center of symmetry, the results, however, remain valid under a very considerable relaxation of this requirement. Let us denote, for a convex body C in containing the origin as an interior

point, by F(e) the distance from the origin to the boundary of C in the direction determined by the unit vector e. The functional fl(C) = max

the max being taken over all unit vectors e measures, in some sense, the centrality of the origin with respect to C. Clearly = 1 if and only if C is symmetric. It has been shown in [4] that we have Let a (0 < a < 1), an integer k> 1 and a function g defined over Then there exists N = N(k; a; g) such that if C is any convex body in E', n N, for which ,9(C)
the Positive integers and satisfying (3) be given.

'The same N(k; a, o;g) as in Theorem 2.

SOME NEAR-SPHERICITY RESULTS

Using the considerations of the preceding section we can make the identical assertion with (1) replaced by (6). Statements analogous to those of Theorems 2, 4, 5 etc. can also be made. 5. As indicated at the end of § 2 an important step in the proof of the results is the fact that if C is a symmetric convex body in E" satisfying (5) then the distance from the origin to the boundary of C is nearly equal in most directions. For the case of the cube a much more precise result was

proved in [4]. THEOREM 7. Let be the distance from the origin to the boundary of the unit cube C" along the line E1, and Put, for > 0, ( / =
\ 2 log n — (1 — e) log log nj

=

I

(

j

\1/2

I

Then we have lim

= urn

=

0

for every e > 0.

We recall that

of a set of lines through the origin is (up to a factor)

the (n — 1)-dimensional measure of the set where these lines intersect the

boundary of B". Thus Theorem 7 states that though the range of variation in most directions the actual value is very close to of r(E') is from 1 to n112(2 log n — log log

This is very reminiscent of the laws of large

numbers of the theory of probability and may, indeed, be interpreted as such a law for certain dependent random variables.

Here we shall consider an analogous question for projections of C' and obtain, quite simply, a result indicating even sharper concentration. THEOREM 8. Let be half the length of the orthogonal projection C" I E1 of the unit cube C" on the line E1, and consider as a random variable on the

space

endowed with the probability measure

Then we have, as n —. oo,

E and a' denote expectation and variance, respectively. From (10) and (11) it follows immediately by Chebyshev's inequality that if t,, is any sequence of positive numbers increasing to infinity however slowly we have

ARYEH DVORETZKY

208

which again varies and sharper consequences may also be drawn. Thus between 1 and exhibits a much greater concentration than is shown for r, in the preceding theorem. To prove Theorem 8 we introduce polar coordinates = p sin 01 = p sin cos = sin =

01, cosO1,

COS

COS

is a unit vector on E' we have

If ea =(Ya,

On grounds of rotational symmetry we have therefore sinO (12)

E(p,,) =

0 dO

=

I) = n

cos'20d0 0

where we have put

__=lj0 1

It is well known that (13)

—

l'(n/2)

—

—

1)12)'

from which (10) follows immediately. To prove (11) we compute

Z

15;<JSn

=

+ 1

+

+ n(n — 1)

E(Iy1Hy1I)

+ n(n — ° %

.10

=÷ by (13).

Hence,

4(n

—

.10

n—i

SOME NEAR.SP}IERICITY RESULTS

= and

since, by (13),

=

—

= (n

—

(3/2)

+

1

— 1) —

+ an

(2nri

easy computation yields (11).

given by (12) is the bound on the proWe remark that the value of jection constant of E' which has been computed in a somewhat different way — by B. Grunbaum [7]. 1) also the distance from the origin to the octahedron i/ n 0' circumis

scribing B'. Hence both Theorems 7 and 8 may be formulated in terms of sections and projections of 0'. 6. It is interesting to know whether our quantitative results, such as the estimate given at the end of Theorem 1, are of the right order of magnitude. Presumably they are very far off, but it seems quite difficult to obtain significantly better results. From the results of the previous section it follows, by the method of proof mentioned at the end of § 2, that the unit cube and unit octahedron have the Thus, property asserted by (9), though for both these bodies a(C) = presumably Theorem 5 (and also Theorems 3 and 4) could be proved under much weaker conditions. There should not be great trouble in proving it for + + I I' 1} (p 1), but we would like to have I x1 I' + C= I

general results. In particular: Does there exist N' = N'(k; such that for all symmetric convex C in E' for n N' there are Ek satisfying both (1) and (6)? It would be enough to show that there are Ek for which both C n Ek and Cl Ek are nearly ellipsoidal from which the result would then easily follow. This in turn raises the question: Does (2) hold for all symmetric convex bodies C if the asphericity is replaced by the unellipsoidality where unellipsoidality is

defined exactly in the same way as asphericity except that, instead of considering concentric balls, homothetic ellipsoids are taken into account? An affirmative answer to this question will lead to a similar dropping of restrictions

in Theorems 4 and 5, and also to a positive answer to the previous problem. Actually the ellipsoidality is, in many respects, more interesting than the sphericity; it is this notion that is important in most applications to functional analysis. It should be interesting to develop the whole theory by integral geometric methods relative to the group of centro-affine transformations, instead of those relative to the rotation group as done in [3] and (4]. 7. The following result was established in [6) for symmetric convex bodies C, but the identical proof holds under the wider conditions given here. (A reproduction of the proof may be found in (2] as well as in [4].)

Let C be a compact set in E' containing the origin as an interior Then there exists a centro-affine transformation T for which B' c T(C) (i = 1, - n) satisfying and such that points e, = (e,.1, . . -, THEoREM 9.

point.

for

ARYEH DVORETZKY

210

may be found on the intersection of B" with the boundary of T(C).

The fact which is used in deducing from this the following result is that, for every k, the unit vectors e1, •, are as nearly orthogonal as we please provided n is sufficiently large. (It may be mentioned that the above theorem, for symmetric convex C, is used in establishing the near-sphericity results in [3; 4].)

From Theorem 9 it is possible to derive the following [5]: TREOREM 10. Let F be an infinite dimensional Fréchet space and let 1,

2, ...)

be

(n =

a sequence of Positive numbers tending to 0 and satisfying

<00. Then there exist (n= 1,2, ..) such that n n0 and the series x, is commutatively convergent.

for all

The proof is similar to that used in [6] to establish the same result for infinite dimensional Banach spaces. We recall that the fact that distinguishes a Fréchet space from a Banach space is that its norm need not be positive homogeneous. This loss of homogeneity results in the "balls" of F not being necessarily convex, and hence the need to use Theorem 9 in the form given here.

Taking, say, = 1/n (n = 1, 2, •) we see that in every infinitely di,nensional Fréchet space there are series which are commutatively but not absolutely convergent. This is a recent result of S. Rolewicz [8]. REFERENCES

1. T. Bonnesen and W. Fenchel, Theorie der konvexen Korper, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 3, 3. Springer, Berlin, 1934. 2. M. M. Day, Norined linear spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, New Series, Vol. 21, Springer.Verlag, 1958. 3. A. Dvoretzky, A theorem on convex bodies and applications to Banach spaces, Proc. Nat. Acad. Sci. U.S. A. 45 (1959), 223-226. 4. , Some results on convex bodies and Banach spaces, Proceedings of the International Symposium on Linear Spaces held at the Hebrew University of Jerusalem, July 5-12, 1960, pp. 123-160, Jerusalem, 1961. 5. , On series in spaces, Bulletin of the Research Council of Israel, to appear. 6. A. Dvoretzky and C. A. Rogers, Absolute and unconditional convergence in normed linear spaces, Proc. Nat. Acad. Sd. U. S. A. 36 (1950), 192-197. T. B. Grunbaum, Projection constants, Trans. Amer. Math. Soc. 95 (1960), 451-465. 8. S. Rolewicz, On a generoiiration of the Dvcretthy-Rogers theorem, Colloq. Math. 8 (1961). 103—106.

9. A Weil, L'intigration dane les groupes topoiogiques et sea applications, Actualités Sci. hid. No. 869, Hermann, Paris, 1940. THE HEBREW UNIVESBITY OF JERUSALEM

ON THE KREIN-MILMAN THEOREW BY

KY FAN 1. Introduction. In this paper, we shall first formulate a simple general lemma (Lemma 3), which is "purely set-theoretical" and to which the KreinMilman theorem [8] can be directly reduced. By "purely set-theoretical" we mean that the statement of Lemma 3 involves neither topological nor vector space structure. Because of the various applications of the Krein-Milman

theorem, Lemma 3 itself seems to be of interest. We also discuss Stone's theorem [11] on ideals of Boolean rings and Wallman's theorem [12) on principal

filters, as both have a strong resemblance to the Krein-Milman theorem. Theorems of Stone and Wailman are reduced to Lemma 2 which is dual to Lemma 3 (but somewhat simpler). Next, 0-convex sets and a-extreme points are defined in terms of an arbitrary set 0 of real-valued functions on a set S. Theorems 4, 5,6 on 0-convex sets require topological structure on S, but not vector space structure. The proof of Theorem 4 is again based on Lemma 3. When S is a locally convex topological vector space and 0 is the set of all continuous linear functionals on S, Theorems 4,5 and 6 become the Krein-Milman theorem, a special case of a result of Jerison [6], and Milman-Rutman's theorem [9], respectively. Theorem 4 is related to a minimum principle of Bauer [2). As we shall see in the final section, Theorem 4 can be proved by applying Bauer's theorem which in turn can be reduced to Lemma 3 (with the modification indicated in Remark). By a vector space we mean a real vector space. For topological vector spaces, we implicitly assume Hausdorif separation axiom. We use the notation 0 for the empty set and for set difference. We are indebted to Professor Klee for calling our attention to Bauer's work

Results on 0-convex sets were obtained before we learned of these papers. Remark on (2.7) has been suggested by Bauer's minimum principle. 2. Some general lemmaa. A family of subsets of a set S is said to be -stable if Se and if any intersection of sets of ..9• is a member of 5. For a ii -stable family ..9 of subsets of S and for any A c S, the intersection of all sets X satisfying Xe .9 and X D A is called the ,.9-hull of A. SimiIii; 21.

larly, a family 9 of subsets of S is said to be u-stable if 0 e 5 and if any union of sets of is a member of For a u-stable family ..9 of subsets of S and for any A S, the union of all sets X satisfying Xe and Xc A is called the of A. A family of subsets of S is said to c be n -inductive if, whenever is a nonempty chain (i.e., for any A, Be %f, 1 This work was supported by the U. S. Atomic Energy Commission at Argonne National Laboratory.

211

KY FAN

212

at least one of the inclusion relations A c B, B c A holds), we have for is said to be u-inductive if UZEr, Xe Similarly, flIEr, X€ subsets of S is called of c family A every nonempty chain and Bc A imply Be hereditary, if A be three families of subsets of a set S such that LEMMA 1. Let 9, is hereditary. For any and n•inductive, 0e is .5- is n-stable, Ae

the

follozving two conditions are equivalent:

(2i) For any Xe

such that A

X, there is a Be

with

Bc A and

If Tc A and Tn

contained C of in A, then the 5hull of A coincides with the 5--hull of T. satisfies (2.1). Let T c A. Assume PROOF. (2.1) (2.2). Assume A (e for every minimal member C of contained in A. Let that T n C is hereditary, 1', Z denote the ,.9Thulls of A, T, respectively. Since contained in A. being Zn C for every minimal member C of n-inductive, every Be contains a minimal member of by Zorn's lemma. is hereditary, it follows that B a for every Be Again, since satisfying Bc A. Then by property (2.1) of A, we have A c Zand therefore Yc Z. Since Tc A, we have also Zc Y. satisfies condition (2.2). Let Xe 5; be such (2.2) (2.1). Assume A (e of T, but that A ci: X. Consider T A a X. Then X contains the of A. So the 5-hulls of A and T are different. By property not the (2.2)

such that Bc A and T T, we have A specialization of Lemma 1 is the case where = {X c SI I contains no member of For the convenience in application, we restate this case of Lemma 1 in its dual form. is LEMMA 2. Let 5; be two families of subsets of a set S such that For any A e the following two u -stable, is u -inductive and

(2.2)of A, there is a minimal member B of

Since

conditions are equivalent: (2.3)

For any Xe5; such that XttA, there is

wilhBDAand such

contains B u X. that no member of (2.4) The 5;-kernel of A coincides with the 5;-hernel of 8(A), where 8(A) denotes the intersection of all maximal members of containing A. We now prove our main lemma which will be used in 4,5,6. LEMMA 3. Let .5- be a a -stable family of subsets of a set S, and let be a a -inductive family of subsets of S such that the following two conditions are fulfilled: and—Xe .5- such that A ci: X, there is a Be (2.5) For any A with

BcAandBflX=Ø.

(2.6)

For

any A e

with AnX*O

—

containing more than one element of S. there is an

Xe .5-

andActX.

Then, for every A the set {x e A {x) coincides with the .5--hull of A.

is nonemPty, and its .5--hull

ON THE KRE!N•MH..MAN THEOREM

mrinductive, every Ac

Paoor.

contains a minimal member of

0 for every To prove that that every minimal member of is a set containing exactly one element of S. Suppose a minimal member C of contains more than one element. By (2.6), we can find an Xe with C n X * 0 and C '3 X. Then, by (2.5), there is a with BcC and BnX= 0. As C isaminimalmember of we

must have B=C. But then CnX* 0 and BnX= 0 contradict

is each other. Since 0 this proves that every minimal member of 0 for every a set formed by a single element of S. Hence {x e A I {x)

Next, let be the family consisting of the empty set alone. Then (2.5) says that every A has property (2.1). Therefore, by Lemma 1, every has property (2.2). In other words, for any Ac the .9-hull of A AG coincides with the 5-hull of Cx A I {x) e REMARK. Without changing the above proof, condition (2.6) in Lemma 3 may be replaced by the following slightly weaker, but less simple, condition: (2.7) For any A e which contains two distinct elements x, y such that

{y)

{x)

there

is an Xe 9 with A n X * 0 and A

X.

This modification of Lemma 3 will be useful in § 6. Note that both (2.6), (2.7) will be satisfied in the case when {x} e 5 for every x e S. It is sometimes (see § 4) useful to generalize Lemma 3 as follows. LEMMA 4. Let 9 bea ti-stable family of subsets ofa setS, and let be a family of nonemPty subsets of S with a partial ordering such that the

following conditions are satisfied: (2.8) (2.9)

impliesAcB. In the Partially ordered set has a lower bound.

(2.10)

For any Ac

(2.11)

For any A

and

(with the Partial ordering

every

Xe 5 such that A X, there is a Be

andBnX=O.

chain with

containing more than one element of S, there is an

Xe5 with An X* 0 and

Then for every A {x) e

Cx) A) is nonempty, and its ,.9-hull coincides with the 5-hull of A. In fact, we can similarly generalize Lemma 1 by replacing the inclusion satisfying (2.8), (2.9). relation for sets of by a partial ordering on Then the proof of Lemma 4 is the same as that of Lemma 3, except that we use this generalized form of Lemma 1. I

and

3, Theorems of Wailman and Stone on filters. There is a striking similarity between the Krein-Milman theorem and theorems of Waliman and Stone. For purpose of comparison, we reduce these two theorems to Lemma 2. Let S be a partially ordered set with the following two properties: (3.1) Any two elements x,y of S have the greatest lower bound x A y. (3.2) $ has the least element 0.

KY FAN

214

A subset A of S is called a filter if A * 0 and if A satisfies the following three conditions:

(3.3) xe A and x y imply y A. xe A and ye A imply x A ye A.

(3.4) (3.5)

A filter is called an ultrafihter if it is not contained in any other filter. For

isclearlya filter. Filters = any aeS such that a * 0, of this type are called principal fillers. For a partially ordered set S having properties (3.1) THEoREM 1 and (3.2), the following two conditions are equivalent: (3.6) For any two distinct elements a, b of S, there is an x e S such that exactly one ofa AX and bAt is equal toO. (3.7)

For every a e S such that a * 0, the principal filter F. is the intersection of all ultrafilters containing F..

Paoor. Let .9 be the family consisting of the empty set 0 and all those subsets X of S such that X is a union of principal filters. Let be the family of all filters. Then is U stable and is ii -inductive. For any 8(A) defined in Lemma 2 is the intersection of all ultrafilters containAc ing A. Since any intersection of filters is a filter, 8(A) is a filter. On the

other hand, since every filter is a union of principal filters, every filter is its own .9-kernel. Thus, condition (3.7) means that if A is a principal filter, then A has property (2.4). Then by Lemma 2, (3.7) is equivalent tp the following condition: (3.8) It

For any Xe5 and aeS such that a * 0 and

there is a filter

B containing F. and such that no filter contains B U X. remains to prove that (3.6) and (3.8) are equivalent. (3.6)

Let 1e9 and a E S be such that a * 0 and

(3.8).

X

a

c

a. By (3.6), we can find y e S such

that exactly one of a AY andcAa Ayis0. Clearlywemusthavea Ay*O,

cAaAy=O. Takex=aAy. ThenaAx*O,cAx=0. LetB=F.A.. but no filter contains BUX, since ceX,xeil and cAX=0.

Then

(3.8) (3.6). Let a, b be two distinct elements of S. We may assume b a. Then a * 0 and b If b = 0, we have a A a * 0, b A a = 0. If b * 0, we have F6 e F6 F. and then, by (3.8), there is a filter B D F. such that no

filter contains B u Fb. This last fact implies the existence of an xc B with

1' A x =0. Since a B, x€ B. we have a A x *0. In the next theorem, the partially ordered set S is required to satisfy (3.1), (3.2) and the following condition: (3.9)

To

each x€S, there is an x'eSsuch that: (i)XAX'=O, (ii) yAx'=O

implies y

x.

THEoREM 2 (STONE).

In a partially ordered set S having Properties (3.1), (3.2)

filter A is the intersection of the ultrafilters containing A. have the same meaning as in the proof of Theorem 1. Paoop. Let 9, Applying Lemma 2, it suffices to show that every filter A in S satisfies conand (3.9),

every

ON THE KREIN-MILMAN THEOREM

215

dition (2.3). (See the beginning part of the proof of Theorem 1.) Let A be a

filter and let 1€ 5' be such that Xct A. Consider any xc X such that x A.

IfyeSandyAx'=O, then we

(since

Hence a A x' (I for every a e A. Hence there is a filter B containing A and x'. Since x X, x' B and x A x' = 0, no filter contains B U X. Thus every filter A verifies (2.3). Wailman's original theorem [12) is restricted to distributive lattices. The above formulation is due to Samuel [10]; see also Birkhoff and Frink [3]. The original formulation of Stone's theorem [11] states that, in a Boolean ring S, every ideal * S is the intersection of all maximal ideals containing it. The above formulation is due to Bourbaki [4, p. 43] and Samuel [101. 4. The Kreln-Milman theorem.

Let A, B be two sets in a vector space. A

is said to be an extreme subset of B if A c B and if x B, y B, 0 < a < 1, and ax +(1 — a)y€A imply xeA,yeA. In case A = {a}, a is called an extreme point of B. The following properties of extreme subsets are well known and can be easily verified.

(4.1) Any union of extreme subsets of a set A is an extreme subset of A. (4.2) Any intersection of extreme subsets of a set A is an extreme subset of A. (4.3) If A is an extreme subset of B, and B is an extreme subset of C, then A is an extreme subset of C. (4.4) If A c B c C and if A is an extreme subset of C, then A is an extreme subset of B. X. If, (4.5) Let be a family of sets in a vector space, and let Y = one is an extreme subset of the other, then of any two members of Y is an extreme subset of each Xe THEOREM 3 (KREIN.MILMAN). In a locally convex topological vector space S. if o(K) is the set of all extreme points of a nonempty compact set K, then K and a(K) have the same closed convex hull. The following reduction of this theorem to Lemma 3 is essentially a rearrangement of the proof given by Kelley [7]. Let 9 be the family of all closed convex sets in S. Let be the family of all nonempty closed extreme subsets of a fixed nonempty compact set K. The n -inductive property of follows from (4.2) and the fact that the intersection of a chain of nonempty compact sets is nonempty compact. (2.6) is trivial since every one-point set is closed convex. (2.5) is verified by separation of convex sets: if A (e is not contained in a closed convex set X, there is a continuous linear functional f on S such that B = {xe A 11(x) = is disjoint from X; B is a nonempty closed extreme subset of A and therefore by (4.3), Be If we use Lemma 4 instead of Lemma 3, then we take as the family of all nonempty compact sets in S and define A B in to mean that A is an extreme subset of B. Then (2.9) is verified by (4.5) and compactness.

5. 0-convex sets and 0-extreme points. Let 0 be a set of real-valued functions defined on a set S. A subset X of S is said to be 0-convex, if either

KY FAN

216

f(x) a), where or X is an intersection of sets each of the form {x is a real number. Thus the 0-convex subsets of S form a n -stable family. For A c S, the intersection of all 0-convex sets containing A is called the 0-convex hull of A. LEMMA 5. Let 0 be a set of real-valued functions defined on a set S and let 0 * A c S. If each fe0 is bounded below on A, then the 0-convex hull of A is X=

S

f€0 and a

a,),

a1 = infgE.1f(a). Psoop. Clearly X is a 0-convex set containing A, and therefore X contains

where

the 0-convex hull of A. Let x0 £ S be not in the 0-convex hull of A. Let Y= be

fl {x eS 1(x)

1€,

a 0-convex set containing A but not x0, where V c 0 and

are real

numbers. Since A c Y, we have a1 for fe !l'. Since x0 E 1', there is an V with fo(x0) < Then f0(x0) < a10 and x0 X. Hence every point not

in the 0-convex hull of A is also not in X, so X is the 0-convex hull of A. Given a set 0 of real-valued functions on a set S and x, y, z€ S, y is said to be 0-between x and z if, for every f€0,f(y) f(x) = 1(y) =1(z). Clearly, if X(cS) is 0-convex and if y is 0-between two

points of X, then yeX. For two subsets A,B of S,A is said to be a 0-

extreme subset of B, if A c B and if, whenever a point ye A is 0-between two points x, z of B, we have x, z€ A. In case A = (a), a is called a 0-extreme point of B. It is easy to see that properties (4.1)—(4.5) remain valid for 0extreme subsets. THEOREM 4. Let 0 be a set of real-valued functions defined on a Hausdorff space 5, and let K be a nonem/,ty compact subset of S such that the following two conditions are fulfilled: (5.1) The restriction of each fcO on K is lower semi-continuous on K. (5.2) For any two distinct Points x, y of K, there is an fe 0 with f(x) * f(y). Then K has 0-extreme points. Furthermore, if a(K) denotes the set of all 0-extreme points of K, then K and a(K) have the same 0-convex hull.

Let

be the n-stable family of all 0-convex subsets of S. Let the family of all nonempty closed 0-extreme subsets of K. is nonempty, as Ke That is n -inductive follows from property (4.2) (for 0-extreme subsets) and compactness. To verify condition (2.5), let A e and Xe .9 be such that A X. Then PROOF.

denote

X*Sand Xis of the form

where I is a non-

empty set of indices, and for each €1, €0 and a real number. Let x0 A and x0 X. Then for some I, we have < infzezf,,0(x). As A c K, the restriction of on A is lower semi-continuous on A. Let B be the set formed by those points of the nonempty compact set A, at which attains its minimum over A. B is nonempty, closed and B n X = 0. We have also Be i.e., B is a 0-extreme subset of K. in fact, if x2€B is

ON THE KRELN-MILMAN THEOREM

217

we must have 0-between two points x1 , x3 of K, then since x2 E B c A and x2 is 0-between the points = x3 A. Then, since £ B. Hence i.e., x1, = = f?0(x2) x3 of A, we must have This verifies condition (2.5). Be

Next, let A e

and

assume that A contains two distinct points a, b. By

(5.2), there is an feD with f(a) * 1(b). Let a max(f(a),f(b)) and X = A n X * 0 and A Ct X. Thus condition (2.6) {xe S 11(x) a). Then Xe is verified. Then the theorem follows from Lemma 3.

Let 0 be a set of real-valued functions defined on a Hausdorff Let {C2} be a sequence of nonempty compact 0-convex subsets of S space S. such that C2 D C2+1, and let C = fl1 Ce.. Let o(C2) be the set of all 0-extreme If conand let D be the topological limit superior of points of ditions (5.1), (5.2) are satisfied by K = C1, then C is the 0-convex hull of D. THEOREM 5.

PROOF. Let EA = U. Then D = Da. Since Da C = Ca, D2 and D are compact, and D c C. Let X denote the 0-convex hull of D. Then Xc C and X = fir5. {x S 11(x) a1), where aj denotes the minimum of 1 over D (Lemma 5). To prove X D C, it amounts to showing that minleDf(x)

for every fe 0. Consider any feD. Since C2 is the 0-convex hull of a(C2) (Theorem 4), and e(C2) c D2 c C8, the minimum of I over Ca is the same as the infimum of lover a(C8), or the minimum of 1 over D8. Let x,, e D,, be such that 1(x2) is the minimum of f over D8 (same as minimum over C8). As D8 D8+1, C2 C, we have 1(x2) f(x.+,) min,ecf(x). Let x0 be a cluster point of (x,.}. Then But by lower semi.continuity of we have f(x0) x, D, so minjeDf(x) min2ecf(x).

Consider again a set 0 of real-valued functions defined on a Hausdorif space S, and a nonempty compact set K in S satisfying (5.1). Condition (5.2) is no longer assumed. For each fe 0, let M(f, K) denote the set of all those points of K, at which I attains its minimum over K. A subset A of K will

be called a 0-supporting subset of K if either A = K, or A * 0 and A is an intersection of sets each of the form M(f, K) (with 1€ 0). By (5.1), each M( f, K) is a nonempty closed subset of K, so every 0-supporting subset of K subsets of K is is nonempty compact. Therefore the family of all n -inductive. The miuimal members of this family are called minimal 0supporting subsets of K. THEOREM 6. Let K be a nonempty compact set in a Hausdorff space S. Let 0 be a set of real-valued functions defined on S and satisfying condition (5.1). If a subset C of K meets every minimal 0-supporting subset of K, then K and C meets every C have the same 0-convex hull. The same conclusion holds, minimal nonempty closed 0-extreme subset of K. consist PROOF. Let 5- be the family of all 0-convex subsets of S, and of the empty set alone. Let be the family of all 0-supporting subsets of K. Assume Xe be such that KCt X. Let x0 e K and x0 X. Then there Bc K is an feD with f(xa) < infzezf(x). Take B = M(f, K). Then Be

KY FAN

218

and B n X 0. Thus condition (2.1) is verified by A = K. Hence the first assertion of the theorem follows from Lemma 1. Again by Lemma 1, the second assertion is proved by considering the same and by taking as the family of all nonempty closed 0 extreme subsets of K. In the proof of Theorem 4, we have already shown, under the hypothesis (5.1) alone, the n-inductive property of and that (2.1) is satisfied by every A e (in particular by A = K).

If S is a locally convex topological vector space and 0 is the set of all continuous linear functionals on S, then the 0-convex sets are precisely the closed convex sets. In this case, Theorem 4 reduces to the Krein-Milman theorem, Theorem 5 becomes a special case (the case C,,c, Cfl+L) of a result of Jerison 16], and the first part of Theorem 6 is Milman-Rutman's theorem E9; p. 79].

5,

6.

Bauer's minimum principle. Our Theorem 4 is closely related to Bauer's

minimum principle [2] stated below as Theorem 7, although they are quite different in formulation. In § 5, we start with a set 0 of real-valued functions and define all the concepts in terms of 0. Bauer begins with a subset of K x .9(K) (where K is a compact space and .9(K) is the power set of K) and his set of functions on K is determined by For the purpose of comparison and unification, we first rearrange Bauer's proof of his theorem to a reduction to our Lemma 3 (and Remark on (2.7)). Then we use Bauer's theorem to give another proof of Theorem 4. THEOREM 7 Let K be a compact space and let .9(K) denote the power set of K (i.e., the set of all subsets of K). Let be a subset of K x .9(K) such that: (6.1) (6.2)

To each x e K, there is at least one Ee .9(K) with (x, E)e (x, E)€ implies E 0.

Let p(K) be the subset of K formed by all Points x having the following (6.3)

Let

(x, E)e implies E = {x}. denote the set of all real-valued, lower semi-continuous functions f on K

satisfying the following condition: (6.4) If (x, E)E and 1(x) f(y) for alt y E, then f(x) = f(y) for ally e E. If, for any two distinct Points x, y of K p(K), there is an f€ with 1(x) f(y), then every function of attains its minimum over K at some Point of p(K). PROOF.

Let 9 be the family of all p-convex subsets of K (in the sense

defined in § 5). Let having the property:

(6.5) x€ A and (x, E) e is nonempty, as K e

the family of all nonempty closed subsets A of K

be

imply E c A. (x) if and only if XE p(K). -

is fl-in-

ductive (see [2, Lemma 1]). To verify condition (2.5) of Lemma 3, assume A XE9 be such that A (tX. Let X {xeKlf(x)_ a,) (see Lemma 5) and we have f0(x0)
ON THE KREIN-MILMAN THEOREM

219

empty closed subset of A and B n X= 0. We claim Be i.e., B has property (6.5). In fact, if XE B and (x, E)e then because xc Bc Ac we have E c A. Then f0(x) f0(y) for all ye E and consequently, by (6.4), fo(x) = fo(Y) for all ye E, i.e., E C B.

Next, we verify condition (2.7). Assume A e and A contains two distinct Then x,, x2 are in K— p(K). There is points x,, x2 such that {x,) {x2} E an!' with f1(x,) * f,(x2). Let a max (f,(x,), f,(x2)) and X {x e K I a}.

Xe X n A * 0 and A Ct X. Consider now any fE Let A = {xE K 11(x) = By (6.4), we have A (see [2, Lemma 2]). Applying Lemma 3 and Remark on (2.7), there exists an x0 e A with {r0) In other words, 1(x0) is the minimum of f over K, and x0 E p(K). Another proof of Theorem 4. Let S, K and 0 be given as in the hypothesis of Theorem 4. For each xe K, let be the set formed by x and all those Then

points ye K such that x is 0-between y and some point of K. Let in Theorem7. Let = {(x,E,)IxeK) anddefine = {fxIfeO},

ix denotes the restriction of / on K. By (5.1) and our definition of c !V and therefore, by (5.2), separates points of K. it is easily seen that Since p(K) is precisely the set a(K) of all 0-extreme points of K, we have by Theorem 7, = minZeCIg)f(x) for every f€ 0. If a1 denotes this common value, then fl155{xeSlf(x) a,) is the 0-convex hull of Kas well where

as of ti(K) (Lemma 5). REFERENCES

1. H. Bauer MinimaisteUen von und Rxtremaipunkte, Arch. Math. 9 (1958), 389-393. 2. , MinimalsteUen von Funktionen und Extremalpunkie. II, Arch. Math. 11 (1960), 200-205.

3. G. Birkhoff and 0. Frink, Jr., Representations of lattices by sets, Trans. Amer. Math. Soc. 64 (1948), 299-316. 4. N. Bourbaki, Topojogie générale, 2e éd., Hermann, Paris, 1951; Chapitres 1, H. 5. M. M. Day, Norin,4 jinear spaces, Springer-Verlag, Berlin, 1958. 6. M. Jerison, A property of extreme points of compact convex sets. Proc. Amer. Math. Soc. 5 (1954), 782-783. 7. J. L. Kelley, Note on a theorem of Krein and Mii,nan, J. Osaka Inst. Sd. Tech.

Part I 3 (1951), 1-2. 8. M. Krein and D. Milman, On extreme points of regularly convex sets, Studia Math. 9 (1940), 133-138.

9. D. P. Milman and M. A. Rutman, On a more precise theorem on the cons pleteness of the system of extreme points of a regularly convex set, Doki. Akad. Nauk SSSR 80 (1948), 25-27. (Russian) 10. P. Samuel, Lfltrafdters and compactification of uniform spaces, Trans. Amer. Math. Soc. 64 (1948), 100-132. 11. M. H. Stone, The theory of representations for Boolean algebras, Trans. Amer. Math. Soc. 40 (1936), 37—ill. 12. H. Wallman, Lattices and topological spaces, Ann. of Math. (2) 39 (1938), 112-126. NORTHWESTERN UNIVERSITY

ON LIPSCHITZIAN MAPPINGS OF CONVEX BODIES' BY

DAVID GALE

Let f be a univalent mapping of one n-dimensional compact convex set C of width w onto another C' of width w' such that (1) IIf(x)—f(y)II k is some positive number and II xli denotes the usual Euclidean norm. The purpose of this note is to prove the inequality This result is intuitively highly plausible, but the only proof we have found

of it, though short, is nonelementary in that it makes essential use of the Borsuk-Ulam Theorem of algebraic topology. Before giving the proof we make some preliminary observations. First, the

hypothesis that f is one-one cannot be omitted, for as shown in the figure below, it is possible to "shrink" a rectangle of width w onto one of width 2w. a

b

f

e

I

2w —÷ 4—2w —' *— 2w —+ w 'I.

d

h

c

b'

c'h'

g

e'

Ii

required mapping cuts off the end rectangles abcd and efgh of C and pastes them together along cd and hg to form the square C', and the middle rectangle bche is then suitably collapsed onto the upper edge of this square The

b'c'h'e'. (This example was suggested by a similar one due to A. H. Cayford.)

It is, of course, immediate that the diameter of C and also the radius of its circumscribed circle will also satisfy inequality (2), and it is trivial to show that this will also be true of the radius of the inscribed circle, though here again the univalent property is necessary, as the above example shows. Univalence is needed here only to insure that the boundary of C is mapped onto the boundary of C', but this implication again involves a moderately 1

This work was supported by a research contract with the Office of Naval Research

under Task NR047-018. 221

DAVID GALE

222

advanced topological argument (invariance of domain). Curiously, the hypothesis that C is convex is not necessary and will not be used. This will turn out to be important later.

To prove the theorem, we first take care of the case in which C' is strictly i.e., each supporting hyperplane to C' meets it in a single point. The general case is then handled by a standard approximation argument. The idea of the proof which was suggested by an argument of Besicovitch for the two-dimensional case (unpublished) is easy to explain. Let us call a pair of points x' and y' of C' antipodes if they lie on distinct parallel supports of C'. Note that for a symmetric body, e.g., a sphere, this definition agrees with the usual one. We will show that there is a chord [x, yJ of C which has the same direction as the width of C and such that the images x' and y' of x and y under f are antipodes. This will establish the result, for we then have IIx—yU w, hence 1k' and since x' andy' are antipodes, C' lies in a strip of width at most II x' — y' II. Q.e.d. To prove the existence of. the chord [x, y] we need, convex,

Let ço be a continuous mapping of the (n — 1)sphere which seParates antipodes. Then for any vector v in there is a Point such that ço(z) — ço(—z) is parallel to v.

ToPoL.oolcAL LEMMA.

into

z in PRoOF.

Define a new mapping .b from Sn_i to R,1 by the rule

Then ç& is clearly a symmetric mapping of Sn-i, i.e., it carries antipodes of

into points which are negatives of each other in R. Now assume, contrary to the conclusion of the lemma, that there is a vector v such that no point in the image of lies on the line through v and the origin. If one now composes .b with the orthogonal projection of R, onto the hyperplane perpendicular to v, one gets a mapping of into an (n — 1)-dimensional space, and one sees easily that this mapping takes antipodes of into distinct points. But this is precisely what the Borsuk-Ulam Theorem says cannot happen, so the lemma is proved. To apply the lemma let be the Unit n-sphere in the space containing

C', and let 0 map the unit vector z onto x', the point of contact of C' with the supporting hyperplane whose outward normal is z (x' is unique because of the strict convexity of C'). Clearly 0 separates antipodes, for otherwise C' would have dimension less than n. Now if = f1 oO then maps S,,_1 onto C and satisfies the conditions of the lemma. Letting v be the direction of the width of C there exists z in such that 60(z) — q'(—z) is parallel to v. Letting x = çe'(z), y = ço(—z) we see that [x, y] is the desired chord. For the case in which C' is not strictly convex we use the fact (see Bonnesen

and Fenchel [1, page 37J) that for any J> 0 there is an convex set B' contained in C' and such that no point of C' is further than 8 from some point of B'. It is then a routine matter to show that the width of C' exceeds that of B' by no more than 28. Let B = '(B') and observe that the pair B, B' satisfies our hypothesis for the strictly convex case. (Note

I

ON LIPSCHITZIAN MAPPINGS OF CONVEX BODIES

that B need not be convex. It is therefore fortunate that this was not needed for our proof!) Hence the inequalities

w' widthB' + 28 kwidthB + 28 kw + 28 hold for all 8, so (2) follows, completing the proof. A slightly stronger result can be proved in which the inequalities (1) and (2) are changed to strict inequalities. The proof goes through as before for the strictly convex case, but the approximation for passing to the general case has to be handled in a slightly more complicated though quite standard way. REFERENCE

1.

T. Bonnesen and W. Fenchel, Theorie

der

Mathematik, Vol. 3, No. 1, J. Springer, Berlin, 1934. BROWN UNIVERSITY

konvezen Körper, Ergebnisse der

NEIGHBORLY AND CYCLIC POLYTOPES BY

DAVID GALE

Introduction. Let S be a finite set of points in n-space. A pair of points p and q of S are called neighbors if the segment joining them is an edge of the convex polytope spanned by S. Some years ago [1), 1 was concerned with

the question of when a set S can have the property that every two of its points are neighbors. It is intuitively clear and easily proved that such

"neighborly" sets in 3-space can have at most 4 points. However, in 4-space it turned out that there are neighborly sets having any finite number of points. To be somewhat more general and also more precise, let S be a finite set in n-space. DEFINITIONS. (1) A subset R of S is called a set of neighbors if there is a supporting hyperplane to S which meets S in R. (Equivalently, R spans a face of the convex hull of S.) (2) The set S is called rn-neighborly if every subset of m points are

neighbors.

The principal result of [1] asserts that in 2rn-space there are rn-neighborly sets of arbitrary finite cardinality. The proof given was somewhat lengthy and indirect. After publication of [11 Professor Coxeter suggested that explicit examples of those rn-neighborly polytopes were perhaps the so-calLed Petrie polytopes [2] (we show this to be the case in §3). An even simpler example of rn-neighborly polytopes can be obtained from the "moment curve" 0 in 2m-space given by

O(t)=(t,tt, In the next section we shall show that any n points on the moment curve form an rn-neighborly seL This fact seems to have been known to many people and probably to Carathéodory as far back as 1911 as indicated in [3). However, the short proof of neighborliness given in the next section has not to my knowledge been published previously. The polytopes obtained as the convex hull of points on the moment curve will be called cyclic polytopes which are a natural generalization of convex

polygons in the plane. We completely describe the face structure of these polytopes and compute the number of their 2rn-faces. This has connections with one of the main open problems in the theory of polytopes, namely, that of determining the maximum number of faces a polytope with n vertices in rn-space can have. It has been conjectured that the cyclic polytopes achieve this maximum. § 2 is concerned with a conjecture of Motzkin that any neighborly polytope

has the same face structure as a cyclic polytope. This can be shown for 225

DAVID GALE

226

polytopes with fewer than 2rn + 4 vertices but there seems to be some difficulty in the general case. In the final section we show that the Petrie polytopes

are neighborly and that they are natural generalizations of the regular polygons in the plane. 1. The cyclic polytope8. Let 1,, •--, t,, be n distinct real numbers and let given by (1) above. be the 2rn-vector ..., 0,,) is m.neighborly. THEOREM 1. The set S = PROOF. Given any m points in S, say 0,, •, 0,,,, we will find a hyperplane through them which supports the moment curve 0, that is, we will find a 2mvector b and a number fib such that

i=l,•,m,

(2)

b• 0(1) + fi,, > 0

for t * t17i = 1, •-, m.

Let p(t) be the polynomial defined by

+1k". Clearly p(t) is non-negative and vanishes only for I = I = 1, - - •, ,n. Letting ", fl2rn-i, 1) it is clear from (3) that b and satisfy (2) as required. We shall call a polytope of the type given above a cyclic Polytope and we

b = (Ps, /3k,

-

proceed to describe the face structure of such polytopes. Our first result applies not only to cyclic but to any neighborly polytope. The Points of an rn-neighborly set S in 2m-space are in general

THEoREM 2.

Position. PRooF. Suppose some set R of 2m + 1 points of S lies in a hyperplane. The set R is rn-neighborly since S is, but R would then be an rn-neighborly set in a space of dimension 2m — 1 and as proved in [1, § 3], such a set can contain at most 2m points, giving a contradiction.

The faces of a neighborly Polytope are simplexes.

We shall next determine exactly when a set of vertices of a cyclic polytope spans a 2m-face. For this purpose it is convenient to assume the numbers

occur in increasing order so that the points 01 , 02, •, 0,, occur in order on the moment curve. Let R be any subset of S containing 2m points. THEOREM 3. The Points of R are neighbors and only if (3) for any two Points 0,, 0 not in R there are an even number of points of and 0,. R between Let R = -. -, We consider the polynomial 2w,

P(t)

Letting b = (SB,,

-

•, thm)

fl(t

— ti,.)

=

it is clear as in Theorem 1 that the hyperplane

through R is given by the equation

NEIGHBORLY AND CYCLIC POLYTOPES

+ Pa = 0

b

P(11) + and this hyperplane will support S if and only if the numbers not in R. Suppose is not in R and p(t,) is are all of the same sign for positive (negative). This means from (4) that there are an even (odd) number But if 0 is also not in R then there will be an greater than of roots, even (odd) number of roots, 1,,, greater than I if and only if there are an even number of these roots between and tj which is precisely the condition in the statement of the theorem. There is a simple schematic representation to illustrate Theorem 3. Imagine

a set of n beads strung on a circular string and suppose 2m of them are black and the rest white. The black beads will then correspond to a face of S if in this necklace they occur in intervals of even length. It is clear from Theorem 3 that all cyclic polytopes in 2rn-space having the same number of vertices are combinatorially isomorphic, that is, there is a one-one correspondence between vertices which induces a one-one correspondence between faces. Motzkin has raised the question as to whether every rn-neigh-

borly set in 2m-space is combinatorially equivalent to a cyclic polytope. He indicated an approach to this problem which yields a partial result to be taken

up in the next section. We conclude the present section by computing the number of (2m — 1)-faces of the cyclic polytopes. THEOREM 4. The number of (2m — in 2m-space is given by the formula

1)-faces

of a cyclic Polytope with n vertices

PROOF. (We assume, of course, that n > 2m.) Let R be a face of S and let


correspond to the points of S which are not in R. We

distinguish two cases. Case 1. i1 is even. Then the roots I, of (4) are divided into n — 2m + 1 and intervals, those less than '$1, those between •--, those greater than We thus have n — 2m + 1 even integers whose sum is 2m, or dividing by 2, we have n — 2m + 1 integers whose sum is m. Conversely any such set of numbers will yield a face. But the well-known formula for the number of ways of writing a non-negative integer a as the sum of non-negative integers gives ((n—2m+l)+rn—l\ — (n—rn (n—rn

\ (n—2m+i)—1

Case 2.

i1 is odd.

)\n—2m/\

It then follows that R is divided into n — 2m + 1 intervals

the first and last of which contain an odd number of elements of R while the rest contain an even number. Thus and belong to R. If we throw 0, and away we are back in case 1 again where n has been reduced by 2 and m has been reduced by 1: The formula thus becomes fn — m — 1

\

rn—i

DAVII) GALE

228

and adding this to the expression in case 1 we get the desired formula. REMARK 1. As already noted, it has been conjectured that (5) gives the maximum number of faces for any polytope with a vertices in 2;n-space. REMARK 2. Mr. Martin Fieldhouse in an unpublished paper has shown that any in-neighborly polytope in 2rn-space must have the number of (2,;z — 1)faces given by (5), and more generally he has computed the number of faces in all dimensions just from knowing that the polytope is in-neighborly. This may be a further indication that all neighborly polytopes are cyclic. 2. Neighborly polytopes which are cyclic. In this section we shall understand by a cyclic polytope one which is combinatorially isomorphic to a polytope described in the previous section. In other words, the set of points S in 2rn-space spans a cyclic polytope if its points can be cyclically ordered in such

a way that a subset of 2rn points are neighbors if and only if they form a collection of disjoint "even intervals" in this ordering, as described in § 1. We shall show that if S is in-neighborly and has fewer than 2m + 4 vertices then it is cyclic. The case n = 2m + I is trivial for in this case we have a 2;n-simplex and if the vertices are ordered in any manner whatever the "evenness condition" of Theorem 3 will be satisfied for all 2;n element subsets. To treat other cases we shall need the following general fact about convex polytopes. LEMMA 1.

space.

For k

Let S = (a1, . . ., a finite Point set in general Position in min a set R of k points arc neighbors if and only if the equations = 0,

(6) (7)

have no nonzero solution with A1

0 for a, e S — R.

The necessity of the condition is clear for if R is a set of neighbors then for some vector b and number p0 b

.

a1

+ Po

for

0

in S

with equality only for a1 in R. Taking scalar product of (6) with b, multiplying (7) by and adding we would then have a positive quantity equal to zero, a contradiction. The proof of sufficiency requires some fairly standard juggling with linear inequalities. For details see [1, Lemma 1). COROLLARY. The set S is k.neighborly if and only if every nonzero solution of (6), (7) has a/least k + I Positive and k + 1 negative coordinates.

PROOF.

Suppose there is a solution with only v negative coordinates, say

A1, .- ., A.,, where v

k

m; then a1,

•,

are not neighbors from the lemma.

If there is a solution with k or fewer positive coordinates then its negative is a solution with k or fewer negative coordinates and the same argument applies.

229

NEIGHBORLY AND CYCLIC POLYTOPES

We

now consider the case of 2rn + 2 points 2rn

a

•,

2rn

unknowns. If S is rn-neighborly then every solution of (6), (7) has exactly rn + 1 positive and rn + 1 negative coordinates and must therefore be unique up to multiplication by a scalar (otherwise we could produce a solution with of (6), at least one coordinate zero). Picking a particular solution (As, -••, (7) we call a, positive or negative according as A, is positive or negative. Now order the a, in any manner as long as the positive and negative terms alternate in the ordering. When will 2m points span a face? From the lemma this will occur precisely when the remaining two points have opposite signs, but because of the way the points were ordered this will happen only when the 2m points in question satisfy the evenness condition of Theorem 3. The case n = 2rn + 3 is somewhat more complicated. Let us denote by A the set of all solutions of (6), (7). Since we have 2m + 1 equations in 2rn + 3 unknowns A has dimension at least 2, and again every vector in A has at least rn + 1 positive and m + 1 negative coordinates. It follows that A cannot contain more than two independent vectors or we could form a linear combination with two coordinates zero. be a basis for A. We may choose c Let b=(j91, 0 for all i and > 0, We now reorder the vectors a, so that so that < 192/7,

< - -. <

this will turn out to be the required ordering. The fact that all ratios in (8) are distinct follows again from the fact that no vectoi in 4 can have

and

more than one zero coordinate. We must now show that the faces of S satisfy the evenness conditiOn with respect to the ordering given by (8). This is done by describing the "sign

configuration" of the vectors in 4, that is, the order in which positive and negative coordinates occur in these vectors. We have already noted that at most one coordinate of a vector in A can vanish. Further for each coordinate there is a vector in A vanishing at that coordinate and this vector is uniquely determined up to a scalar multiple, for if not we could again produce a vector with two zero coordinates. Calling a vector with a zero coordinate a singular vector we will show LEMMA 2. The singular vectors of 4 have coordinates which alternate in sign, i.e., deleting the zero coordinate leaves a vector whose adjacent coordinates have

oPposite signs.

Assuming

the lemma for the moment let us establish that S satisfies the

evenness condition. First, knowing the sign configuration of the singular vectors we easily obtain the sign configuration for all vectors of A since any such vector can be obtained by "perturbing" a singular vector in such a way

that none of the nonzero coordinates changes sign. It then follows from Lemma 2 that any vector in A has coordinates which alternate in sign except for one pair of adjacent coordinates which have the same sign. It is also clear that all possible sign configurations satisfying this condition can be

DAVID GALE

230

realized by suitably perturbing the appropriate singular vector. It is now easy to verify the evenness condition. From Lemma 1 we know that given any three points , a2 the remaining points will be neighbors provided that no vector in A has 0. Now if the subscripts i1, i1 satisfy the evenness condition then the corresponding coordinates cannot all be positive because from what we have seen concerning sign configurations at least one pair, say and A,2, are end points of an even interval on which coordinates alternate in sign. If and do not satisfy the evenness condition then two of the three intervals into which the a, are divided will contain an odd number of points. We can then find a vector in A with 0, for if, say, the interval from i1 to i2 is even we choose a vector in A having adjacent coordinates of the same sign for some pair on this interval. It follows that and will have the same sign, as will since the interval i,, i1 is odd. Hence by Lemma 1 again the complement of the set a,2) will not be a face. PROOF OF LEMMA 2. Let A be a real number and define the vector d(A)

b — Ac

=

(ö1(A),

.

. .,

For A small, in fact for A d(A) will have the same sign configuration as c. As A increases the coordinates of d(A) will change sign successively as A passes through the points Pi/ra, PuTt, etc. Since is positive it follows that ö1(A) will change from positive to negative as A passes through Next .4 passes through PuTt and o3(A) changes sign, necessarily from negative

to positive for if it also changed from positive to negative we would have encountered two successive sign changes from positive to negative. But since we started out with at most m + 2 positive coordinates we would then be reduced to m positive coordinates which from the corollary to Lemma 1 cannot happen for a neighborly polytope. Thus changes from negative to positive as .4 passes through P2/Ti and by the same argument goes from positive to negative as .4 passes through !9s/Ts, and so on. This argument shows that at each stage the coordinates of d(A) alternate in sign except that if thir,
(

\

cosO —sin0

sin0 cosO

Let e be the unit vector (1, 0). Then the vertices of

are the vectors

e, eR, el?, . . ., eR"', or in coordinate form (1, 0), (cos 6, sin 0), (cos20, sin 20),..., (cos (n — 1)0, sin (n — 1)6). It is clear that is carried onto itself by the

NEIGHBORLY AND CYCLIC POLYTOPES

cyclic group generated by the rotation R. Again letting 0 =

We now define an analogous notion in 2rn-space.

we

define the rotation R by the matrix

cos8 sin0 —sinO cos8 cos28 sin 28 —sin 28 cos28

cosm8 sin mO —sinmO cosm0

One easily verifies that the matrix is orthogonal and has order Let e be the vector (1, 0, 1, 0, --• 1, 0) in 2m-space and let be the polytope spanned by e, eR, - ••, A simple inductive xerification will show that (4)

eRk

=

(cos ho, sin ho, cos 2k0, sin 2*8, -• -, cos rnkO,

sin mM).

It is again clear that is carried into itself by the cyclic group of rigid motions generated by R. We shall now show THEOREM 4.

is rn-neighborly.

PROOF. We must find a supporting hyperplane to passing through any m of its vertices. We shall as in § 1 prove something more general. Consider the curve ço in 2m•space defined by coO) = (cos 1, sin 1, cos 21, sin 21, - •, cos

ml, sin ,nt)

and let , . •-, be any rn-numbers on [0, Then there is a supporting hyperplane to meeting 'p in exactly the points co(lz), co(tz), .. -, To see this we consider the trigonometric polynomial P0) = [1(1 clearly P(t)

—

cos(t —

0 and equality holds only if I = Ik, k =

1,

-

•, rn.

trigonometric algebra

But by elementary

+a0 for suitable numbers ak and It then follows that the hyperplane with the desired properties is defined by

a-x+a0=0

wherea=(al,pl,at,p2,-.-,aM,.PM).

It is also possible to show by an argument similar to that of Theorem 3 that the polytope defined here is cyclic. We omit the proof.

232

DAVID GALE BIBLIOGRAPHY

1. D. Gale, Neighboring vertices on a convex polytope, Linear inequalities and related systems, pp. 255-263, Princeton Univ. Press, Princeton, N. J., 1958. 2. H. S. M. Coxeter, Introduction to geometry, Wiley, New York, 1961, p. 412. 3. C. Carathéodory, Uber den Variabiiitdtsbereich der Fourierechen Konstanten von positiven harmonisehen Funktionen, Rend. Circ. Mat. Palermo 32 (1911), 193-217. BRowN UNIVERSiTY

MEASURES OF SYMMETRY FOR CONVEX SETS1 BY

BRANKO GRUNBAUM 1. Introduction. In the study of convex sets in Euclidean spaces the term "measure of asymmetry" (or "symmetry") has appeared quite frequently (see, e.g., Besicovitch Eli, Eggleston [11, Stein (1]). Although no definition of

"measure of asymmetry" was given in any of those papers, it is intuitively clear that the term was meant to denote a non-negative real-valued function defined for convex sets, with value 0 for centrally symmetric sets and only for such sets, and possibly satisfying some additional conditions. One of those supplementary conditions should apparently be that the value of the function for a set which is "less symmetric" be greater than for a "more symmetric" one. The interpretation of the terms in quotation marks is left vague on purpose, except for supposing that the maximum can be attained for triangles (in the plane), or simplices in general, and for them only. An examination of the literature reveals quickly that the number of measures of asymmetry actually studied was much greater than the frequency of the appearance of the term. The first (and, in a certain sense, most important) such "undeclared" measure of asymmetry goes back to Minkowski [1] (see below). Ever since, particular measures of asymmetry, declared or undeclared as such, have been considered by various authors. Many facts were discovered, and forgotten and rediscovered, again and again. One of the main objectives of the present paper is an attempt to summarize those known results, and to give appropriate references. Despite all his efforts aimed at completeness, the present author has no illusions about the in this endeavour. As a consequence, any communication degree of of supplementary references (and corrections of any kind) would be greatly appreciated. A glance at the references given in § 6 will show how much effort and time, at least potentially usable to better purposes, was spent during the years on rediscovering known proofs of known results. Another aim of the present paper is to give a more precise definition of and to examine some general procedures for de"measure of fining measures of asymmetry. At least a part.way systematization of certain types of measures of asymmetry is given, leading to appreciable savings in space in the summary of known results. As in any other domain, a collection of known results points out, by contrast,

the holes in the present knowledge. Since the gaps will be obvious to the reader in the same way they were to the writer, the sentence "it would be interesting to find •.," or any other formal statement of open problems of This work was supported in part by a grant from the National Science Foundation, USA (NSF-G18975). 1

233

BRANKO GRtJNBAUM

234

this kind, was avoided. On the other hand, the nature of the subject leads quite naturally to connections with many other fields and problems, beyond the rather specialized topic of measures of asymmetry. The author felt free to digress at some length into those areas, and to point out some of the problems he believes to be still open and seem interesting to him. Some measures of asymmetry may be defined for all compact convex sets;

for sake of simplicity and uniformity we shall restrict our attention to the Euclidean space of some fixed dimension n, and consider measures of asymof all convex bodies (i.e., compact convex sets metry defined on the family with nonempty interior) in Euclidean n-space Mainly for reasons of subjective preference, we shall discuss measures of sym,netry instead of measures of asymmetry. In any given case, the transi-

tion from a measure of symmetry to a related measure of asymmetry, and vice versa, is easily accomplished.

One way of classifying measures of symmetry is by the groups of transwhich leave the measure of symmetry of every formations (acting on member of r invariant. Though there is no a priori reason for it (nor any other, altogether, except pragmatic ones; see the Appendix), only measures of symmetry invariant under regular affine transformations, and those invariant under all similarity transformations, have been considered in the past and will be considered here. We proceed, finally, to define measures of symmetry.

A rest-valued function I defined on the family sr of all convex bodies in is an affine [similarity] invariant measure of symmetry provided: 1 for all KeSr; 0 has a center of symmetry; (2) 1(K) = 1 if and only if K€ (3) 1(K) =f( T(K)) for every K .Q" and every nonsingular affine [similarity] transformation T of onto itself; (4) 1(K) is a continuous function of K.

(1)

Condition (4) obviously implies that a topological structure is imposed on of classes of members r, or, rather more precisely, on the space 3e: [resp.

of r, consisting of convex bodies equivalent under regular affine [resp. onto itself. A discussion of the topology similarity] transformations of and E is deferred to § 2. (and metric) of It is to be noted that if h(2) is any homeomorphism of the closed interval [0, 11 into itself, with h(1) = 1, and if f is any measure of symmetry, then h of will also be a measure of symmetry. In view of the generality of the above definition of measures of symmetry, it seems necessary to adopt one of the procedures described below in order to obtain results of any geometric significance. One course, actually followed in all previous investigations of measures of symmetry, is to use only functions derived from geometric quantities determined by members of W', in such a fashion as to measure—in one intuitively appealing way or another—the departure of a set from symmetry. § 6 of the present paper deals with this approach to the problem and with

MEASURES OF SYMMETRY FOR CONVEX SETS

results obtained in this direction. Another way would consist in trying to subject "measures of symmetry" to additional restrictions, with a two-fold purpose: first, to be able to derive some general properties of measures of symmetry satisfying the additional

conditions; second, to be able to distinguish, among the many "geometrically" definable measures, those whose behavior is more or less similar. In § 4 we shall discuss an attempt in this direction, and some rather interesting related problems. But already now it seems worth while to point out that no additional condition is known which is satisfied by all the intuitively "proper" measures of symmetry treated in the literature. §3 deals with "invariant points and sets," a notion which is useful in discussing measures of symmetry and is also of some independent interest. In § 5 we give a short outline of the most important general methods used to define "geometrically" measures of symmetry, and specify some points of terminology. This allows a relative condensation of § 6, in which we present the known results on special measures of symmetry. The rest of the paper deals with some functionals which, possibly, are measures of symmetry, and

some which (somewhat surprisingly) are not. In the Appendix some generalizations of the main topic are discussed. 2. Distance-functions for spaces of convex sets. The set (li" of all compact convex sets in E' is usually metrized by the Hausdorif distance, in either of the following two (topologically equivalent) variants (B" denotes the solid unit ball in E"):

(i)

d(K1, K3)= min{A

K, c K + 2B";

c K1 + AB") (see, e.g., Blaschke

12));

(ii)

e.g.. Eggleston 151). For the topology determined by either of these distance-functions the wellknown "selection theorem" of Blaschke [21 asserts the compactness of the set of all members of (1" contained in a (fixed) compact set.

Another distance-function (strictly speaking, on the space of measurable subsets of E" having finite and positive measure) was considered by Dinghas [1]; on the space = of convex bodies in E" it is defined by d(K, , V(KI u IC,) — V(K1 n K,), where V(K) denotes the n-dimensional volume of K. These distance-functions do not seem to transfer in any usable form to the spaces S? and E of classes of affinely [resp. similarity) equivalent convex bodies in E" (though Blaschke's selection theorem is used in establishing the compactness of E). Better adapted to those spaces are the distance-functions described in the following lines. 1. For K,, K, 3?' let J,(K,, K,) = jnf { V( T(K,))/ V(K,): K, c T(K,), T regular affine map of E' onto itself}. Then ln o,(K,, K,) is an asymmetric distance function, and d,(K,, K,) in o,(K,, K,) + ln J,(K,, K,) is a distance-function on It was considered, e.g., by Macbeath [1) in the proof of the compactness

of E, and by Levi [1]. 2. Of a similar natuie is the following procedure, considered by Asplund

BRANKO GRUNBAUM

[lj for centrally symmetric sets (see also § 7). For K1, K2 E s", let ö2(K1, K2) = a translate of T(K2), T regular inf {I a translate of T(K2), aK1

affine map of E" onto itself}. Then d2(K1, K2) = In o2(K1, K2) is a distancefunction on E. (Obviously, d1(K1, K2) nd2 (K1 , K2); it is easy to see in a direct way that d1 and d2 define the same topology, a fact which follows also from this inequality and general properties of compact spaces.) 3. Another procedure was recently described (with generalizations to sets which are finite unions of members of by Shephard [1]. For K1, K2 let and d3(K1 , K2) = inf {ln (1 + 2ô3(K1, T(K2))): T regular affine map of

onto

It is not hard to see that the topology induced in by d3 coincides with that determined by d1 or d2. The determination of the diameter of E in either of these distance-functions (or the variety of other distance-functions which may be devised along similar lines) seems to be exceedingly difficult; the problems are open even in the case n = 2. For the d1-diameter of E the upper bound 2n In n results from Macheath [1]; for n = 2 it was given also in Levi [11 and Fulton-Stein [I]. In the twodimensional case this estimate may be improved to ln 9 by noticing that where H denotes the regular hexagon. The d1(K, H) In 3 for every Ke last inequality is obvious on account of the existence of an affine-regular hexagon inscribed into each Ke (see § 3) and Figure 1. itself}.

FIGURE 1

A similar argument, using inscribed affine-regular octagons (GrUnbaum (1]; are centrally symmetric then d1(K1, K2) see also § 3), shows that if K1 , K2 2), improving the estimate ln 4 given by Levi [1]. ln (3/2 +

Many special problems related to distances of certain geometrically interesting sets to other sets have received attention. Let K* [Ma] denote any 4*, [centrally symmetric] member of s", let be the cube, simplex, ball, resp., of dimension n, and let H be the regular hexagon. Then

MEASURES OF SYMMETRY FOR CONVEX SETS 42)

o1(C2, K2)

o,(C3, K3)

9/2,

K')

n',

=

2

Fulton-Stein [1], Süss [41; Bielecki-Radziszewski [11;

,

Macbeath Li), }ladwiger (1); Estermann [21, Levi [i];2

o1(K2, C2) o1(42, C2) = 2 , o1(M2, C2) o1(H, C2) = 4/3

o1(K', C')

237

Petty [1);

,

Radziszewski [1] for

,

Macbeath [1).

Hadwiger [1] strengthened this to the assertion that for every K' there exists

a rectangular parallelotope containing K', with volume not greater than n! V(K').

(*)

The same result was obtained by Kosiñski [1).

o1(K2, 42)

ô1(C2, 42) = 2

Gross

,

For ô1(K', 4') only the trivial upper bound n' seems to be known. ö1(4", K')

o1(4', B')

d1(K', if)

d1(B',

Gross [1], Knothe [1], Macbeath Ill.

4') = n In n While max o1(K', 4') is not known, it is interesting to note the following difference between approximations of convex bodies by inscribed simplices, and by circumscribed ones. According to (**) above, the ball B' is, for all n, worst to approximate by inscribed simplices. For circumscribed simplices, however, one easily finds ö1(C', 4') n"/n!, and 1) 4') n"2(n +

which is greater than n'/n! for n 8.

Thus, while according to (*) parallelograms are the worst plane sets for approximations by circumscribed simplices, in sufficiently high dimensions balls are worse than hypercubes. But even in high dimensions it seems to be rash to conjecture that max .31(K', 4") = 51(B', 4'). Let Q' denote the n-dimensional octahedron ("=cross-polytope); we were unable to find for o1(Q', 4') any upper bound smaller than the rather trivial (n — l)'/2''(n — 2) (for n 3) which is, for large n, much bigger than ôt(B', 4'). On the other hand, the conjecture that 0' is worst in all dimensions is contradicted by o1(Q8, 42)

2 < i51(B3, 43) =

For similar results on o2 and d2 see § 7.

If the maps T occurring in the definitions of d1, d2, d3 are restricted to similarities, distance-functions on are obtained. Again, the same topology is induced by all of them. In contrast to E, the space n> 1, is locally compact but not compact. is the union of countably many compact sets; to obtain a representation of this type it is sufficient to consider those elements

for whose members the ratio of the diameters of a circumsphere and an insphere is at most k, for k = 1,2,3, '..) of

The compactness of E implies that the bounds of any continuous functional on E are attained; in particular, for each affine invariant measure of symmetry

2 See Note 1 on p.

264.

BRANKO GRUNBAUM

there exist sets in E ("extremal sets") which have minimal measure of symmetry. This property does not hold for similarity invariant measures of symmetry (see § 6).

Invariant points and sets. a convex body Ke .Q" is an example of the follow3.1. The centroid g(K) 3.

ing concept.

Let a point p(K)EE* be determined for each convex body Ker in such a way that (i) if T is any nonsingular affine map of E" onto itself, then p(T(K)) = T(p(K)); (ii) p(K) depends continuously on K (in the topology of E); then p(K) shall be called an affine-invariant (a.i.) point of K. a closed subset of Similarly, if p(K) is, for each convex body satisfying (i) and (jj*) p(K) is a lower semi-continuous function of K; then p(K) will be called an affine-invariant (a.i.) set of K. Restricting the transformations T in (i) to similarities (Le., orthogonality

preserving maps), one obtains the definitions of similarity invariant (s.i.) points and sets of K. Many extremal problems lead to points or sets which are affine or similarity invariants of K. Such are, e.g., the critical points and sets, and the level surfaces, of the measures of symmetry discussed below. Also, the (unique) ball [resp. ellipsoidj of minimal volume containing K is an s.i. [resp. a.i.J set of K; therefore, their centers are correspondingly invariant points of K. An example of an s.i. set which satisifes (jj*) but not (ii) is the set of centers of all balls of maximal volume contained in K. Before proceeding to an enumeration of the a.i. and s.i. points and sets which shall find mention in connection with measures of symmetry of convex bodies, we shalt discuss some other problems about these points and sets. 3.2. For convex bodies ,in the plane there exists an s.i. point k(K) which is, moreover, additive, (iii)

satisfies

k(K + K') = k(K) + k(K').

This point, first considered by Steiner [1], is the curvature-centroid of K (i.e., the centroid of a masé-distribution on the perimeter of K, whose density is determined by the curvature of the boundary of K). The curvature centroid

was shown by Steiner to be related to solutions of a number of extremal problems; other results of the same nature were obtained by Su [1). Kubota [JJ proved that k(K) can be characterized by the vanishing of the coefficients of cos 0 and sin 0 in the Fourier-series development of the support function of K. From this it is easy to deduce the additivity property of k(K). In higher dimensions, the expansion of the support function of a convex body K in a series of spherical harmonics leads to an analogously defined point k(K) such that the coefficients of the first-degree members vanish if k(K) is taken as origin. This k(K) is obviously an s.i. point of K, and satisfies

MEASURES OF SYMMETRY FOR CONVEX SETS

(iii).

In V the point k(K) is the centroid of the Gauss curvature of the

boundary of K (Gericke [2J). The geometric properties of k(K) in higher dimensions do not seem to have been investigated. Another problem which seems to be open is whether (in the plane, or in higher dimensions) k(K) is the only s.i. point satisfying (iii). The additivity of k(K) is of interest in connection with certain properties of some similarity invariant measures of symmetry (see 4, 5,6). Considering the related properties of affine invariant measures of symmetry, one is naturally led to the question whether there exist a.i. points p(K) with the additivity property (iii). The negative answer follows at once from the following remarks. (a) The only a.i. point of a triangle, or a square, is its centroid g(K). denote the sets represented in Figure (b) Let T..0, T,.. and

2. Obviously 7'.., + T,,. = K.+, and therefore, if p(K) had property (iii), we When a —*0 then p(K.+p) — = ((a + (a + would have while tends to a square S for which p(S) =

(40)

(p,0

0)

FIGURE 2

Open, and more difficult, is the question whether there exists an a.i. set (with p(K) not identical with K) which satisfies (iii), or the weaker

P(K)

property

p(K+K')='p(K)+p(K'). If K is a convex body such that there exists a group of affine maps of K onto itself which leave one and only one point of K fixed, then obviously all the a.i. points of K coincide (with that fixed point). Similarly for s.i. points. This implies, in particular, that centrally symmetric sets, and simplices, have a unique a.i. point, an observation which may be used to compute the value of some measures of symmetry for simplices (see, e.g., Ffiry-Rédei [1J). The

question whether the cOnverse of the above statement is true seems to be The same holds for the following problems. If K is a convex body let K° denote the set of all points of E* (individually) fixed under every affine map of K onto itself. Is the family of all a.i. points open.

BRANKO GRONBAUM

= {p(K))? In particular, if K = E', is then a, = 1, m, and reals a1 with or K, I I is another al. point of K. Does there exist,

p(K) big enough to ensure K°

Given a.i. points the combination

among all the a.i. points of K, a finite "basis" such that each a.i. point is a combination of them' The same types of problems arise with respect to s.i. points of k. An additional set of problems is exemplified by the following. For every convex body Ke Ste, with centroid g and circumellipse-center c, it is easy to show that g + 2(c — g) K; on the other hand, it is easy to verify that for the set K on Figure 3, g + 6(q — c) E K. What is the greatest interval [a, P1 the relation g ÷ 2(g — c) K such that for every 2 E [a, j3] and every holds?

(3,6)

(12,0) (6,0)

(3,—C) FIGURE 3

A similar problem was suggested by Fenchel (see Pal [1]): In a convex set K€ .Q' of diameter 1, how far can the centroid of K be from the center of the circumcircle? A partial answer was given by Pal [I); for a related problem, and an improvement of Pal's result, see Grtinbaum [7). 3.3. The most important a.i. and s.i. points considered in the literature are the centroids (centers of gravity) of different mass-distributions related to geometric properties of convex bodies. (1) The centroid g = g(K) of a convex body Ke Sr is the centroid of a homogeneous distribution of mass throughout the convex body K. A summary of older results on the properties of the centroid g(K) is given by BonnesenFenchel El, p. 53]. It would lead us too far to attempt to summarize here

the more recent results on this subject (see, however, pp. 249

and 255 for some open problems). (ii) The surface-area cent roid a(K) of a convex body Ke Sr is the centroid

of a homogeneous distribution of mass on the boundary of K. The surface-

MEASURES OF SYMMETRY FOR CONVEX SETS

241

centroid received much less attention than g(K). Of the relevant results, not contained in Bonnesen-Fenchel [1], one should mention those by Bose and area

Roy (published in a series of papers in Bull. Calcutta Math. Soc., starting with Bose-Roy [11) which generalize and unify many previously known theorems. It seems quite certain that the results of Bose and Roy are susceptible to far-reaching generalizations.' A problem, representative of a whole class of questions about whose solu-

tions nothing seems to be known, is the following. Does there exist, for each convex body K c a nonsingular affine transformation T of onto itself such that a(T(K)) = (iii) The curvature centroid k(K) of a plane convex body K is the centroid

of a mass-distribution on the boundary of K, with density proportional to the curvature. (See p. 238 for further properties.) For convex bodies in there are n — 1 analogs of k(K); very little seems to be known about their properties.

circumellipsoid-center c(K) and inellipsoid-center i(K).

It is well exists a unique ellipsoid of minimal volume containing K,—the Lowner-ellipsoid of K. Its center c(K) is an a.i. point of K. The existence of the Löwner-ellipsoids was established by different authors (Bebrend [1; 2) for n = 2; in the general case, John [1], 3.4.

known that for each convex body K€ r

there

Danzer-Laugwitz.Lenz [1], Zaguskin [1]).

In a sense dual to Lowner's ellipsoid is the unique ellipsoid of maximal volume contained in the convex body K This ellipsoid was considered by Behrend [1; 2] for n = 2, and by Danzer-Laugwitz-Lenz [1] and Zaguskin (1] for all n. Its center i(K) is an a.i. point of K. Of considerable interest is the following theorem, first established by John 11] (see also Leichtweiss [2]): For each [centrally symmetric) convex body Ke .Q" there exists an ellipsoid F c K such that the concentric ellipsoid F5, homothetic to E in the ratio n [resp. I contains K. If K is not a simplex, the minimal ratio is less than n. (In other words, the d, distance (see § 2) of the ball to any element of W' is ln n, while the distance of B* to any centrally symmetric element of R" is In v'i. See also § 7.) It is well known (see John [1], Leichtweiss [2]) that the Löwner-ellipsojd of K may be taken for F5 in John's theorem. On the other hand, it is an easy exercise in analytic geometry to prove that the ellipsoid of maximal volume contained in K verifies John's theorem if taken as E. This remark obviously and implies the statements about (on p. 247). 3.5. The following two a.i. sets should be considered in connection with measures of symmetry of convex bodies in the plane. (i) The set of sixpartite points s(K). R. C. Buck and E. F. Buck [1] called six-partite point of K each point s with the property that there exist three straight lines through s such that each of the six wedges determined by them Contains 1/6 of the area of K. The existence of at least one six partite point follows from simple Continuity arguments applicable not only to

'

See

Note

on p. 264.

242

BRANKO GRUNBAUM

the partition of the area of a convex body, but also to the partition of an arbitrary, continuous, distribution of non-negative masses; with a slight change

in the definition ("each of the six closed wedges contains at least 1/6 of the total mass") it applies to arbitrary distributions of non-negative masses. Sixpartite points, and some of their properties, have been considered by YaglomBoltyanskii [11, Eggleston [2; 3; 51, Newman [11. A convex set may have more than one six-partite point (Eggleston [3, p. 126]);

no characterization of convex bodies with a unique six-partite point, or of sets which may serve as the set of all six-partite points of a convex body, seem to be known. No generalizations of the notion of six-partite points to higher dimensions have been established; for some problems in this direction see Grunbaum [3]. One analogous notion seems to deserve mention. In the plane, for each continuous mass-distribution there exists at least one pair of mutually perpen(licular straight lines such that each of the four wedges contains 1/4 of the

total mass. Does this generalize in any way to mass-distributions in E" and to the 2" wedges formed by n concurrent hyperplanes? The problem seems to be open even for E3. The only known result is that of Zindler [1]: For every convex body K e and for every plane P1 bisecting the surface-area

of K, there exist planes P2 and P3 such that each of the eight wedges determined by P1, P2. P3 contains 1/8 of the surface area of K. ii The set of hexagonal points h(K). An affine-regular hexagon is the image of a regular hexagon under a nonsingular afline transformation. Simple arguments prove that each convex body Ke.Rt admits at least one

affine-regular hexagon (i.e., such that its vertices belong to the boundary of K). This result was established by very many authors, usually a lemma needed for other results in the geometry of numbers or Minkowski see Besicovitch [1], Fáry [I), Laugwitz [fl, [1) for typical applications. The center of any affine-regular hexagon inscribed into K is

called a hexagonal point h(K) of K. The set of all hexagonal points of K is an a.i. set of K. For a centrally symmetric convex body KE each boundary point of K is the vertex of at least one inscribed affine-regular hexagon. (For further results on inscribed affine-regular hexagons and other

regular or affine-regular sets see, e.g., Asplund-Grünbaum Li), Böhme [1], Eggleston [41, GrUnbaum [ii, Kelly [1], Pucci ElI.)

No characterization of convex sets with exactly one inscribed affine-regular hexagon, or of those with only one hexagonal point, seems to be known; the set of all hexagonal points may be a 1-dimensional continuum, but the general nature of such sets is still undetermined. Another unsolved problem, whose affirmative solution would probably be of interest in various branches of combinatorial geometry, is as follows. The affine-regular hexagon is the vector sum of a triangle and its- negative. In analogy with the above, does there exist for each convex body KE ,!rl" at least one inscribed "hyper-hexagon" (i.e., vector sum of a simplex and its negative)? The question is open even in E3, where the "hyper-hexagon" is a nonsingular affine transform of the Archimedean "cubooctahedron."

MEASURES OF SYMMETRY FOR CONVEX SETS

243

It should be mentioned that for circumscribed affine-regular hexagons results

completely analogous to those mentioned above hold (L. Danzer, oral communication; it seems that these results are not to be found in the literature). In the same way, the results (Bthme [1], Grünbaum [1]) on inscribed aflineregular pentagons and octagons have their analogues for circumscribed ones.

Another direction which seems to be completely unexplored is that of existence of inscribed affine-regular hexagons (and similar sets) into arbitrary closed Jordan curves. 4. A property of some measures of symmetry. In trying to find additional conditions which are to be satisfied by a function in order to qualify for the

name "measure of symmetry," the following property was accepted as "reasonable" by quite a few of the people we consulted. "The addition of a 'more' symmetric set to a 'less' symmetric one should not decrease the measure of symmetry of the latter set." More precisely, this "superminimality" condition is: For all K1, K2 W', mm {f(K,), f(K2)}.

f(K1 + K2)

Though an "objective" justification for imposing, or trying to impose, this condition is missing (and seems rather likely to remain so), the intuitive appeal of (*) seems to be sufficient to explain our interest in it. One's enthusiasm is likely to cool off somewhat after noticing that such intuitively "right" measures of symmetry as that of Kovner-Besicovitch (see § 6) fail to have the superminimality property.

We feel that, despite this, the superminimality is of more than marginal interest. Our enlarging on it is due mostly to the fact that (*) is stringent

enough to insure that every measure of symmetry (as defined by conditions (1) to (4) in § 1) satisfying it has at least a certain resemblance to the tive notion of a measure of symmetry. More precisely, the following theorem holds: (A) For every affine [similarity] invariant measure of symmetry for which satisfies (t) we have mm {f(K): Ke = f(simplex) [resp. inf {f(K): Ke =

inf{f(T): Ta simplex cE)J.

Theorem (A) is an immediate consequence of (B) Every convex body in Et is the limit (in the Hausdorff metric) of finite

FIGURE 4

BRANKO GRONBAUM

244

sums of triangles.

The last assertion follows at once from the well-known and elementary fact (Blaschke [5, p. 250], Yaglom-Boltyanskii [I]): (C) Every convex Polygon in E2 is the sum of finitely many segments and triangles. (See Figure 4.) Theorem (A) allows us, for measures of symmetry satisfying the mality condition, to determine the lower bound of f(K) for KE by considering triangles only; in some cases this is a far-reaching simplification of the problem (see § 6). The following remarks seem to be in order. 1. For all the affine invariant measures of symmetry on discussed in the literature, the simplex (triangle) is the only extremal body (see, however, p. 256). This brings near the question whether the simplex is the only cxtremal body for every measure of symmetry satisfying (*), or perhaps also the additional condition (**) f(K1 + K2) = mm {f(K2),f(K2)} implies that either K1 is similar to K2,

orf(K1)=f(K2)= 1. 2. Theorem (A) is conceivably valid for measures of symmetry on any Sr. but the method of proof indicated above does not extend beyond n = 2. In fact, as noted by Gale already in E3 there exist convex polyhedra K, different from a simplex, which are "indecomposable," Le., such that K = K1 + K2, with K1, K2 is possible only if K1 and K2 are homothetic to K. Such polyhedra are, e.g., the octahedron, the icosahedron, the body obtained by attaching a regular simplex on one of the faces of an octahedron, etc. This brings us to an apparently quite hard problem: Determine all indecomposable polyhedra in E3 (or Es). A related problem bears on "irreducible" sets. A set Ke Sr is irreducible if it has the origin as center of symmetry and if 2K = K* + (_K*) is possible only when K* is a translate of K. It is well known (Groemer [1]) that hypercubes of any dimension are irreducible (though they are not indecom-

posable); in E2 parallelograms are the only irreducible sets (Grünbaum [4]), but

the situation is more complicated already in E3; no characterization of irreducible convex bodies (or polyhedra) seems to be known. 3. Though (C) does not generalize to higher dimensions, it is conceivable that the following generalization of (B) is true (it would obviously imply the truth of (A) in higher dimensions): (B*) Every convex body KE Sr is the limit of finite sums of simj.'lices.

The validity of (B*) is uncertain even if K is the octahedron in E'.' For a generalization of (C) to higher dimensions (using an addition of convex bodies different from the vector-addition considered here) see FireyGrUnbaum El]. 4. The following conditions, weaker than superminimality, are possibly of some interest: (***) f(K + M) f(K) for all K, Me .W', M = —M. See Note 3 on p. 264.

MEASURES OF SYMMETRY FOR CONVEX SETS (****)

f((1 — A)K + A(—K)) is, for every K€ i", increasing for 0 1/2. The Kovner-Besicovitch measure of symmetry does not have property (***) (see §6); it is not known whether it has property (****) A

5. General methods for geometric definitions of measures of symmetry. In a rather obvious fashion, every real-valued continuous functional on the space which assumes one of its extremal values for all centrally or symmetric convex bodies, and for such sets only, gives rise to measures of

symmetry. To mention one example, let

//" denote the subset of .Q" consisting of all centrally symmetric sets, and let d be any distance-function on or ,Q'; then f(K) = exp[—d(K, //")] is a measure of symmetry. White this approach has not been actually used in the literature, it is of interest to note that the Minkowski measure of symmetry (see § 6.1) on can be obtained in that fashion from the distance-function d2 considered in § 2. (For another functional related to d2, which possibly is also a measure of symmetry, see §7.) More attention received another approach. In some well-determined way a centrally symmetric convex body is associated with every convex body, the set associated to a member M of ._-//" being M itself. Then some geometrically significant quantity (such as the volume) of the convex body and of the set associated to it are compared. The most characteristic instance of this

approach is the measure of symmetry derived from the "difference body" (see §6); with a slight modification of the above procedure, the RogersShephard measure of symmetry 6) is another example. Probably the most interesting method is the following. For every convex body Ke .Q" and every point x K a number f(x, K) is defined (by appropriate geometric considerations), with 0 f(x, K) 1. A functional Jon is then defined by

f(K) = max {f(x, K): Xe K). With suitable choices of f(x, K), the functional f can be made affinely or similarity invariant, and even a measure of symmetry. Many of the measures of symmetry discussed in § 6 are of this type. A wealth of extremal problems,

and measures of symmetry, can be derived from such functionals by considering (instead of (*))

J"(K)=f(p(K),K) where sidering

is an affine [resp. similarity] invariant point of K, or by conJP(K) =

mm

{f(p, K): PE P(K)}

where P(K) is an affine [similarity] invariant set of K. (In some instances max may be used instead, or besides, mm.) Such "derived measures of symmetry," especially in case = g(K) is the centroid 'of K, have been frequently considered; details on the results found in the literature are given in §6.

BRANKO GRUNBAUM

The interest of the method we just described for defining measures of symmetry is due, at least in part, to the fact that it gives rise to a number of geometrically interesting notions and problems.

Thus, for each 2 with 0 2 1, the set ct'A(K,f)={x€K:f(x,K)2f(K)) is a is an affine [resp. similarityJ invariant set. The boundary of level (niveau) surface of f(x, K), and is also an a.i. [resp. s.i.1 set; for many

of the usually considered measures of symmetry (of this type) the sets = = are convex (see § 6). In particular, the set cf1(K,f) is known as the "critical set" of K (with respect to the measure of symmetry f). For some measures of symmetry consists, for each K, of a single point, called the "critical point," or "f-critical point," of K. The available data in this direction are also summarized in § 6. Obviously, many other more or less general methods for constructing measures of symmetry could be devised. Indeed, for almost every one of the measures of symmetry discussed in the literature, it is possible to describe a "general method" corresponding to it. We refrain from formulating them since no results justifying the definitions seem to be known. We also avoid the presentation of a long list of functionals which are, or which possibly are, measures of symmetry but about which we may report nothing else. The three examples of "suspected" measures of symmetry (in § 7) are given because they seemed to possess some unusual features. Known results on special measures of symmetry. 6.1. MINK0wSKI's MEASURE OF SYMMETRY. 1. For each point p belonging to

6.

and for each hyperplane H with p H, let H1 and H2 a convex body K€ be the two supporting hyperplanes of K parallel to H. Let f(H,p) be the ratio, not exceeding 1, in which H divides the distance between H1 and H2 -

FIGURE 5

define 1(p) = min{f(H,p): HBp} and F,(K)= max{f(p):peK}. The first estimates for for n = 2, 3, go back to Minkowski [1; 2; 3); Radon 11] seems to have been the first to establish F1(K) 1/n for K€ Se". (More We

precisely, both authors considered the derived measure Ft(K).) Simpler proofs have since been given by different authors (see below). The same measure of symmetry has been studied, even more extensively,

MEASURES OF SYMMETRY FOR CONVEX SETS

247

FIGURE 6

in a different formulation. For each point P Ke Sr and for each chord D of K with p e D, let g(D, p) be the ratio, not exceeding 1, in which p divides D. Let g(p) = mm {g(D, p): D 3/}. An obvious argument shows that g(p) = f(p) for each p and K. The measure F1(K) = max {g(p): p e K) was first considered

by Neumann [1) for n = 2. Estimates of F1(K) for general n, using this definition, were given by Stiss [2), Hammer [1], Klee [1], Leichtweiss [1], [1]. Birch [1); see also For another definition of F1(K) see p. 245. From the second definition it is obvious that F1(K) =

1

if and only if K

has a center. Using the first defitiition it is equally immediate that the levelsurfaces of F1(K) are convex. So is the fact that F1(K) has the superminimality property F1(K1 + K2) mm {F1(K1), F1(K2)) (see § 4).

The critical sets consist, for n =

2,

of one point only (Neumann [1]; see

also Stiss [8]). In general, the following inequality between F1(K) and the F1) was established by Klee [1]: dimension of the critical set F1(K)

+ dim

F1)

n.

F1) As an immediate consequence it follows that dim n — 2 for Ke sr, that each Ke Sr with F1(K) > (n — 1Y1 has a unique critical point.

and

The only extremal bodies are the simplices (Neumann [1) for n = 2; Stiss [21, Hammer [1], Klee [1], Leichtweiss [11, Birch [1] in the general case; see also Hamnaer-Sobczyk [11). For some properties of the F1-level-surfaces and certain related sets see Hammer [21. For a generalization of the above concepts and results see Motzkjn [1]. 2. Affine-invariant derived measures. Quite a few measures of symmetry, derived from F1 by the method mentioned in § 5, have been considered. (i) The derived measure Ft based on the centroid g = g(K). The inequality F!(K) 1/n for Ke Sr was established by Minkowski (11 for n = 2,3, and by Radon [1] for all n. Other proofs were given by Estermann [2] (n = 3), Siiss [1) (n = 3), Siiss [2], Hammer [1], Yaglom-Boltyanskii [1] (n = 2,3), Ehrhart [2], Leichtweiss [1], Birch [1]. Most of these authors also proved that the only extremal bodies in E" are pyramids with arbitrary (convex) (it — 1)-dimensional base.

BRANKO GRUNBAUM

248

based on the centers of the circumand (ii) The derived measures scribed (resp. inscribed) ellipsoids. In view of John's theorem (John [1), F(K) 1/n Leichtweiss [2); see also p. 241) it follows that for all K€ The only extremal bodies are the simplices. based on the hexagonal points. The func(iii) The derived measure max {f(h): h a hexagonal point of tional F'(K) defined for Ke by 1/2. (See Figure 7.) The simiK) is a measure of symmetry, with larly defined functional = mm {f(h): h a hexagonal point of K) is also 1/2 for all KE Se'. In both cases there are a measure of symmetry; extremal bodies different from triangles (see Figure 8).

FIGURE 7

FIGURE 8

(iv) Another proof of F,(K) 1/n for Ke was given by Rado [1], by establishing that f(t) un, where t is the centroid of any simplex of maximal volume inscribed in K. It may be mentioned that the functional max t the centroid of a simplex of maximal volume contained in K) is not a measure of symmetry since it has values less than 1 for centrally symmetric sets. E.g.

(Figure 9), for a square its value is 1/2; similarly, among the bodies in E' for which the lower bound 1/n is attained there are centrally symmetric ones.

K

FIGURE 9

Similarity-invariant derived measures have also been considered. (i) The derived measure F1' based on the centroid a = a(K) of the surfacearea of K. For convex bodies Ke .R" the inequality > (2n — 1)_1 holds (Sz.-Nagy [IJ for n = 2,3; Ehrhart [4) for n = 2; Asplund-Bredon-Griinbaum [1] for all n). The bound is the best possible, sufficiently flat pyramids (on 3.

MEASURES OF SYMMETRY FOR CONVEX SETS

arbitrary convex (n — 1)-dimensional base) have

249

arbitrarily close to (2n — 1Y'.

(This solves in the negative a conjecture of Ehrhart [4].)

(ii) The derived measure F based on the curvature centroid k = k(K) of The best estimates is F'(K) > 0; e.g., for the triangle T. of Figure = a/(,r — a) f or a ,r/3. One interesting property of F is 10 we have

FIGURE 10

its superminimality F(K1 + K2) mm 1K(K1), F(K2)}, which follows at once from the additivity of k(K) and that of the support function. 4. Remarks. (i) In the plane, the F'-critical point of the convex body K divides at least three different chords in the ratio F1(K). This was first

proved by Neumann [1]; it may also be easily derived from Helly's theorem on intersections of convex sets (see, e.g., Yaglom-Boltyanskii [1)). A simple continuity argument implies that the F1-critical point of Ke bisects at least three different chords of K. As is easily verified, the same assertion can be made about the centroid of K (Bose [1], Ehrhart [1], Viet [1]) and, obviously, about each hexagonal point. (The assertion in Bose [1], that the number of chords bisected by the centroid is odd whenever finite,. is to be taken cum grano salis: "doubly-counted" chords may occur; indeed, it is easy to find examples showing that the number of different chords bisected by the centroid may be even.) The following question appears to be quite natural: What properties has the set of all points of Ke R2, each of which bisects at least three different Is it connected, or even convex? Let K be a convex'body in E". Consider (n — 1)-dimensional hyperplanes H intersecting mt K, and centroids of the sections H n K. For n = 3 some properties of these centroids have been studied by Ascoli, Tricomi, Fenchel (see Bonnesen-Fenchel [1, p. 13] for references) and, more recently, by Steinhaus [2). Each point of mt K is the centroid of at least one section of

K (see Grünbaum [5] for extensions to arbitrary n, Stein [2] for some related results, Kosiñski [4] for further generalizations). Steirihaus [2J also proved, for n = 3, that for each K either some point of mt K is the centroid of continuum many different sections, or continuum many points are centroids of at least two different sections each; in view of GrUnbaum [5] and the nonexistence of a homeomorphic image of the projective n-space in for n > 1, Stein-

haus' proof applies as well in higher dimensions.

BRANKO GRUNBAUM

250

Among problems related to these theorems and generalizing the elementary we mention some raised in Griinbaum [5]: If K c E' is a convex results on

body, n 3, does there exist a point x e mt K which is the centroid of at least n + 1 different sections of K? (For an affirmative answer to this problem, see § 6.2.) What properties does the set of all such points x have? Does it contain the centroid of K?

The above results of Steinhaus are true also if, instead of centroids of sections, other continuous point-functions of convex sets are considered (such as surface-area centroids, centers of minimal circumellipsoids or circumspheres, etc. (see Grünbaum (5], Steinhaus [21)), and with respect to each such pointfunction the foregoing questions may be raised. (ii) Neumann [1] considered also certain other functionals, related to F1, which are easily modified in such a way as to become measures of symmetry. Following his notations, let p1, 0 i 2m, be endpoints of chords of and

all passing through a point xeK, and such that p' is etc. (See Figure 11 for m = 2.) Let = mm LIP.

between q,,,+1 — xIII p,

—

I

FIGURE ii

over all choices of p1 , q1 satisfying the above conditions, and let = max XE K}. Then m p2m+i(K) m + 1/2, where ptN+I(K) = m charac-

terizes triangles, whire

= m + 1/2 if and only if K has a center of

symmetry. For m 1 this result can easily be obtained from the properties of F1 (Neumann derived F1(K) 1/2 from p5(K) 1); it seems also possible to derive the estimate p2m+t(K) rn from 1/2. (iii) Similarly defined are the following measures of symmetry. For KE W' let x1, be points of hdKand let Let y, be

the other endpoint of the chord of K determined by x and x1, and let

c(x)=min

1

+1

—

xI

MEASURES OF SYMMETRY FOR CONVEX SETS

for all families {x1) with x E cony

Let G(K) = max {c(x): XE K), ir(K) =

K). From the results on F1 it follows that 1/n a(K) 1 and = 1 if and only if K ir(K) 1. It is obvious that o(K) = 1 or has a center of symmetry, and that the lower bound can be attained, if at max {lr(x): x

= all, only by the simplex 4". For ir it was proved recently that (Berkes [1], Schopp [1]); the assertion u(4") = 1/n is also easily established. 1. For each point p of a convex each hyperplane H with p e H, let f(H, p) be the ratio, not exceeding 1, of the volumes of the two parts of K determined by H. We define 1(P) = mm {f(H, p): Hsp) and F2(K) max {f(p): p K).

6.2.

WINTERNITZ' MEASURE OF SYMMETRY.

body Ke sr,

and

This measure of symmetry, or rather the derived measure F(K), was studied by different authors (see below); the bounds for F2(K) are the same as those for F(K). The only F2-extremal bodies are the simplices (Grünbaum It is easy to establish that the level surfaces of F2 are convex, and that there exists a critical point for each K€ Se" (Hammer [4]; also Süss [31 for n = 2). Applying a variant of Helly's theorem (see, e.g., Ortinbaum [6, Lemma 3]) it follows that there exist n + 1 or more hyperplanes H through the critical point '(K) of K each of which divides the volume of K in the ratio F2(K). Since for each such H the point is centroid of H n K (see l3laschke [3] for n = 3; the proof applies to all n), an affirmative answer to one of the problems mentioned on p. 250 follows. The property of F2 most difficult to establish is that F2(K) = 1 only if K has a center of symmetry. Though this fact is immediate for n = 2, the known proofs for = 3 (Funk [1], see Blaschke [5, p. 250], Bonnesen-Fenchel 11, P. 25)) use integral equations, or spherical harmonics. (The author regrets the historically incorrect statement made in this connection at the end of Gilinbaum [3].) Although it is possible to extend these methods to > 3,. no

[3], Hammer [4]).

such proof seems to have been published. Worth mentioning In this connection

is the interesting result of Ungar (1].' 2. The following results are known on affine-invariant measures derived from F2.

(i) The derived measure fl based on the centroid g = g(K). The history of bounds for F(K) is quite interesting. The case KE was first investigated by A. Winternitz, who established the inequality 4/5 and the fact that triangles are the only extremal bodies. Winternitz's results were published in 1923 in Blaschke's book [5, pp. 54-55); they seem to have been unnoticed and were rediscovered independently by Lavrent'ev-Lyusternik [1] (in 1935), Neumann [2] (in 1945), Yaglom-Boltyanskii [1) (in 1951), Ehrhart (1) (in 1955), and Newman [1] (in 1958). They were ascribed to Winternitz (but

without date or reference) in the German translation of Yaglom-Boltyanskii Li] (in 1956).

The inequality F(K) n"J[(n + 1)' — n"] for Ke .Q' was conjectured by Ehrhart 11) and established by Ehrhart [21 for n = 3 and by Grünbaum [31 and Hammer (41 for all n. The bound is attained if and only if K is a pyramid on any (n — 1)-dimensional convex base. See Note 4 on p. 264.

BRANKO GRONBAUM

252

(ii) Eggleston (2] (reproduced in (3] and in [51) proved for each six-partite point s of K the inequality = f(s) 4/5. From this and the uniqueness of F2-critical points follows at once the uniqueness of the sixpartite points for triangles (proved by a different method in Eggleston [2)). 3. Generalization of F1 to mass-distributions. Given any distribution p of in complete positive masses in E', with p(E*) < 00, one may define analogy to the definition of F2(K), except for substituting the mass contained in the half-spaces for the volumes used in the definition of F1(K). Obviously, F2(p) is not a measure of symmetry since F2(p) = 1 is possible for essentially noncentral mass distributions. On the other hand, F1(p) 1/n for any massFor n 2 this result was first proved by Neumann [2], distribution p in and rediscovered by [1] and Newman [1]. For arbitrary n the inequality was established by Rado [2], Birch (1], Grilnbaum [3];? see also Danzer-Grtinbaum-Klee [1]. The bound 1/n is obviously the best possible. A characterization of distributions for which it is attained is given in GrUnbaum

For a discussion of F1 in the plane, and some related notions, see

[31.

Neumann [31. 6.3.

Tnn SURFACE-AREA ANALOGUE OP

MPASURE.

Let F3(K), for

Ke sr, be defined in the same way as F2(K) except that f(H, P) is the ratio in which the surface-area of K is divided by H. From the results on F2(u) it follows at once that F3(K)> 1/n for Ke SI". Nothing more is known about the functional F8(K) for Ke sr with n 3. In particular, it is not known whether F3(K) = 1 only if K has a center. For n 2, it is easy to establish that F(K) < 1 if K does not have a center. Neumann [2] conjectured that the lower bound of F1(K) for K€ 512 (i/T — 1)/2; elementary computations show that the lower bound may not exceed (VT — 1)/2, since F2(K) comes arbitrarily close to this number if K is a sufficiently narrow isosceles triangle. Ehrhart [1] mentions that F3'(K)> 1/3k for Ke 512, but the proof seems not to have been published.

K

FIGURE 12

I

See Note 10 on p. 264. See Note 5 on p. 264. See Note 6 on p. 264.

MEASURES OF SYMMETRY FOR CONVEX SETS SYMMETRY. 1. Let p be a point of the interior 6.4. Two MEASURES of a convex body Ke r, H a hyperplane through p, H1 and H2 hyperplanes parallel and equidistant from H, and H1 supporting K. If K'(H3) denotes the part of K contained between H1 and H2, let , p) = V(K'(H1))I V(K) and = mm {f(H1 , p): H1 supporting K). We define F4(K) = max (1(p): pe K). Since F4(K) = 1 implies Fi(K) = 1, it follows that F4(K) = 1 iff K ha8 a center. that f4(K) 8/9 Fvom the known resultsftbotjj F:(K) (see below) it of for K€ with triangles as only extremal bodies. The exa4

F4(K), for Ke r, with n 3, are not known; we conjecture that F4(K) l)/(n + 1)]' for Ke R', with simplices as the only extremal bodies. The derived measure F41(K) was consideIed in Asplund-Grosswald-Grtinbaum for Ke it" cannot be determined in closed of 111. The lower bound 1 — [(n —

form for n 3, but may be computed_-numerically (e.g., a3 = 0.8746 "); the extremal bodies are pyramids on arbitrary (n — 1)-dimensional convex bases,

truncated in a suitable ratio (depending on n; e.g., 0.2-•- for n = 3). For the plane, a1 and the only extremal bodies are triangles. It is of some interest to note thiL denoting by 4% the n-dimensional simplex, we have =1— 0.857 ... = 0.865Let K be a convex body in E", p an interior point of K, and H1, H1 a pair n K, of parallel supporting hyperplanes of K. For each pair of points 1x1, x1] and containing P. let C(x3, x1) be the chord of K paralleito the 2.

FIGURE 13

H' — and

f(p) =

mm

(length C(x1, x3)

£7

length [x1 , x2)

(1(1', He): H1,

supporting

K).

—1 S—i,

We

define

F5(K) =

max {f(p): pe mt K).

The assertion "F1(K) = 1 if and only if K has a center" is equivalent to Theorem 1 of Hammer [3). It is not hard to verify that F5(K) satisfies the superminimality condition; therefore it is immediate that F5(K) 2/3 for Ke.Q2, with equality only for triangles.10 The lower bounds of F1 in higher 10

See Note 7 on p. 264. See Note 6 on p. 264.

BRANKO GRUNBAUM

254

dimensions seem not to have been determined; it may be conjectured that simplices are the extremal bodies (this would imply that for n = 3, F5(K) 1/2). For the derived measure F.' it is easy to show, using methods similar to those applied in the proofs of F,'(K) 4/5 for KG that FT(K) 2/3 for Ke The bounds of F1(K) for Xe r are not known for n 3. 3.

Some problems on affine diameters. If K is a convex body in E', a

chord C of K is said to be an affine diameter of K if and only if there exists a pair of (different) parallel supporting hyperplanes of K, each containing one of the endpoints of C. (The segments [x1, x5] used in the definition of F5 are affine diameters.) It seems to be well known that each point of K belongs to at least one affine diameter. This result was announced by Hammer (3], but his proof was not published. It may be easily derived from Theorem 1 of Grunbaum 15] and the remark following it. (It is also easy to establish that for each direction there is an affine diameter parallel to it.) The following

problems on affine diameters are similar to those on centroids of sections mentioned on p. 249. (For generalizations of the above results, and partial answers to the problems below, see Kosiflski 12; 3].)

For a convex body K in E*, does there exist at least one point contained in at least n + 1 different affine diameters? In particular, is the centroid of K such a point? Does each K have either continuum many points each belonging to more than one affine diameter, or a point belonging to continuum many affine diameters?

For n = 2 the affirmative answer to the last question is trivial. The answer to the first two is also affirmative; it is not hard to show that both the centroid of K and the F1-critical point of K belong to at least three different affine diameters. 6.5. and a

MEASURE OF SYMMETRY.

1.

For a convex body Ke r

point peK let K(p) = Kn (2/' — K) be the maximal subset of K with center at p. We define to be the ratio of the volumes of K(p) and K, i.e., = V(K(p))/ V(K); the Kovner-Besicovitch measure of symmetry is F.(K) = max Pc K). The symmetry measure F. has been the object of many investigations. It seems that the first published proof of F,,(K) 2/3 for is to be found in Lavrent'ev-Lyusternik [1, pp. 358-367), where it is credited to S. S. Kovner. Independently, simpler proofs of this inequality were given by Besicovitch [1], by Fáry (1), and, for derived measures, by other authors (see below). Fáry [1] and others also noted that F,(K) = 2/3 characterizes triangles. Very few results are known on F. in higher dimensions. Bielecki and Radziszewski [1] proved that every convex body Ke

contains a parallelepiped

P with V(P) (2/9) V(K). It follows that F.(K) 2/9 for Ke Ffiry and Rédei (1] proved that every KeSr has a unique F,-critical point, and that the surfaces are convex. The same results were obtained independently by Stein [1]; for n = 2 they were established already in Lavrent'evLyusternik El]. Ffiry and Rêdei [1] established also that

MEASURES OF SYMMETRY FOR CONVEX SETS

F.(4') = F:(4') = (n

=

+ ir E

(_1)'('t

+

255 1—

2ir

r(sintr'dt Jo\ t

/

(The numerical value given for F,(46) is incorrect.) Stein [1] established also

the remarkable fact that the average of F(K), for x ranging over Ke a", is 2*; i.e., From this it follows that F1(K) > 2" for

KeJr.

Special investigations were made on the lower bound of F.(K) for plane sets K of constant width. Besicovitch [2] showed that each such set K has measure of symmetry F6(K) 0.840.... This result was extended by Eggleston [1].

The following examply, due to W. J. Firey, shows that the KovnerBesicovitch measure of symmetry does not satisfy the superminimality condition (*) (see p. 243), and not even the weaker condition (***) (p. 244). Let KA

be the ."truncated" square (see Figure 14); then, for A

1/2, it

is easy to

1

FIGURE 14

show that F.(KA) =

(2 — 2A)/(2 — At)

disc, Fe(KA + öBt) will be less than

On the other hand, if B6 is the unit provided ö is small enough and

A<1— 2.

Quite a few derived measures of F. have been considered in the litera-

ture.

(i) The derived measure F.' based on the centroid g = g(K). The bound F((K) 2/3 for KE was conjectured, and established under some restrictions on the sets K considered, by Ehrhart [3]. For all Ke this inequality was established independently by Stewart [1] and Kozinec [1]. Stewart also gives an example which shows that this estimate is not a consequence of the estimates of F.' and F' (see below) and the convexity of the level-lines. The bounds of F(K) for higher-dimensional K have been considered by Levi [1]; he established F.'(K)> 21(1 + n') for Ke .R".

BRANKO GRUNBAUM

(ii) The derived measure F based on the hexagonal points of Ke.Rt. This derived measure was the first variant of F6 to be studied. Both Besicovitch [1] and Fâry [1] proved F(K) 2/3 for each hexagonal point h of Ke and in this way established F6(K) 2/3. The same estimate of F(K)

was obtained also by Yagloin-Boltyanskii [1] and Linis [1]. The result, as established by these authors, is even stronger: Every affine-regular hexagon inscribed in has an area of at least 2/3 the area of K. (iii) The derived measure F6' based on sixpartite points was considered by Eggleston ([2], reproduced in [3; 5J); he proved F(K) 2/3 for every sixpartite point s of (iv) The derived, measure F based on the center of the circumellipsoid seems to be the simplest example of a geometrically "plausible" measure of symmetry for which, even in E2, the simplices (= triangles) are not the sets with minimal measure of symmetry. Though the lower bound of for Ke seems to be unknown, the set K of Figure 15 can be shown to satisfy =

+ 2ir

FIGURE 15

6.6.

For any convex body Ke SQ" and be the minimal convex body with center p. containing K. We define F71'(K) = V(K)/ V(K[P]) and F7(K) = max {Ff(K): P E K). The lower bound F7(K) 1/2 for K€ seems to have been established first by Estermann [2]; he also proved that triangles are the only extremal sets. These results were obtained also by Levi [1], and the bound 1/2 also by Fáry [1] and Yaglom-Boltyanskii [1J. Estermann [2] asked whether F7(K) 1/3 ESTERMANN'S MEASURE OF SYMMETRY.

any point p e K let

MEASURES OF SYMMETRY FOR CONVEX SETS

257

The estimate F7(K) 1/6 for follows from a result on circumscribed parallelotopes (see § 2). for K e Sr was established by Rogers and Shephard The bound F7(K) > [1; 3]. As pointed out by Ffiry-Rêdei [1] (and also Rogers-Shephard [3]) the F7-critical set of K€ Sr may be n-dimensional (e.g., if K is a simplex); they also found some regularity conditions on K which guarantee that the critical set of K consists of one point only. For the n-dimensional simplex Fáry-Rédei [1] determined

the problem is still open.

for

F7(S*)

= Levi The only derived measure of F7 which has been considered is for KG Sr. [1] proved that F/'(K)> For some extremal problems related to Estermann's measure of symmetry

see Rogers-Shephard [2]. 6.7.

SOME ANALOGUES OF THE KOYNER-BESICOVITCH AND ESTERMANN MEASURES OF

and be defined similarly For a convex body KG let to F6(K) and F7(K), except that the perimeters of K and K(J.') [resp. KEp)J areas). Due to the additivity of perimeterare compared (instead of length, both F and F satisfy the superminimality condition. Therefore it is for triangles, in order enough to determine the lower bounds of F' and It is trivial that F.4'(K) 2/3, with to have the lower bounds for all K€ SYMMETRY.

2/3 only if K is an equilateral triangle. Probably (v. ah equality only if K is an equilateral triangle), but the inequality does

F7 (K) = r.

seem to have been proved yet.1' The

sets may be 2-dimensional,

'd they (and also the level-lines in general) are convex (see Figure 16). It may be remarked that a straightforward application of Bieberbach's [1] isoperimetric theorem yields F6'(K) > 2/sr for Ke Se'. The fact that each of

K

FIGURE 16

" See Note 6 on p. 264.

BRANKO GRONBAUM

258

the conditions F."'(K) 1 or F7(K) = 1 implies that K has a center follows at once from the well-khown strict monotonicity of perimeter-length of convex sets.

The measure F." was proposed (oral communication) by W. J. Firey; he also posed the analogous questions for higher dimensions. We did not reproduce them here since no results about them seem to be known. 6.8. THE DIFFERENCE-BODY MEASURE OF SYMMETRY. For a convex body Ke R' let K* = (1/2)[K + (—K)]. (The set

is usually called the difference-body,

or the vector-body, of K.) Then V(K*) V(K), with equality if and only if K is centrally symmetric. It follows that F.(K) = V(K)/ V(K) is an affine'invariant measure of symmetry. Blaschke [4] raised the problem of determining the lower bounds of F,(K) for Ke W'. For n = 2 the relation F.(K) Rademacher [1] and Estermann (1; 2]. The case n — 3 2/3 was established was settled by Estermann (2] and SUss [1]. Partial results on the general problem were obtained by Bonnesen-Fenchel [1, p. 105] and Godbersen [1]. The complete solution was found by Rogers and Shephard [1], which proved that F,(K)

for Ke 2".

The bounds are attained for simplices only. Modifications of their proof are given in Rogers-Shephard (3] and Eggleston [5]. Groemer [2] proved that F.(K) 2''/(2" — 1) whenever Ke SI' is a pyramid; equality holds if the base is centrally symmetric; for n = 3 this result is due to Estermann [21. 69. THE ROGERS-SHEPHARD MEASURE OF SYMMETRY. Let K be a convex body orthogoin E'; let B' be imbedded in E"1 and let e be a unit vector in nal to E". Rogers and Shephard (3] defined the "associated" (n + 1)-dimensional convex body K* as the convex hull of K U {e + (—K)). Then F,(K) = is an affine-invariant measure of symmetry. Rogers and Shephard 131 established F.(K) (n + 1)2" for Ke 2", with equality if K is

a simplex.

Some extremal problems which possibly lead to measures of symmetry. The distance-function d on the space E of affinely equivalent classes of convex bodies in B" (see § 2) leads to many problems, some of which are relevant in functional analysis. The following results on d2 are known (4 denotes a simplex, C hypercube, Q hyper-octahedron, B ball, M any centrally symmetric convex body, K any convex body; superscripts denote dimensions; H is the regular hexagon): (1) d,(B', K') d2(B', 4") = In d2(B", M') C') = d2(B', Q') = 1/2 ln n, John [11, Leichtweiss [1], (2) d2(C', M') Inn; M') In n , Day [1], Lenz (1], Taylor [1], (3) d2(Ct, M2) H) = in 3/2; d2(H, M') in 3/2, Asplund [1], 7.

1.

(4)

K2)

d8(C2, 4) = in 2,

MEASURES OF SYMMETRY FOR CONVEX SETS (5)

d2(if, K')

M') =

In n.

To prove (4), it is sufficient to verify the easy equality d2(C, I) = in 2 and, for K * 42, to consider the quadrangle A of maximal area contained in K;

then the parallels to the diagonals of A through the vertices of A are

supporting lines of K; they form a parallelogram T(C') such that a translate of T(C)/2 is contained in A C K. In order to prove the inequality in (5) one has only to consider a simplex 4' of maximal volume contained in K; a translate of —nd" then contains K. To prove that d,(4', M") = in n, we remark first that if M" contains some T(4") then it contains also a translate of — T(4"); but the smallest I A I for

which a translate of AT(f) contains both T(4') and a translate of — T(4") is IAI = n. Some of the interesting open problems are: Is the d,-diameter of 51 equal in n? Is max d,(C', K') = In n? What is the d,-diameter of the space (consisting of all centrally symmetric members of .Q)? Is max d,(M", C') = in 2.) in (n + 1)/2? (It can be shown that d,(C', Q3)

The relation of the distance-function d, to a measure of symmetry arises from the following conjecture, which is open even for n = 2: d,(4', K') = In n if and only if K" has a center of symmetry.1 If this conjecture is true, then f(K) = d,(K4')/ln n is a measure of symmetry derived in a particularly simple way from an aft me-invariant distance-function (see § 5 for a more complicated relationship of a similar kind).

FIGURE 17

A great variety of problems may be posed about functionais similar to d,. We mention the following two: (a) Define K,) in the same way as d,(K1, K,) was defined, but with the additional assumption that the centroids of K, and T(AK1) coincide. The 42) is are not known even for E'. The value of seen from Figure 17 to be not greater than in 5/2; it can be shown that actually

bounds on 1

See

Note 8 on p. 264.

BRANKO GRUNBAUM

This is probably the least upper bound for d(K2, 42), and possibly also for d'(KL K). (b) Consider the following procedure, originating from a problem on projections in Minkowski spaces. (We note that it is easy to use a slight modification of the following definition in order to obtain stilL another distance for the space of affine classes of convex bodies in E*; but the corresponding problems seem to be extremely unmanageable.) For a convex body K = — K c E* let P(K) be the greatest lower bound of positive reals 2 for which L a1 = 1, such there exist affine transformations T,, and reals 0 C AK. that K c and It is known (Grünbaum [2], Rutovitz Li)) that d*(Cz, 42) = in 5/2.

= n.21h1(

= 1)/21)

and

PIB*\

Determine I',,

max P(K) for K =

—K C E*.

In particular, is P2 = 4/3, results on d3 that P <,t.) The notion of sixpartite points of plane sets leads to many extremal

P3 = 3/2, P,, 2.

nI(n/2)

—

(It follows from

problems, among which we mention the following two, which probably determine measures of symmetry. (a)

For a plane convex body K consider all partitions of K by three concurrent lines such that the areas of the six regions satisfy (see Figure 18)

FIGURE 18

= a and A(K3) = A(K4) = A(K.) = b. Let f(K) = mm a/b for all partitions of this type. We conjecture that 1/2 f(K) 1 for all K c E2, with f(K) = 1 characterizing centrally symmetric K, and 1(K) = 1/2 A(K,) = A(K3) =

for triangles only.

Three chords of a plane convex body K divide K, in general, into 7 If the outer six regions have all the same area, a, then the area b of the inner region (see Figure 19) satisfies b a/8. This conjecture of Buck and Buck [1] was proved by Sholander [1] and Eggleston [2; 3; 51. Thus, putting f(K) = mm (1 — b/a), the minimum being over all partitions of K of (b)

regions.

MEASURES OF SYMMETRY FOR CONVEX SETS

261

FIGURE 19

1, where equality on the left the considered type, we have 7/8 characterizes triangles, while equality on the right holds for all centrally symmetric K. Probably f(K) = 1 characterizes centrally symmetric sets; if so, then f(K) is a measure of symmetry. 3. Let K be a convex body in E*, and H a hyperplane intersecting K in such a position that it is equidistant from the two supporting hyperplanes of K parallel to it. If fv(H, K) is the ratio (not exceeding 1) of the volumes of the two parts of K determined by H, let = minfv(H, K) for all H with the above property. Similarly, considering the ratio of the surface-areas instead of the volumes, another functional, fA(K), is obtained. It is obvious that — 1) for K c E' where equality holds for pyramids on an arbitrary (n — 1)-dimensional basis, and that f4(K)> 11(2' — 1) for K c E" is the best possible estimate for The interesting problem in connection with and f4 is whether they are = 1 implies that K is measures of symmetry, i.e. whether fy(K) = 1 or centrally symmetric. The problems seem to be open even for n = 2. 8. An interesting functional. The following functional fails, surprisingly enough, to lead to a measure of symmetry. For a convex body K c E2 let L be a chord of K bisecting the area of K; let PL be the ratio, not exceeding 1, in which L divides the perimeter of K, and let p(K) = mm PL. It is known that p(K) > 1/3 for all Kc E2 is the best possible estimate (see Pleijel Iii for a more general result, which determines also the bound for the related problem about the ratio in which the area of K is divided by chords bisecting the perimeter; the bound in this case is 0). The question of characterizing those K c E2 for which p(K) = 1 was treated repeatedly. Matumura claimed to have proved that p(K) = 1 characterizes centrally symmetric sets K. But already Zindler [1J has examples to the contrary; Auerbach determined all the sets with p(K) = 1 (and found a very close relationship between them and sets of constant width). Figure 20 is one of Auerbach's examples (reproduced also in Steinhaus (1]) of a set with p(K) = 1 which does not have a center of symmetry.

Auerbach [I] proved also that for sets K with p(K) = I and without a

262

BRANKO GRUNBAUM

FIGURE 20

center of symmetry, all the chords bisecting the area (and perimeter) of K have equal lengths. This theorem forms the connection to the following group of results and problems. For generalizations of this result see Strubecker [lJ where also additional references and comments may be found. 1-lirakawa El] established the following result: If a convex body K c Et has the property that whenever two of its chords have the same length, they

divide the area (or the perimeter) of K in the same ratio, then K is a circle. (A simpler proof of Hirakawa's theorem was given by Gericke Li].) Auerbach's

theorem shows that Hirakawa's does not remain valid in general if only chords of an arbitrary fixed length are considered (in contradiction to a claim of Miyatake [1]). On the other hand, Salkowski [1] has shown that if a convex body K is such that, for some integer k> 2, all the chords which cut off the i/k part of its perimeter have a fixed length, then K is a circle. One notices

immediately a certain similarity between this situation and the problems related to the theorems of Radon [2], Funk [1] and Ungar [1] (cf. p. 251), and eigenvalue problems in general. All the problems mentioned above have obvious analogues in higher dimensions, sections of K c E" by (n — 1)-dimensional hyperplanes corresponding to the chords in the plane case. No estimates seem to be known for p(K).

The most intriguing problem is related to the possibility that, for K c with n 3, p(K) = 1 is characteristic for centrally symmetric K. An affirmat:ve answer to this problem would bring to light another "singular" property of E2, similar to the existence of Radon curves is E2, which do not generalize (except trivially) to higher dimensions (Blaschke [1]). Appendix: Some generalizations. Many aspects' of the definition of measures of symmetry for convex bodies given in § 1 are, admittedly, of a quite

arbitrary nature. A wide variety of notions, logically as sound as that of

MEASURES OF SYMMETRY FOR CONVEX SETS

"measure of symmetry for convex bodies," may be forwarded. The main reason for placing these other notions in a separate appendix is the same as that which induced the author to refrain from any reference to them in the first draft of the paper: the scarcity, not to say absence, of significant results which would justify our proposing the general definitions. The definition of measures of symmetry as functionals involved several notions; to be specific, the measures of symmetry were required to be

(i)

continuous;

(ii) defined for all convex bodies in (iii) Euclidean n-space E'; (iv) to be invariant under the group of regular affine transformations or under its subgroup (similarities); (v) to achieve the maximum precisely for classes of sets containing a member invariant under the two-element subgroup of generated by the reflection in the origin. Each of these aspects (and conceivably some additional ones) may be subjected to meaningful variations. As a very simple example, one could change in (iii) E' into S', the n-dimensional sphere, and take in (iv) the group of motions of S'. One could certainly develop in this setting a theory quite analogous to that for E": invariant points, general methods for the definition of measures of symmetry, particular measures analogous to those in § 6. As far as known to the author, none of this has been done. For another example, only (v) is to be changed: instead of one could take the group of rotations around the origin. As particular such "measures of ellipticity" (resp. "sphericity") for convex bodies K' in E' one could take, e.g., simple functions of ö1(B', K'), B'), or o2(K', B') (see 2 and 7). See in this connection Fejes.TOth [11. For a deep result on the existence of "nearly-spherical" sections of convex bodies see Dvoretzky [1]. In a rather obvious fashion, one could similarly define "measures of axiality,"

"symmetry with respect to a hyperplane," "trigonality" or "pentagonality," or "symmetry" with respect to any other subgroup of While some of the problems thus posed may turn out to be quite interesting, the results known at the present seem to form a subset of the empty set. A variant of possibly greater interest than the above is obtained by substituting in (ii) "measurable sets of finite and positive measure" or "distributions of positive masses" for "convex bodies." The functional discussed in 6.2.3 may be viewed in this light, though it is not a measure of symmetry. Another unexplored point in this connection seems to be the meaning which is to be attached to the continuity condition (i). How can the space of affine. equivalence classes of compact, measurable sets of finite and positive measure be metrized; is it compact in a suitable metric? One may modify the Kovner-Besicovitch measure of symmetry (as well as some other measures discussed in § 6) to be applicable in this situation. But See Note 9

on p. 264.

BRANKO GRUNBAUM

264

in each case in which we tried to do so, the lower bound was 0. Is there any "natural" measure of symmetry where the lower bound is positive? Notes added in proof, September 10, 1962 For another proof, and for some related results, see G. Pólya and G. Szegö, Isoperi. metric inequalities in mathematical phyeics, Princeton Univ. Press, Princeton, N.J., 1

1951, 109 pp.

C. J. Gerriets and G. Pólya (On the location of the centroid of certain solids, Amer. Math. Monthly 66 (1959), 875-879) consider a class of polyhedra ('pyramidoids") which share with pyramids some properties of the centroids. The answer is negative. B. Asplund proved (private communication) that an are affine transoctahedron is not the limit of bodies in any family where formations and is any finite set of nonoctahedral bodies. 'See also Radon [2, p. 275J. The proof given in Grünbaum [31 applies only to continuous mass-distributions; the general case is then deducible by standard methods. 0 A proof was given by Mrs. Y. Kovetz in her M. Sc. Thesis, Some extresnal problems on convex bodies, Hebrew University, Jerusalem, 1962. (Hebrew) 1 The proof given in Hammer [3) is invalid. The theorem was proved in Mrs. Kovetz's thesis (see Note 6). E. Asplund showed the conjecture to be false, even for n = 2. The axial symmetry of centrally symmetric convex bodies in the plane was recently O

investigated by W. Nohi (Die innere axiale Symmetrie zentriacher Eibereiche der Euklidi.clsen Ebene, Elem. Math. 17 (1962), 59-63). 10 The following result of K. Zarankiewicz (Bisection of plane convex sets by lines, Wiadom. Mat. (2) 2 (1959), 228-234 (Polish)) is worth mentioning in this connection: A plane convex body K has a center of symmetry if and only if there is a unique point through which pass three (different) chords of K each of which halves the area of K. REFERENCES

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3. Problems in Euclidean space: Application of convexity, Pergamon Press, New York, 1957. 4. Figures inscribed in convex sets, Amer. Math. Monthly 65 (1958), 76-80. 5. Convexity, Cambridge Univ. Press, Cambridge, 1958.

E. Ehrhart 1. (Jne généraheation du thêorème de Minkoweki, C. R. Acad. Sci. Paris 240 (1955), 483-485.

2. Sur les ovales et lea ovoides, C. R. Acad. Sci. Paris 240 (1955), 583-585. 3. Propriétea arithmo-geometriques des ovales, C. R. Acad. Sd. Paris 241 (1955), 274-276.

4. Stir lea polygonea sties ovales, C. R. Acad. Sci. Paris 242 (1956), 332-334. T. Estermann 1. Zwei neue Beweise einss Satzes von Biaschke und Rademacher, Jber. Deutsch. Math. Verein. 38 (1927), 197-200. 2. Ueber den V.ktorenbereich sines konvexen Körpers, Math. Z. 28 (1928), 471-475. I. Fáry 1. Stir Ia densitd des réaeaux de doniaines convexes, Bull. Soc. Math. France 78 (1950), 152-161.

1. Fáry and L. Rédei 1. Der zentralsymmetriache Kern und die zentralsymmetriache Hülle von konvexen Korpern, Math. Ann. 122 (1950), 205-220. L.

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On ellipsoids circumscribed and inscribed to polyhedra, Acta. Sci. Math. Szeged 11 (1948), 225-228.

W. J. Firey and B. GrUnbaum 1. A decomposition theorem for convex polyhedra, (in preparation). C. M. Fulton and S. K. Stein 1.

Parallelograms in8crt.bed in convex curves, Amer. Math. Monthly 87 (1960), 257-258.

P. Funk 1. Ueber ems geometrische Anwendung der Abeischen Intergralgleichung, Math. Ann. 77 (1915/16), 129-135. D. Gale 1. Irreducible convex sets, Proc. Internat. Congr. Math., Amsterdam, 1954, Vol. 2, pp. 217-218.

P. Ganapathi 1. The vector region of convex closed bodies, Math. Z. 38 (1934), 488-489. H. Gericke 1. Einige kennzewhnende Eigenechaften des Kreises, Math. Z. 40 (1936), 417-420. 2. Ueber stützbare Fldchen und ihre Entwicklung nach Kugelfunktionen, Math. Z. 46 (1940), 55-61. C. Godbersen 1. Der Satz von Vektorbereich in Rilumen beliebiger Dimensionen, Dissertation, tingen, 1938, 48 pp.

MEASURES OF SYMMETRY FOR CONVEX SETS

H. Groemer 1. AbscMtzungen fur die Ansahi der konvexen Kör per, die einen konvexen Kör per

beruhren, Monatsh. Math. 65 (1961), 74-81. 2. Ems Bemerkung über Lagerungen konvexer Kegel, Arch. Math. 12 (1961), 78-80. W. Gross 1.

Ueber affine Geometrie XIII: Eine Minimumeigensehaft der Ellipse und des ElKI. '70 (1918), 38-54. lipsoids, Ber. Verh. Sachs. Akad. Wiss. Leipzig.

B. Grünbaum

1. Affine.reguiar polygons inscribed in plane convex sets, Riveon Lematematika 13 (1959), 20-24.

2. Projection constants, Trans. Amer. Math. Soc. 95 (1960), 451-465. 3. Partitions of maw distributions and of convex bodies by hyper planes, Pacific J.

Math. 10 (1960), 1257-1261. 4. On a conjecture of H. Hadwiger, Pacific J. Math. 11 (1961). 215-219. 5. On some properties of convex sets, Colloq. Math. 8 (1961), 39-42. 6. The dimension of intersections of convex 8ets, Pacific .J. Math. (to appearl. 7. The distance between circumcircie center and cent roid in plane convex sets, (in preparation). H. Hadwiger

für die einen Eikörper uberdeckenden und unterdeckenden Paredielotope, Elem. Math. 10 122-124. P. C. Hammer 1. The centroid of a convex body, Proc. Amer. Math. Soc. 2 (1951), 522-525. 2. Convex bodies associated with a convex body, Proc. Amer. Math. Soc. 2 (1951), 1.

781-793.

3. Diameters of convex bodies, Proc. Amer. Math. Soc. 5 (1954), 304.306. 4. Volumes cut from convex bodies by planes, Mathematika (to appear).

P. C. Hammer and A. Sobczyk 1. Critical points of a convex body, Abstract 112, Bull. Amer. Math. Soc. 57 (1951), 127. J. Hirakawa 1. On a characteristic property of the circle, Tôhoku Math. J. 37 (1933), 175-178. F. John 1. Extrensum problems with inequalities as subsidiary conditions, Courant Anniv. Volume, 1948, pp. 187-204.

P. J. Kelly 1. A property of Minkowskian circles, Amer. Math. Monthly 57 (1950). 677-678. V. L. KIee, Jr. 1. The critical set of a convex body, Amer. J. Math. 75 (1953), 178-188. H. Knothe 1.

Ems kennzeichnende Eigenschaft der Ellipse, Math. Z. 60 (1954), 235-242.

A.

1. A proof of an Ulam theorem on convex bodies, CoIloq. Math. 4 (1957), 216-218. 2. On involution and families of conlpacta, Bull. Acad. Polon. Sci. CI. III 5 (1957), 1055-1059.

3. On a problem of Steinhaus, Fund. Math. 46 (1958), 47-59. 4. A theorem on families of acyclic sets and its applications, (to appear). B. N. Kozinec

1.

the area of the kernel of an oval, Leningrad. Gos. Univ. Nauk. 1958, 83-89. (Russian) On

Zap. Ser. Mat.

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T. Kubota 1.

Ueber

die &hwerpunkte der konvezen geachiossenen Kurven und Flãchen, Tôhoku

Math. J. 14

(1918), 20-27.

D. Laugwitz 1. Konvexe Mittelpunktsberezche und

Rtlume, Math. Z. 61

(1958), 235-244.

M. A. Lavrent'ev and L. A. Lyusternik 1. Elements of the calculus of varlatu)n8, Vol. 1, Part II, Moscow, 1935.

K. Leichtweiss 1. Zwei Extremaiprobleme der Min/cowski-Geometrie, Math. Z. 62 (1955), 37-49. 2. Ueber die affine Exzentrizität konvexer Kör per, Arch. Math. 10 (1959), 187-199. H. Lenz 1. Einc Kennzeichnung des Ellipsoids, Arch. Math. 8 (1957), 125-129.

F. W. Levi 1.

Ueber

zwei Satze von Herrn Besicovitch, Arch. Math. 3 (1952), 125-129.

V. Linis 1.

Problem 45-46.

E1170, Amer. Math. Monthly 62

(1955), 365;

Solution, ibid. 63 (1956),

A. M. Macbeath 1. A compactness theorem for affine equivale,we-classes of convex regions, Canad. J. Math. 3 (1951), 54-61. S. Matumura 1.

Ueber Ftdchen und Kurven XXII. Einige Bemerkungen über die Theorie der

Konvexkurven Mem. Fac. Sci. Agr. Taihoku Univ. 28 (1940), 259-297. H. Minkowski 1. AIls gemeine Lehrs.Itze über konvexe Polye4er, Nachr. Ges. Wiss. Gottingen 1897,

2.

(=Ges. Abh., Vol. 2, pp. 103-121, 1911). Abh. 2(1911), 230Voiumen und Oberftäche, Math. Ann. 57 (1903), 447-495

3.

276). Theorie der konvexen KUr per, inebesondere Begründung ihres Oberfiachenbegriffs,

198-219

Ges. Abh., Vol. 2, pp. 131-229, Leipzig.Berlin, 1911. 0. Miyatake 1. On a characteristic property of the circle, Tbhoku Math. J. 45 (1939), 245-248. (See also the review by H. Gericke in Jber. Fortschr. Math. 65 (1939), 823.) T. S. Motzkin 1. Endovectors, these Proceedings, pp. 361-387. B. H. Neumann 1. On some affine invariants of closed convex regions, J. London Math. Soc. 14 (1939), 262-272.

2.

On

an invariant of plane regions and mass distributions, J. London Math. Soc.

20

(1945), 226-237.

3. Sharing ham and eggs, Iota (Manchester) 1 (1959), 14-18.

D. J. Newman 1. Partitioning of areas by straight lines, Abstract 548-108, Notices Amer. Math. Soc. 5 (1958), 510.

J. F. Pal 1. Om konvekse figurers tyngdepunkter, Mat. Tidsskrift B 1937, C. M. Petty 1.

101-103.

On the geometry of the Minkowski plane, Riv. Mat. Univ. Parma 6 (1955), 269-292.

MEASURES OF SYMMETRY FOR CONVEX SETS

A. Pleijel 1. Ueber die Teilung von ebenen konvexen Berewhen durch Sehnen, Math. Scand. 2 (1954), 74-82.

C. Pucci 1.

Sulla inscrimbihtâ di un ottaedro regolare in un insiemo convesso limitato deUo spazio ordinario, Atti Accad. Naz. Lincei. Rend. CI. Sci. Fis. Mat. Nat. (8) 21 (1956), 61-65.

H. Rademacher 1. Ueber den Vektorenbereich eznes konvexen ebenen Bereiches, Jber. Deutsch. Math. Verein. 34 (1925), 64-79. R. Rado 1. Some covering theorems, I, Proc. London Math. Soc. (2) 51 (1949-50), 232-264.

2. A theorem on general measure, J. London Math. Soc. 21 (1946), 291-300. j. Radon I. Ueber sine Erweiterung des Begriffs der konvexen Funktionen, mit einer Anwendung auf die Theorie der konvexen Körper, S. -B. Akad. Wiss. Wien 125 (1916), 241-258.

2.

(Jeber die Bestimmung von Funktionen durch ihre Integraiwerte Lange gewisser Mannigfaltigkeiten, Ber. Verh. Akad. Wiss. Leipzig. Math.-Nat. KI. éø (1917), 262-277.

K. Radziszewski 1. Sur tin problème extremal relatif aux figure8 inscrites et circonscrite aux figures convexes, Ann. Univ. Mariae Curie-Sklodowska Sect. A 6 (1952), 5-18. Yu. G.

1. An extremal problem from the theory of convex curves, Uspehi Mat. Nauk 8 (1953), no. 6 (58). 125-126. (Russian). C. A. Rogers and G. C. Shephard 1. The difference body of a convex body, Arch. Math. 8 (1957), 220-233. 2. Some extremal problems for convex bodies, Mathematika 5 (1958), 93-102. 3. Convex bodies associated with a given convex body, J. London Math. Soc. 33 (1958), 270-281.

D. Rutovitz 1. Parameters associated with Minkowsici spaces, J. London Math. Soc. (to appear). E. Salkowski 1.

Eine Kennzeichnende Eigenschaft des Kreise8, S.-B. Heidelberger Akad. Wiss.

Math. J. Schopp 1.

Nat. KI. No. 14 (1934).

Ueber sine Extremaleigenschaft des Simplex im n-dimensionalen Raum, Elem. Math. 13 (1958), 106-107.

G. C. Shephard 1.

Inequalities

between mixed volumes of convex sets, Mathematika 7 (1960),

125-138.

M. Sholander 1. S.

Proof of a conjecture of R. S. and E. F. Buck, Math.

Mag.

24 (1950), 7-10.

Stein

1. The symmetry function in a convex body, Pacific. J. Math. 6 (1956), 145-148. 2. An appiwation of topology to convex bodies, Math. Ann. 132 (1956), 148-149. 3. Steiner 1. Von dem Krümmungsschwerpunkte ebener Curven, J. Reine Angew. Math. 21 101 122 (=Ges. Werke Bd. 2, pp. 99-159, C. Reimer, Berlin, 1882). (1840),

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H. Steinhaus 1. Mathe,natical snapshots, Oxford Univ. Press, New York, 1950. 2. Quelques applications des principee topologiques a La géo,nétrie des corps convexea, Fund. Math, 41 (1955), 284-290. B. M. Stewart 1. Asymmetry of a plane convex set with respect to its centroid, Pacific J. Math. 8 (1958), 335-337.

K. Strubecker 1. DifferentiaL geometrie ties isotropen Raumes. Zur Tlieor-ie der Eilinien, Math. Z. 51 (1949), 525-573. B. Su 1. On Steiner's curvature-centroid, Japan. J. Math. 4 (1927), 195-201. W. Süss 1.

tJeber den Vektorenbereich eines Eikörperg, Jber. Deutsch. Math. Verein. 37 (1928), 87-90.

2. Ueber ems Affininvariante von Eibereichen, Arch. Math. 1 (1948), 127-128. 3. Ueber Eibereiche mit Mitteipunkt, Semesterber. 1 (1950), 273-287. 4. Ueber ParaUelogramme und Rechtecke, die rich ebenen Eibereichen einbeschreiben iezssen, Rend. Mat. e AppI. (5) 14 (1955), 338-341. B. V. Sz.-Nagy 1. Schwerpunkt von konvezen Kurven und von konvexen Fldchen, Portugal. Math. $ (1949), 17-22.

A. E. Taylor 1. A geometric theorem and its application to biorthogonat sy8tems, Bull. Amer. Math. Soc. 53 (1947), 614-616. P. Ungar 1. Freak theorem about functions on a sphere, J. London Math. Soc. 29 (1954), 100103.

U. Viet 1. Umkehrung eines Satzes von H. Brunn uSer Mittelpunktseibereichs, Semesterber. 5 (1956), 141-142. 1. M. Yaglom and V. G. Boltyanskii 1. Convex figures, Gos. lzd. Tehn.-Teor. Lit., Moscow-Leningrad, 1951. (Russian); German translation published 1956, English translation published 1961. V. L. Zaguskin 1. On circumscribed and inscribed ellipsoids of extremal volume, Uspehi Mat. Nauk 13 (1958), no. 6 (84), 89-92. (Russian). K. Zindler 1. Usber konvexe Osbilde. I, II, 111, Monatsh. Math. 30 (1920), 87-102; 31 (1921), 2556; 32 (1922), 107-138. UNIVERSITY 01 WA8HINCTON AND HEBREW UNIVERSITY, JERUSALEM

BORSUK'S PROBLEM AND RELATED QUESTIONS1 BY

BRANKO GRUNBAUM 1. Introduction. If the regular simplex is partitioned into n sets, space =

of diameter1 1 in Euclidean n-

A,, then at least one of the. sets A4 has diameter 1. This assertion is completely obvious since at least contains two or more vertices of one of the sets A much deeper result is that of Borsuk [1]: If the n-dimensional solid ball

B' of diameter 1 is partitioned into n parts at least one of these sets has diameter 1. This is a consequence of the following result (cf. LusternikSchnirelmann [1], Borsuk [1; 2]): If the (n — 1)-dimensional sphere S"1 (the boundary of a solid ball in E') is covered by the union of n closed sets, then at least one of these sets contains a pair of antipodal (i.e., diametrally opposite) points of S*_I. These facts led Borsuk [2] to formulate the following problem: Is it possible to partition every bounded set A c E*, with diam A > 0, into n + 1 sets, each of diameter less than that of A? The present paper is devoted to a summary of the known partial answers to Borsuk's problem, and to some related questions. One of the most interesting aspects of this subject is the great discrepancy between the elementary formulation of the problem and the, at least at the present, unsurmountable difficulties of its solution. Also, Borsuk's problem

and the results that let to it may be examined from many various viewpoints, and yield in different directions very appealing possibilities of gener-

alization. The author felt free to mention many such variants as well as

other problems that seemed to fit in the context, but clearly no attempt can be made at completeness in such a vaguely defined field as that indicated by the title of the present paper. 2. ReductIons of the problem. Borsuk's partition problem refers to all subsets of E. Obvious arguments show that one may consider, without loss of generality, only sets A which are closed, and of diameter 1. Moreover, due to the well-known fact that the diameter of any set is the same as that of its convex hull1, it is enough to establish the solution of Borsuk's problem for compact, convex sets A. An additional assumption, which can be made whenever convenient, is that A is a convex body, i.e., a compact convex set with This work was supported by a grant from the National Science Foundation, USA (NSFG18975). * The diameter diam A of a set A in a metric space is the l.u.b. of the distances between pairs of points belonging to A. * The convex hull of a set A is the intersection of all the convex sets containing A. The convex hull of a bounded and closed (i.e., compact) set in E is compact.

271

BRANKO GRUNBAUM

212

nonempty interior. Given a convex body A in

of diameter 1, if it is possible to partition the boundary B of A into n + 1 parts, each of diameter less than 1, the same kind of partition is possible for A. Indeed, the decomposition of B being given by B = B1, and x being an interior point of A, the sets A1 = u B1) determine a partition of A with diam A1 < 1 for all i. (Strictly speaking, the point x should be contained in only one of the sets A, in order to have a partition of A; but since a covering of a set can always be changed into a partition of the set, and the diameters of the parts do not increase under this procedure, we shall in the sequel be satisfied by finding covers of A by sets of diameters less than diam A.) One more reduction of Borsuk's problem is possible, due to a result in the theory of convex sets. For a convex body K in and a unit vector u, the width of K in direction u is defined as the length of the orthogonal projection of K onto straight line parallel to u. (Equivalently, it is the least distance between a pair of hyperplanes orthogonal to u and enclosing K.) A convex body K which has the same width c in every direction u is said to be of constant width c; obviously, for every such K also diam K = c. Pal [1] proved (for n = 2; see Bonnesen-Fenchel [1] or Eggleston [5] for a proof in the general case and for further references) the following result, relevant to Borsuk's problem: Every set A c E* is a subset of a convex body K of constant width equal to diamA. In view of Pal's result and the previous remarks, Rorsuk's problem is equivalent to the following: May the boundary of every convex body K c of constant width 1 be partitioned into (or covered by) n + 1 sets, each of diameter less than 1? We digress to mention some open problems on sets of constant width. (i) Is every smooth4 convex body K c contained in a smpoth convex body of constant width diam K? (L. Danzer, private communication.)5 (ii) What is the greatest real number c such that every convex body K c EW which has width at least 1 in every direction Contains a subset of constant width c? Eggleston (2] proved that c = (3 + )16 for n = 2. For n> 2 the only related result known seems to be that of Steinhagen [1) (see also Santaló [1]) to the effect that every convex body K c with minimal width 1, contains a ball of diameter at least (n + 2)11t/(n + 1) for even n, and for odd n. 3.

PartIal solutioni of Borsuk's problem. At the present, no complete

solution of Borsuk's problem is known. The following theorem is representative of the established facts in (see Hadwiger [4; 5; 6], Perkal [1], Lenz [1] for this and some closely related results): 4 A convex body K is smooth if every boundary point of K is contained in exactly one supporting hyperplane of K. See Note 1 on p. 284.

BORSUK'S PROBLEM AND RELATED QUESTIONS

273

Every smooth convex body K c E* may be partitioned into n + 1 sets, each of diameter less than diam K.

For a sharpening of this result to some not necessarily smooth sets see Anderson•Klee [1).

The proof of the theorem uses a mapping f, which assigns to each point x in the boundary of a convex body K c E* a subset f(x) of the (n — 1)-dibelongs to 1(x) if and in the following way: ye mensional sphere at y is parallel to a supporting only if the supporting hyperplane to and K contained in the corresponding halfhyperplane of K at x (with spaces). The affirmative solution of Borsuk's problem for smooth K follows x and easily from the observation that if x, may equal diam K only if the points f(x) and f(x*) are antipodal on S'-1, and from the possibility of partition of Sw.I into n + 1 closed sets, none of which contains a pair of antipodal points. An affirmative solution to Borsuk's problem, without any restrictions, is known for V. The first proof was published by Eggleston 11] in 1955; it seems that a very similar proof was found by J. Perkal and presented in a talk to the Polish Mathematical Society in Warszaw in 1947 (Perkal [1] and private communication). The proof also uses• the mapping f described above, for an arbitrary K of constant width; the difficulty, in the absence of smooth-

ness, is in the proof of the existence of a covering of S by four sets S1, none containing pairs of antipodal points, and such that if an S has a nonempty intersection with some 1(x) (which in general may be a 2-dimensional subset of S'), then f(x) c S1. (For details see Eggleston [1] or (3, pp. 77-92].)

Due to difficulties of a topological nature, no extension of this method to higher dimensions is known. 4. Universal covers. For dimensions 3 or less, an approach much more elementary than that described above leads to results even stronger than the affirmative solution of Borsuk's problem. The idea is to find first a suitable universal cover C for all sets of diameter 1 (i.e., a compact convex set C

such that for every set A with diam A 1 there exists a subset B of C congruent to A), and then to describe an appropriate partition of C. This method has been successfully applied to Borsuk's problem in 2 and 3 dimensions, as well as to some variants of Borsuk's problem (see § 5). In the

FIGURE 1

Z74

BRANKO GR(YNBAUM

plane, one possibility is to establish, first, Pal's [1] result that the regular is a universal cover for sets hexagon H of width 1 (i.e., of diameter of diameter 1 (see, e.g., Gale [1], YaglomBoltyanskii [1], Eggleston [5]), and then to observe that H may be partitioned into three congruent pentagons F, each of diameter (see Figure 1). For a similar approach in the plane, applicable in a more general situation, see § 6.

It is worth noticing that the above result that every plane set of diameter 1 may be partitioned into three sets, each of diameter is the best possible. Indeed, it is easily seen that in every partition of the circular disc of diameter 1 into three parts, at least one of the parts has a diameter In three dimensions the method of universal covers is still applicable, though

it does not seem to lead to any "best possible" result. The universal cover for sets of diameter 1 in E' is a truncated octahedron T, obtained from the regular octahedron of width 1 (i.e., of diameter by cutting off three of its vertices by planes perpendicular to diameters and at distance 1/2 from the

The resulting polyhedron T has diameter V'T. Two somewhat different partitions of T into four parts, each of diameter <1, have been described in the literature; neither is optimal even as a partition of T. The diameters of the parts are <0.9977 in the partition of Heppes [2], while in center.

that of Grunbaum 12] the diameters are <0.9887. The polyhedron T is not, in any sense, a minimal universal cover, and it

is probable that one could reduce somewhat the bounds given above by additional truncations of T and meticulous partitioning of the resulting sets. It seems safe to predict that only small improvements can be obtained in this way, and it is highly improbable that the method of universal covers will lead to a solution of Borsuk's problem in dimensions greater than 3. (See however the comments on pp. 78, 91-92 of Eggleston [3].) A few problems on universal covers seem to deserve mention. One is rather old; in 1914 Lebesgue (see Pal [1]) asked for the universal cover (for sets of diameter 1 in the plane) with minimal area, or with minimal perimeter length. The problem is still unsolved; Pal 1.11 proved that the minimal area of a universal cover is between 0.8257 = + and 0.8454 = 2(3 — and 3.382 = 8 — the minimal perimeter is between 3.302 = Another problem, due to V. Klee (oral communication) is new. A minimal

cover is a universal cover no closed, convex proper subset of which is a universal cover. Does there exist a number c,, depending on the dimension n

only such that the diameter of every minimal cover does not exceed While the answer is easily seen to be affirmative for n = 2 (with c2 <3), the problem is open for n 3, as is the question of the exact value of c2. 5. Other resulta on partitions. As a generalization of Borsuk's problem one may consider the following question. For a convex body K c EM, with diam K = 1, and a positive integer k, let d(K, k) be the g.l.b. of reals a such that K may be partitioned into k subsets, each of diameter Let also Using familiar compactness arguments, Borsuk's d%(k) = l.u.b. {d(K, k) : Kc problem is easily seen to be equivalent to the question whether + 1) < 1

BORSIJK'S PROBLEM AND RELATED QUESTIONS

for every n. The problem of determining

275

k) (where B* denotes the solid

n-dimensional Euclidean ball of diameter 1) was mentioned by Knaster [1];

but even the problem of finding d(B", n + 1) seems still to be open. The n + 1) does not shows that

obvious simplicial decomposition of exceed

/n+1

r

forevenn,

n+2

/n—1

V

2

for oddn.

2Vn+3

Hadwiger [7) proved that equality holds in these estimates for n established for n 4 only d(B",n + 1)

3,

but

+

In connection with Borsuk's problem it is worth mentioning that for every K c E2 different from B2 the inequality d(K, 3) < d(B2, 3) = holds (Lenz [2]). In no K is known with

d(K,4)> d(B*,4)= 17(3 + The conjecture wiger [7], Gale [1]).

+ 1) = d(B', n + 1) has been repeatedly proposed (Had-

The situation in the plane was investigated in some detail by Lenz [2; 3]; he obtained the following exact bounds: d2(3) =

d2(4) =

and d2(7) =

1/2.

The bound for d2(3) was established already by Gale [1], while that for d2(4) was found also by Self ridge [1]. In each of these cases, the bound is attained only for the circular disc B2. For other values of k, estimates of d2(k) are given by Lenz [2; 3]; he also deals with some related questions. The problem whether d2(k) = d(Bt, k) for all k was posed in Self ridge [1); for k = 5, 6 also in Lenz [2]. At least for k = 6 it is easy to see that the answer is negative. If L denotes the regular il-gon of diameter 1, we have d2(6) d(L, 6) = 1/2 cos = 0.505 > d(B2, 6) = 1/2. Problems similar to that of may be considered also in metric spaces different from E*. For two-dimensional Minkowski spaces with unit cell different from a parallelogram, a simple reasoning due to E. Shamir (see GrUnbaum [3]) shows that the analog of d2(3) is less than 1. For related problems on the sphere S2 see Fejes-Tóth [1] and the references given there. Returning to the Euclidean case, one may rephrase Borsuk's problem as follows. Let k = Mn) be the smallest integer such that < 1; is k(n) n + 1? The best estimate known for general n seems to be Mn) due

+ 2)- (2 +

to L. Danzer [2]. This bound is derived by considering coverings by

BRANKO GRUNBAUM

276

balls (see next section); it is remarkable that no reasonable bounds for Mn) seem to be known even for n = 4. 6. Coverings by translates. Borsuk's problem may be formulated also in the following way. Is it possible to cover every set K c E', with diam K = 1,

by the union of n + 1 sets, each of diameter <1? From this formulation one is easily led to the question whether such a covering may also be accomplished

if the n + 1 members of the cover are assumed to be related to each other in some specific way. In order to simplify the exposition of known results and open problems in this direction, we shalt use the following terminology. Given a convex body Kc E", a set A c E" will be said to be of K-diameter if for every pair of points of A there exists a translate of aK containing those two points. Obviously, if K is a convex body of constant width 1, the notion of K-diameter coincides with that of the usual diameter. On the other hand, if K has a ceilter of symmetry, then the K-diameter of a set A Coincides with the diameter of A in the Minkowski metric which has 2K as unit cell.

We define ô(m, K) to be the minimum of o > 0 such that every set of Kdiameter less than or equal to 1 may be covered by m translates of 6K. Let G(K) = mm {m: ô(m, K) 1}; an equivalent definition of G(K) is: G(K) is the least integer m with the property that whenever a family of transhave a nonempty lates of K is given such that every two members of intersection, it is possible to partition into m subfamilies 31', in such a way that for each j the intersection of all the members of 31', be nonempty.

We also put L(K) to be the least m such that K may be covered by m translates of aK, for some a < 1, and L(n) = max {L(K) : K c E'}. For the unit disc B' it was proved by Lenz [21 that 6(3, B') = v'112, 6(4, B') = 6(7, B') = 1/2, and t3(m, B')

i/t/2

On the other hand, for every centrally symmetric K c E' different from a parallelogram, 6(3, K) < 1 (Grünbaum 13)); an open problem in this connection is whether, as indicated by "experimental evidence," 5(3, K) = 1/6(1, K).

Bounds for 6(1, K) were first considered by Jung (1] for the case of B"; he determined o(i, B") (see

=

Chapter 6 of Danzer-Grünbaum-Klee Eli for additional references and for

related notions and results). For centrally symmetric Kc E' the bound 2n1(n + 1) was first established by Bohnenblust Li]; for other proofs see Leichtweiss [1], Eggleston [4], Volkov [11, Grunbaum [4]; the last paper also contains a discussion of some related concepts, and the bound 5(1, K) n for an arbitrary convex body K c E". For sets K" c E" of constant width 1, one may also consider the constant S*(m, K") defined in the same way as S(m, K") except for allowing rotations of the sets 6K". (Obviously, S*(m, B") = S(m, B') and, for m > n, the equality between S probably characterizes B".) We conjecture that J(n + 1, K") <1 6(1, K)

BORSUK'S PROBLEM AND RELATED QUESTIONS

277

FIGURE 2

(ab =

1)

+ 1, K") S(n + 1, B") for every K" of constant width 1. (The and that conjecture 5(3, K2) < 1 was presented by the author at a problem-session of the Symposium on Convexity.) The assertion 5*(3, K2) < 1 is very easy to establish using Pal's universal cover 4); indeed, every K2 of constant width and therefore also the set R 1 contains a circular disc of radius 1 — (Figure 2). But R, and even 0.95R, contains the pentagon P (see Figure 1), and thus o*(3, K2) <0.95.6

was first established The relation G(K) 3, for centrally symmetric K c for K = B2 by Hadwiger-Debrunner [1; 2]; for the general case see Grünbaum 3 for all convex bodies K c E2 is still un[5]. The question whether G(K) solved, as is the more general conjecture that G(K) n + 1 for every K c E" (Grünbaum [5)). For related topics and a summary of known results on different variants of G(K) see Chapter 6 of Danzer-GrUnbaum-Klee [1]. See also Schopp [1). The relation G(K) = 1 characterizes K as a parallelotope (Sz.-Nagy [1], Hanner [1]). The alternative definition of G(K) makes sense for not necessarily convex sets. The question whether parallelotopes are the only compact sets K with G(K) = 1 was raised in Nachbin [1] by gremlins (private communication), and quoted by Sz.-Nagy [1]. The simplest counter-example is furnished

by a set K consisting of two distinct points. It seems reasonable, however, to conjecture that G(K) = 1 characterizes parallelotopes among compact and connected sets. For K c E2 and different from a parallelogram Levi [2] proved that L(K) = 3; for a parallelogram K obviously L(K) = 4; thus L(2) = 4. For centrally sym-

metric K c E2 this result was strengthened in Grünbaum [3): K may be covered by three translates of (5(1, K))1 . K, but not by three translates of aK with a . 5(1, K) < 1. No similar generalization seems to be known for either centrally symmetric K in higher dimensions, or for general K in the plane. 6

For some notions related to L(K) and L(n) see Levi [1].

See Note

2 on p.

284.

BRANKOGR(JNBAUM

278

one may conjecture that The n-dimensional cube shows that L(n) (Hadwiger [9]). The problem is open even for n 3•7 For smooth Kc E" it is easily seen that L(K) n + 1 (Levi [21); actually, for such K we have L(K) = n + 1 (see § 8). A related conjecture is that of Hadwiger [11]: Every convex body K c E" is contained in the interior of the union of at most For a sets, each of which is the convex hull of K and some point of discussion of other variants of L(n) see Klee [1]. L(n) =

The following result may be considered as a corroboration of the conjecture

be called antipodal if Let two points of a convex body Kc they are contained in a pair of (distinct) parallel supporting hyperplanes of K; let V(K) be the maximal number of points of K, each two of which are for every K c E" (Danzer-Grünbaum [11). This antipodal. Then V(K) result, which solves a problem of Klee (1], settles at the same time a question of Erdös [2): What is the maximal number E(n) of points in E" such that it is easy to all the angles determined by triples of these points are see that E(n) is at most equal to max {V(K): Kc E*}, and E(n) = follows. Worth mentioning is the still open problem (Erdös [2]) about the maximal number E*(n) of points in E* such that all the angles determined by triples L(n) =

of those points be acute. In the plane, E*(2) = 3 is easily established; E*(3) = 5

has been proved only recently (Croft [1]), while for n > 3 there seems to be even no conjecture about the value of E*(n). 7. Finite setB. A completely different approach, also of an elementary nature, was used in order to establish an affirmative solution to Borsuk's problem for finite sets in E2 and E (but the problem is still open for finite sets in E", n > 3). with diameter A = 1, Considering a set A, consisting of k points in

one may ask how many pairs of points of A are at distance 1. Let us denote

the maximal possible number of pairs by F(k, n). A very simple proof of F(k, 2) = k was given by Erdös [I), where also the question about F(k, 3) is raised and a conjecture of A. Vazsonyi that F(k, 3) = 2k — 2 is quoted. In the

wake of Erdös' travels in 1956, three independent proofs of Vázsonyi's conjecture were published (Grtinbaum [1], Heppes [1], Straszewicz [1]). From the values F(k, 2) = k [resp. F(k, 3) = 2k — 2] it is easy to derive the a direct affirmative answer to Borsuk's problem for finite sets in E2 and proof for the three-dimensional case, along similar lines, was given by HeppesRêvész [1).

In higher dimensions the values of F(k, n) are not known (Hadwiger [8)).

For every k> n

2 it is easy to construct in E* sets which show that

F(k, n) (n — 1)k — (n + lXn — 2)/2 (Hadwiger (8, Appendix)).8 Even if the conjecture that F(k, n) equals to this expression were established, it probably would not lead to an answer to Borsuk's problem; this approach seems feasible only for those n for which F(k, n) k(n + 1)12, i.e., n 3.

The problem of determining F(k, n) for n > 3 seems to be very difficult, 8

See Note 3 on p. 284. See Note 4 on p. 284.

BORSUK'S PROBLEM AND RELATED QUESTIONS

mainly because of the essential difference in the strucfure of the graphs determined by the edges of convex polyhedra in E' for n 3 and n > 3 (noticed already in Carathéodory [1]; for a review of the results known about such 'polyhedral graphs' see Griinbaum-Motzkin (1]). 8. Other related problemB. The theorem of Borsuk-Lusternik-Schnirelmann may be formulated as follows: If S" is covered by the union of n + 1 closed sets, at least one of the sets contains a pair of points the spherical distance between which is it. This result may be considered as a special case of the following problem (in which the number of alternatives could be increased easily).

What is the greatest integer m such that for every covering of (1)

sets

by m (2) closed sets (3)

M

closed congruent sets

at least one of the sets M. contains, for (1)

a given

every fixed9 (3) all (2)

positive 0 not exceeding the diameter of the space, a pair of points at distance 0.

To give an example of the rather self-explanatory notation we shall use, the theorem of Borsuk-Lusternik-Schnirelmann could be formulated, with 0 = it, 2, 1) n + 1; the simplicial decomposition of shows that equality as

holds in this relation. The conjecture m(S*, 2,3) = n + 1 (i.e., in every covering of S" by n + 1 closed sets, the points of at least one of the sets realize all distances, and the number n + 1 is maximal with respect to that property) has been forwarded by Hadwiger [2; 10], and established by Hadwiger [1] for n = 1,2. The weaker proposition 2,2) = n + 1 was proved (in a more general setting) by H. Hopf [1]. Hadwiger [31 was able to show that 2, 2) 3n — 1, where the asterisk * indicates that only distances o with cos 0 —1/n are considered. Under the same additional condition on 0, Hadwiger [2] showed 2,3) = n + 1. that For Euclidean spaces, Hadwiger [2] proved that 2,3) n + 1. Except

for n = 1, it is not known whether n + 1 is the best bound; the best upper estimate for n = 2 seems to be m(E2, 2, 3) m(E*, 3,3) 6 (Hadwiger-Debrunner [2]).

A related result due to Hadwiger [3] is

n

2.

For a discussion of other variants in case of dimensions one and two see Hadwiger-Debrunner [2].

Another unsolved problem is the determination of question whether .' Mc depenthng on ö.

2, 2) =

2, 3) seems to be open.

2,2); even the

280

BRANKOGRUNBAUM

Problems like that of m(E*, 1, 1) concerning coverings by (or decompositions

into) arbitrary sets may be rephrased conveniently in terms of graphs and their colorings. Let a 0-graph be any graph whose nodes are (some or all) points of ER (or Sn), two nodes being connected by an edge if and only if 1, 1) their distance is 0. A question equivalent to the determination of is to find c,, = 1 + m(E*, 1, 1), the maximum of the color-numbers of 0-graphs in ER (The color-number of a graph is the least integer k such that the nodes of the graph may be divided into k classes, with no class containing two nodes connected by an edge.) By a theorem of de Bruijn-Erdös [1], it is sufficient to determine the color-numbers of finite 0-graphs. A simple example (Moser-Moser [1J) shows that c2 4 (see Figure 3). Similar examples in higher dimensions show that n + 2. The exact value of c,, is not known even for n = 2; the best upper bound known in this case is c2 7 by the result of

FIGURE 3

Hadwiger-Debrunner t21 mentioned before (since 1 + m(E2, 1, 1) m(E2, 3, 3) + 1 7). For additional comments on c,, see Klee [1), where the

question about the value of c, is attributed to E. Nelson. The numbers c defined for S" in analogy to the definition of depend = 2, while on 0; thus obviously fl + 2 if C,, satisfies cos 0,, = —1/n. It seems even not to be known whether C,, = l.u.b. {c(O): 0 <0 is finite (except for n = 1, since trivially C1 = 3). A different series of results related to the theorem of Borsuk-LusternikSchnirelmann concerns involutions of S" more general than the mapping of a point into its antipode. For example, the following result was proved (as a consequence of more general results) by various authors (see Yang [1], Jaworowski [1], where references to other papers may be found): For every continuous involution f of and for every covering of by n + 1 closed sets, at least one of the sets contains a point x and its image f(x). For n = 2 a stronger result was obtained by Shkliarsky [1]: For an arbitrary homeomorphism f of onto itself, and for any covering of S2 by three closed sets, at least one of the sets contains a point x and its image f(x). We conclude with a problem due to Knaster, and solved by him in case n = 2, 3 (Knaster [1]).

281

BORSUK'S PROBLEM AND RELATED QUESTIONS

of those topological spaces T which allow fixCharacterize the class point free involutions and have the property: For every fix-point free invo-

lution I of T, and for every covering of T by n the sets contains a point x and its image 1(x).

closed

sets, at least one of

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Notes added in proof, August 27, 1962. 1. The answer is affirmative. = 0.904. 2. L. Danzer proved (private communication) that o*(3, KZ) + 2"(n log n ÷ n loglog n + 3. C. A. Rogers proved (private communication) that L(K) 5n) for K — K c E"; the same estimate holds also for the number of translates 4.

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Tud. Akad. Mat. Kutató Int. Közl. 5 4. (1/2) — (2[n/2D' for every n

ON POLYHEDRAL GRAPHSt BY

BRANKO GRUNBAUM AND THEODORE S. MOTZKIN 1. Introduction. Tait [8] seems to have been the first to point out the desirability of studying the particular class of graphs he called "true polyhedra", which are determined by the vertices and edges of convex polyhedra. In recent years these "polyhedral" graphs received renewed (see [41 and the references given there). While it is intuitively clear (and very easy to prove) that a graph arising from a polygon (i.e., polyhedron in the plane) can not arise also from a polyhedron in E2, the result of Carathéodory [21, Gale [3] and Motzkin [612 (see below) shows that this is not necessarily the case in higher dimensions. In the present paper we study some of the ramifications of the notion "the dimension of a polyhedral graph". 2. Polyhedrality. In all that follows, the only graphs admitted are finite and connected graphs without 1- or 2-circuits. We recall that a graph with at least 2 nodes is n-connected if each pair of different nodes may be connected by at least n paths, disjoint except for the

endpoints. DEFINITIoN 1.

A graph G is called polyhedral [resp. k-polyhedral for a positive

integer k 1] if the nodes and edges of G may be identified with the vertices and edges of a [k-dimensional] convex polyhedron P in some Euclidean space. The polyhedron P is called a realization, or an imbedding, of G, while G is called the graph of P. EXAMPLES. 10. A graph is 1-polyhedral if and only if it is the complete graph with 2 nodes. We shall denote by for every positive integer n, the complete graph with n nodes.

2°. A graph is 2-polyhedral if and only if it is a circuit. A graph is 3-polyhedral if and only if it is a 3-connected planar graph. (This is an easy consequence of the characterization of 3-polyhedral graphs given in [4], which, in turn, follows from the results of Steinitz [7J. A graph is planar if it does not Contain as subgraph any refinement of or of C6 — 2C3, the graph obtained from C6 by deleting the edges of two disjoint 30•

C3's, see Kuratowski [5].)

The complete graph with n nodes is (n — 1)-polyhedral. P being any On the other hand, the following surprising result was established by Carathéodory [2], Gale [3] and Motzkin [6]:2 Every C,, with 4°.

(n — 1)-simplex.

5 is 4-polyhedral.

n 3.

Formation of polyhedral graphs. No general procedures seem to be

I The preparation of this work was sponsored in part by the National Science Foundation (NSF-G18975) and by the Office of Naval Research. See Note on p. 290.

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BRANKO GRUNBAUM AND T. S. MOTZKIN

286

known for the formation of polyhedral graphs as such. The only way at our disposal is to generate new polyhedral graphs, via realizations by polyhedra, from given realizations of given polyhedral graphs. The methods consist in performing, on the realizations, one or more of the following operations. (1) (2)

Dualization.

From a polyhedron A of dimension n, cut off a k-face (k < n — 1) by a hyperplane sufficiently close to it; this replaces the k-face by a new (n — 1). face and has other easily recognizable effects on the combinatorial structure of A. (3) Joining two polyhedra A and B (each of dimension n) along a common (n — 1)-face. (Note that the formation of a nonconvex union of A and B can always be avoided by applying suitable, convexity preserving projective transformations to A and B prior to joining them.) A particular case (also obtain. able by combining procedures (1) and (2)) is the formation of the convex hull of a polyhedron A and a point not in A but sufficiently near to an interior point of some (n — 1)-face of A. (4) The convex hull of two polyhedra A and B, of dimensions n resp. m, contained in a pair of skew linear subspaces. As a particular case we mention the formation of pyramids. (5) The direct product of two polyhedra A and B, of dimensions respectively n and A particular case is the formation of prisms. The dimension of the resulting polyhedron is n in (1), (2), and (3), n + m + 1 in (4), and n + m in (5). 4. Ambiguity. The last example in § 2 shows that a graph may be k-polyhedral for more than one value of k. In discussing the properties of polyhedral graphs we find it convenient to introduce the following terminology. DEFINITION 2. A polyhedral graph G is

(i) dimensionally ambiguous, resp.

(ii) strongly ambiguous, resp. (iii) weakly ambiguous

if and only if there exist realizations of G by convex polyhedra P1 and P, such that (i) the dimensions of P1 and P2 are different; resp. (ii) the polyhedra P1 and P2 have the same dimension but are combinatonally different (i.e., differ in the kind, number, or incidence relations, of their faces); resp. (iii) the polyhedra P1 and P2 are combinatorially equivalent, but for some subset of nodes of G the corresponding vertices of P2 determine a face of F1, while the corresponding vertices of P2 fail to determine a face of P2 A polyhedral graph is unambiguous if it is not ambiguous in any of the above meanings (i), (ii), (iii). An unambiguous graph G may be realized by a polyhedron P which is unique (up to combinatorial equivalence), and the correspondence between G and P is also unique (up to combinatorial automorphisms).

Every k-polyhedral graph, k =

2,3, is unambiguous. REMARK. By Example 4, Theorem 1 fails for k-polyhedral graphs with k 4. THEOREM 1.

1,

ON POLYHEDRAL GRAPHS

Before proving Theorem 1 we discuss the known necessary conditions for k-polyhedrality. 5.

Condltlona for polyhedrality.

The difficulties in deciding whether a given

graph is polyhedral (or k-polyhedral for a certain k) result from the absence of a characterization of such graphs by intrinsic (i.e., graph-theoretic) properties for k 4. For k = 2 or 3, a complete characterization is known (see Examples 2 and 3). THEOREM 2 (Bxi.usici [1)).

Every k-polyhedral graph is k-connected.

Every k-polyhedral graph contains as subgraph a refinement of (i.e., a graph obtained by inserting additional nodes in the edges of Ck+i).

THn0REM 3.

In other words, there exist in every k-polyhedral graph k + 1 nodes, all connected by pairwise disjoint paths. 6.

Proof of Theorem 3. We find it convenient to prove a sharper state-

ment, for the formulation of which we need the following notions. A face-chain of an n-dimensional convex polyhedron P is a set of faces P, where P. = P, each Pk is k-dimensional for 0 k n, and of vertices of P is canonical n. A set Ph_lcPk for 1 with respect to a face-chain {Pk) if (i) Vk e Pk for 0 k n; (ii) Vk Pk-1 for 1 k n; (iii) every Vk, k = 1,2,. •,n, is incident to an edge of P conare contained in a k-dimensional face of Pk+j taining V0; and (iv) Pk-i and for 1 k < n. Obviously, every face-chain contains a canonical set of vertices.

Let C be the graph of P. A subgraph Gt of G is a representation of C,,+1 in P canonical relatively to a face-chain {Pk} and a canonical set of vertices {

if Gt consists of a set of paths in G, connecting all pairs of vertices

V1, V,, pairwise disjoint except for the comm9n endpoints, and such that the and V1, with i <j, is contained in P,, while having only path connecting in common with P,. We shall prove Theorem 3 by establishing inductively the following proposition. Given a face-chain {Pk: 0 k n) and a canonical set of vertices {

0 k

of an n-dimensional convex polyhedron P, and given a representation c

n) of

C,, in P,,.-, canonical relatively to a representation of C,,÷1 in P canonical relatively to {Pk: 0

n) and (V*: 0 k n), extending the given representation of C,,. The proposition being obvious for n 3, we proceed by induction, assuming its truth for all integers less than n. Let P* be the (n — 1)-face of P determined by P,_1 and V. - By assumption, the representation of C,,-1 in I',,-, is and canonical relatively to {Pi:k= 0,1, - - -, n —2) U {Pt) is a face-chain in Pt and the set of vertices {Vk: k = -, n — 2, n) is canonical with respect to it. It follows that there exists 0, 1, k

in P a representation * of C,, canonical relatively to those faces and vertices. But the union of Ct with C,,t and with a path connecting V,,_, and V. while missing P,,.-1 U Pt (except at the endpoints), is the desired representation of C,,+1 in P. (The existence of a path between V,,_, and V with the

288

BRANKO GRUNBAUM AND T. S. MOTZKIN

above property is easily established by considering a two-dimensional projection of P obtained by a suitable projective transformation.) This ends the proof of the proposition, and thus also of Theorem 3. 7. Remarks. Theorems 2 and 3 contain the only known necessary conditions for k-polyhedrality. These two conditions are independent: Antiprisms in are realizations of 3-polyhedral 4-connected graphs not containing as subgraph need not be n-connected. any refinement of C.; obviously, a refinement of

The properties of k-polyhedral graphs, for k 4, differ essentially from those of 2- or 3-polyhedral graphs. The latter graphs may not contain "too many" edges, namely, a refinement of C. [resp. C.]. This feature is lost for k 4, since is k-polyhedral for every n > k 4. On the other hand, any 3-connected subgraph of a 3-polyhedral graph is again 3-polyhedral; as easily seen from the examples given below, a subgraph of a 4-polyhedral graph may be 4-connected (and contain a refinement of C.) and fail to be 4-polyhedral. The conditions of Theorems 2 and 3, while necessary for the k-polyhedrality of a graph, k 4, are not sufficient, at least for k = 4. The graph G obtained from C. by deleting the edges of a simple 7-circuit is, as easily checked, 4connected and contains as subgraph a refinement of C. - Because of this last property, it is not a planar graph (Kuratowski [51); since it does not contain a refinement of C., it is not k-polyhedral for k 5. Thus, jf G were polyhedral it would be 4-polyhedral. Assume that Pc E' is a realization of G. Let V0 be the vertex of P corresponding to that node N. of G which is not incident to the deleted 7-circuit. Each vertex V of P, different from V., is contained in at least one 3-face P. of P not incident to V0. The graph of P. is a subgraph of G obtained by deleting N. and possibly other nodes, and the edges incident to them. An easy check reveals that the only subgraph C0 of G of this type which by deleting from G exactly is 3-polyhedral is one node in addition to N. - The only realization P0

of C. is a prismlike polyhedron with six faces (4 triangles and 2 quadrangles) and six vertices. Thus

P. does not contain all the vertices of P different from V0, and therefore P contains at least two 3-faces

not incident to V., each combinatorially equivalent to P.. But those two faces (each containing 6 of the given 7 vertices) have 5 vertices in common which is impossible since P. does not contain pentagonal faces. Therefore C is not polyhedral, as claimed. In a similar way it may be shown that the graph C, — C., obtained from C. by deleting the edges of a 3-circuit, is not polyhedral. 8. Proof of Theorem 1. The assertion is obvious for 1-polyhedral graphs. Since a circuit does not contain any refinement of C4, the assertion about 2polyhedral graphs is also obvious. If G is 3-polyhedral it is not dimensionally ambiguous; indeed, if it were also k-polyhedral for k 4, it would, by Theorem 3, contain a refinement of C. which is impossible (Kuratowski [5]). Thus we

have only to show that each 3-polyhedral graph G contains as subgraphs a well-determined family of circuits, which in every realization P of G bound

ON POLYHEDRAL GRAPHS

the faces of P. To find such a family, we take any realization P of G and we take as the family of circuits the boundaries of the 2-faces of P. Let C be any other realization of G, and let C be the be any such circuit, let circuit on P5 corresponding to C. Assume, if possible, that C5 does not of P* separated by Ct, bound a face of P5; then there exist vertices V$, contains at least one vertex and i.e., such that every path connecting of Ct. (Note that every edge of G which has both endpoints in C is contained in C.) But this is a contradiction since in P, which is also a realization of may be connected by a path missing G, the corresponding vertices V1 and C. This ends the proof of Theorem 1. 9. Examples. We illustrate the relations between the different types of ambiguity of polyhedral graphs (necessarily k-polyhedral with k 4) by the following examples. 5°. The convex hull of n distinct points on the curve given parametrically in E4 by (t, t', is a realization of C,,. As easily verified, already C. is weakly ambiguous; C. is also dimensionally ambiguous, but (see below) it is not strongly ambiguous. 6°. The following possibilities exist for realizations of n-polyhedral graphs with n + 2 nodes, n 2. Let P be an n-dimensional convex polyhedron in E9, with n + 2 vertices x1, 0 i n + 1. Since the x1's are dependent, there exist coefficients a8, not all 0, such that 94-2 9+1 = 0. = 0,

The structure of the graph G of P is determined by the coefficients a8. If

P is such that relation (5) holds with all a8's different from zero, then we may assume the first r of them to be positive and the rest negative, r (n + 3)/2. If r = 1, then x0 is in the convex hull of the remaining vertices of F, in contradiction to the assumption that x. is a vertex of P. Thus r 2. If r = 2, then P is the convex hull of the segment [x0, x,] and the (n — 1)dimensional simplex determined by x4, i 2, which is intersected by [xo, x2J in one point, relatively interior to both. Therefore the graph G is — C2 (the only edge of C,,+2 missing in C being that determined by x. and x1) for n 3, and C4 — 2C2 (i.e., a 4-circuit) for n = 2. (By C,, — 2C. we denote the graph obtained from C,, by deleting 2 edges not incident to one node. The

graph obtained from C. by deleting two edges with a common node is not polyhedral at all.) If r 3, P is the convex hull of an r-simplex and an (n — r)-simplex which intersect in a single point relatively interior to both. Thus G is C,,+2. In case some a in (5) is 0, the remaining n + I points are in an (n — diMensional space. By induction we see that they determine a graph C,,+1 —PC2, whose P is 0, 1, or 2, and therefore G is C,,+2 — PC..

It is not hard to verify that C,,+. — 2C. fails to be, as an n-polyhedral graph, strongly ambiguous. On the other hand, C,1+2 — C. has n — 2 strongly nonisomorphic realizations in E9, while C9+2 has [((n — 2)12)21 strongly nonisomorphic realizations. In particular, C7 is strongly ambiguous as a 5-poly-

BRANKO GRONBAIJM AND T. S. MOTZKIN

hedral graph (though one of the realizations may be considered to be the limiting case of realizations of the other type; but C, is strongly ambiguous in E', with realizations of types which are not limiting cases of other real• izations). 10.

Problems. The following table summarizes our knowledge on the

possible combinations of the different types of ambiguity (blank spaces denote unsolved problems). +

+

+

+

—

—

—

—

strong

+

+-

—

—

+

+

—

—

weak

+

—

-4-

-

+

—

+

—

ambiguity: dimensional

example

C7

C,

C, — C.

Ce—2C.

4, is k-polyhedral Each of the graphs C,,+1, — 2C,, with n — C,, for every k in the range 4 k n. It is not known whether for every polyhedral graph C the set of all k for which G is k-polyhedral consists of consecutive integers. Is the set of all realizations of a given graph in a give'- "i'nension always connected, i.e., can one always join -two given by 3 continuous family of realizations? For another open question on polyhedral graphs (the i led to the present paper) see Problem in the problem-section o tins ,ehirne. Note added in proof, December 14, 1962. We just found another (and prior) occurrence: M. Bruckner, Ueber die Ableitung der aligemeinen FcZytope und die nach Isomorphismus versckiedenen Type,, dci' ailgemeinen Achizelle (Oktato/'e),

Verb. Nederl. Akad. Wetensch. Afd. Natuurk. Sect. I 10 (1909), no. 1, 27 pp. +2 plates. REFERENCES

1. M. L. Balinski, On the graph structure of convex polyhedra in n-apace, Pacific j. Math. ii (1961), 431-434. 2. C. Carathéodory, (Iber den Variabilititabereich der Fourierachen Konstanten von positiven harmonischen Funktionen, Rend. Circ. Mat. Palermo 32 (1911), 193-217 (=Ges. Math. Schriften. III, pp. 78-110, MUnchen, 1955). 3. D. Gale, Neighboring vertices on a convex polyhedron, Annals of Math. Studies, No. 38, Princeton Univ. Press, Princeton, N. J., 1956. 4. B. GrUnbaum and T. S. Motzkin, Longest simple paths in polyhedral graphs, J. London Math. Soc. 37 (1962), 152-160. 5. C. Kuratowski, Sur La problems des courbes gauches en topoiogie,, Fund. Math. 15 (1930), 271-283.

6. T. S. Motzkin, Comonotone curves and polyhedra, Bull. Amer. Math. Soc. 63 (1957), 35.

7. E. Steinitz, Vorleaungen über die Theorie der Polyeder, Springer, Berlin, 1934. 8. P. G. Tait, Listing's "Topologie", Philos. Mag. (5) 17 (1884), 30-46 Papers, Vol. 2, Cambridge Univ. Press, Cambridge, pp. 85-98). UNIVERSITY OF WASHINGTON AND UNIVERSITY OF CALIFORNIA, Los ANGELES

CONVEX CURVES OF CONSTANT MINKOWSKI BREADTH BY

PRESTON C. HAMMER 1. IntroductIon. Euler, in 1778, published a first account of convex curves of constant breadth which were not circles. Since then many mathematicians have contributed to the theory of constant breadth curves in papers or monographs. Among earlier contributors are included Reuleaux, Minkowski, Meissner, Blaschke, Lebesgue, Jordan, Fiedler, ard Schilling. These curves

appear in applications to kinematics, in geometrical probability, and in the study of diameters of sets. If a bounded closed set X in Eft has the property that adjoining one other point will increase the diameter, then X is a constant breadth convex body (or a point). This characteristic property serves to give a means of generalizing constant breadth to metric spaces. Despite the numerous published results concerning constant breadth convex curves, a satisfactory general representation of these was not published until 1953 when Hammer and Sobczyk (22] demonstrated that they were orthogonal trajectories of families of lines which generalize pencils of lines. Later we

made use of this geometrical characterization to give a complete analytical representation (261. This use of directly defined line families enabled us to avoid the restriction of unnecessary class properties used previously and even later in the discussion of constant breadth curves. In this paper we extend the previous results to give a complete representation theory for all convex curves constant breadth relating to a given Minkowskian metric based on a centrally symmetric convex curve as unit circle. Although this application of outwardly simple line families is sufficient to indicate the manner in which they supplement the powerful support function theory of Minkowski, we feel that the theory of differential geometry may be even better served by further study of the more basic concept of differential equivalence which is only illustrated here. We have written but not published an extendable representation of constant breadth surfaces in E1 as transverse surfaces of suitable line families. Dr. H. J. Rebassoo, in his doctoral thesis, has given a complete representation of constant breadth curves on the sphere extending the work done for the plane. Similar considerations for other metric spaces remain to be applied. The work here we have not yet applied to Minkowskian metrics in for n > 2. We believe it is possible but the problem requires much more detailed analysis than is required for the planar case. In order to make this account comparatively self-contained, we will review, with geometrical emphasis, the theory of diametral lines which led us to define the outwardly simple line families. We also present certain relevant 291

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P. C. HAMMER

results proved in Planar line families. I and Planar line families. II 122; 23], and we use other results proved by T. J. Smith in his thesis (1961). It is necessary to summarize our work on associated convex bodies, AB [211, and we alter the form here. While we stick to the central theme in general, we could not bring ourselves to omit the characterizations of classes of equivalent breadth curves by arc length functions. The finite construction of relatively constant breadth curves geometrically could have been omitted. However, these constructions illuminate the theory and permit constructions in elementary geometry. The details of carrying out the limiting process are omitted. II. Notations and conventions. Except as noted we are concerned here with the real affine plane as a vector space in which we assume a Cartesian coordinate system. Lower case letters, a, b, c, d, v, x, y, z, are used to designate

points (vectors) or vector functions. The letter u is specifically reserved to represent points on the unit circle usually in form u(O) = (cos 0, sin 0). Then u'(O) = u(0 + t/2) = (— sin 0, cos 0), and the dot product u(6) . u'(O) = 0. Real numbers or real-valued functions are designated by f, g, h, p, q, i-, s, I. Curves representable in the usual polar coordinate system are given by p(0)u(6) where p(O) is a real-valued function.

Sets except lines, line segments, and curves, are indicated by capitals X, Y, Z, while convex bodies are designated by A, B, C, D and their boundary

curves are indicated by the corresponding Greek letters a,

T,ö.

Scalar

multiplication and direct sum of sets are defined by

IX= {tx:xeX,t a real number), X+ Y= Y}, We note that t0X0 + t,X1 is a convex set when X0 and K1 are convex sets.

We indicate a line parallel to the vector u(0) by m(0). Such a line is uniquely determined by a real number such that P(0)u'(8) is the foot of the perpendicular from the origin 0 to m(6). If p(0) is a function of 0 defined on (—co, 00) then determines a family of lines and p is called the pedal function of F(p). If p has a derivative at each value 0 then the set of points v(8) p(8)u'(O) — P'(8)u(0) is called the envelope of F(p). A S is the closed set bounded by two parallel lines or, in degenerate cases, a strip may be a line. If K is a bounded set then in each direction u there exists a minimal closed strip S(u) containing X with sides orthogonal to u. This is called a supporting strip of X. The Euclidean breadth of X in the direction u is the distance between the parallel sides of S(u). Two bounded sets X and Y are of equivalent breadth provided each supporting strip of one is a translate of a supporting strip of the other. Note that X and Y are of equivalent breadth if and only if their closed convex hulls are of equivalent breadth. Observe that equivalence of breadth is an affine invariant independent of a definition of breadth. III. Diametral chords and central symmetrization. Let C be a convex

CONVEX CURVES OF CONSTANT MINKOWSKI BREADTH

293

body with boundary curve r. In each complete family of parallel chords of C there is at least one of maximal (relative) length. Such a maximal length chord is called a diametral chord of C. Equivalently any chord of C which crosses a supporting strip of C is a diametral chord. 3.1 THEOREM (H.1IIMER AB). The diametral chords of a convex body C cover the body. PROOF. Since every boundary point of C lies on a supporting line of C it is clear that the diametral chords of C cover r. If x is an interior point of C let r be the maximum positive number such that (1 + r)x — rC C and let b be a common boundary point of C and (1 + r)x — rC. Then define a by (1 + r)x — ra = b. It follows that ab is a diametral chord of C through x since if m is a supporting line of C through b then the line m1 such that (1 + r)x — rm1 = m is a supporting line of C through a and parallel to m. Hence C is covered by its diametral chords. Q.E.D. REMARKS. This proof holds for n-space. Note that (1 + r)x — C is the reflection of C through x. DEFINITION. The set = (X — X)/2 is called the symmetroid of the set X and the transformation is called central or pdint symmetrization. 3.2 THEOREM. Let C be •a convex body and let C be its symmetroid. Let r and r5 be the respective boundary curves. The following state,nents hold.

(1) C and

C

breadth

as a unit circle. (2) Let ab be a diametral chord of C crossing a supporting strip S of C. Then [(a — b)/2, (b — a)/2] is a diamefral chord of and S — (a + b)12 is a supporting strip of C5. Hence parallel diametral chords of C and C* have the Minkowsk ian metric based on

the same length. (3) Let D be a convex body with symmetroid D5. Then C and D are equivalent in breadth if and only if C' = D'. Moreover C and D are equivalent in

breadth if and only if their Parallel diametral chords have the same length always. PRooF. (1) and (2). If S is a supporting strip of C crossed by a diametral chord ab then S' since S C. But S5 is a translate, say, S — (a + b)/2 of S and since (a — b)/2, (a + b)12 C5 and the chord joining these points cross S5. we have that S is a supporting strip of C5. Hence C and C5

are equivalent in breadth and have parallel diametral chords of equal length. (3) This result is a corollary of (1) and (2) since D is equivalent in breadth to D5 and hence to C5 if C and D are equivalent in breadth. But two symmetrical convex bodies with the same center point are obviously equivalent in breadth if and only if they are identical. Q.E.D. REMARKS. That C and C * are equivalent in breadth was certainly known to Meissner and quite likely to Minkowski. The use of diametral chords we

have not seen outside our work, although in the theory they play the role comparable to the binormals of constant breadth curves. That parallel

P. C. HAMMER

294

diametral chords go with parallel supporting lines for equivalent breadth curves

is a feature of the differential equivalence we exploit later. be a convex curve sym,netric with respect to 0 and let Let 3.3 B be the convex body P determines. Then a necessary and sufficient condition that a convex body C be of constant breadth in the Minkowskian metric with as unit circle is thaf there be a Positive number r such that C * = rB. Then

length of chord of the

PROOF.

a

diametral chord of C is r times that of a parallel diametral

If the breadth of C relative to

is 2r > 0, then the symmetroid of breadth 2r relative to and hence rB = C*.

C

Q.E.D. IV. Diametral lines and associated convex bodies. A line containing a

diametral choed of a convex body C is called a diametral line of C. 4.1 THEOREM (HAMMER AB). Let C be a convex body gnd let F be the family of diametral lines of C. The following statements hold. (1) F covers the extended plane. (2) F simply covers the exterior of C, including infinite points, if and only if r contains no pair of parallel opposite line segments. COMMENTS. We state this theorem since it shows that we need to select, in certain cases, a subfamily of the diametral line family. A family F0 of diametral lines of a convex body C is said to represent C provided it simply covers the exterior of C, including infinite points. If r contains no parallel

opposite line segments, then F0 must contain all diametral lines of C. Otherwise, there is an infinity of families of diametral lines representing C. For some purposes we want a unique family of diametral lines. The best choice

from several points of view results when we choose F0 as near a pencil as possible. That is, suppose ab and cd are parallel opposite line segments in r so that a — b and c — d have the same sense. Then ad and be are diametral chords of C which intersect in a point x interior to C. We choose those diametral lines of C which pass through x to represent those which cross between ab and cd. With this choice always made, F0 is called the essential diametral line family of C. 4.2 THEOREM.

Let p be the pedal function of a diametral line family F0

which represents the convex body C. such that r is represented by x(O)

Then there exists a unique function f(O)

p(8)u'(O) + f(O)u(8)

where f(O + iv) + f(O) > 0. The function f satisfies f(O) + p'(O) for which p has a derivative.

0 for all 6

Paoor. Since the diametral line family F0 covers the extended plane and since on each line in F there are two end points of a diametral chord, we

have chosen f(O) so that x(8) is the point on m(6) making

CONVEX CURVES OF CONSTANT MINKOWSKI BREADTH

= [f(0) + f(8 + ir)]u(0) have the same sense as u(8). Now, C contains the intersection points of line pairs from F0 by choice of F0. However, as a theorem [Theorem 5.21 esx(8) — x(0 + it)

tablished by T. J. Smith shows, the closed convex hull D of the set of intersection points for F0 is the same as the closed convex hull of the set (p(0)u'(6) — P'(0)u(8)}

for all 0 for which p has a derivative. Hence f(0)

—

p'(0) necessarily since

Q.E.D. No'rE. As the referee kindly pointed out, we should mention that the function f(0) in the representation (1) is necessarily Lipschitzian and, since p and I are periodic and Lipschitzian, they are uniformly Lipschitzian. 4.3 THEOREM. Let r be a convex curve represented by x(0) as in (1) Theorem 4.2. Then for almost all (2) I x'(O + it) I x'(O) + I x'(8) x'(O + it) = 0 PROOF. Since r is convex the difference quotient vector (x(0 + 40) — x(0))/40 will approach a limiting direction (or be 0) as 48 approaches zero through positive (or negative) values. However, the derivative x'(O) may not exist also because of the fact that a unique limiting length may not exist. Now, as proved in Planar line families. II, p is a Lipschitzian function. Hence p has a derivative almost everywhere. Then if we exclude from consideration those values of 0 where P has no derivative and also those values of 0, 8 + ,v where the left and right difference quotients of x(8) do not agree in limiting direction, we have excluded at most a set of measure zero. At the remaining values of 0 we have that x'(8) exists. If x'(O) = 0 or x'(O + it) = 0, (2) holds by default. Otherwise, neither x'(8) = 0 nor x'(O + iv) = 0. Then x'(O) and x'(O + iv) are para:lel and in the opposite sense since x(0), x(6 + iv) are I

end points of a diametral chord and hence (2) holds almost everywhere. Q.E.D. 4.4

THEOREM.

Let

be

the boundary of the symmetroid C* of a convex body be represented

Let F0 be a diametral line family representing C and let by (1) as in Theorem 4.2. Then is represented by C.

y(8)

= x'(O) =

+ f(8 + ir)Ju(8)

y'(8)

almost everywhere.

REMARKS. The above theorem is a corollary of Theorem 3.1 and the representation Theorem 4.2. The equation (4) expresses the fact that for corresponding values of 8, and have usually parallel supporting lines (or

296

P. C. HAMMER

else x'(e) = 0). Note that I y'(8) I is bounded from zero and y'(O) exists except

for possibly a countable set of values of 0. DEFINITION. Let C = C(1) be a convex body. Let C(r) be the set obtained from C by extending each of its diametral chords about its midpoint by a constant ratio r> 1. Then C(r) is called an associated convex body of C and of C(r) is called a parallel curve of r r(l). the boundary curve It is clear that the supporting strips of C(r) are obtained by expanding the supporting strips of C about their midlines by the ratio r. Hence C(r) is a convex body and C(r) is of constant breadth (=2r) relative to T*(1) as unit circle. It is of interest to state the relationship of the boundary of rfr) to r = r(l). 4.5 THEOREM. Let C(1) be a convex body and let C(r) be an associated convex body of C(1) for some r> 1. The following statements hold. (1) The curves i(r) and r(l) are a distance r — 1 apart relative to T*(1) as unit circle. Hence C(r) = C(l) + ( — 1)C*(1). (2) The set ((r — 1)/2)C*(1) may be translated around the closed ring between r(l) and r(r) touching both curves always. Hence this set is the maximal symmetric set with this property. Hence also 1 1+r V C*(1) : xe C(r) = C(1) U {x +

)}

(3) (4)

A diametral line of C(1) is a diametral line of C(r).

Lim,-... (1/r)Cfr) = C*(1). REMARKS. The foregoing statements are readily verified and may be directly

extended to n-space. Now we consider internal associated convex bodies of C = C(1). By reducing each diametral chord of C (and each supporting strip)

about its midpoint (midline) with a ratio r < 1, we obtain a convex body D(r) as the intersection of the derived supporting strips. But, in general, the symmetroid D*(r) c rC*(1). There exists a unique minimal number r0 0 such that D(ro) = roC*(1). Then we define C(r) = D(r) for r0 r 1, and

we call these bodies the internal associated convex bodies of C. Then the body C(r,) is called the irreducible subbody of C(1) unless r0 = 0, in which case C(1) is symmetrical with respect to a point [= C(r0)J. Letting A = C(ro) if r0 > 0 we have that the external associated bodies A(r) of A for r> 1 are given by A(r) = C(rro) and A*(r) = rroC*(1). The diametral lines of A(1) are diametral lines of A(r) for all r> 1. In Hammer AB it is shown that the essential diametral line family of A(r) is the same as that of A(1) = A. To generate equivalent breadth convex bodies from others, the following theorem is stated. 4.6 THEoREM. If A is equivalent in breadth to B and C is equivalent in breadth to D, then s0A + s1C is equivalent in breadth to s0B + s1D.

Let A and C be of constant breadth relative to a symmetrical convex set Then s0A + s1C is of constant breadth relative to B for all real s0, s1. V. Outwardly simple line families. A family of lines which simply covers

B.

CONVEX CURVES OF CONSTANT MINKOWSKI BREADTH

the exterior of some bounded region, including infinite points, is called outwardly simple or a quasipencil. In higher dimensions the concepts must include continuity and the quasipencil is a more restricted type of line family than the outwardly simple type. In the plane continuity is implied in the definition and quasipencils coincide with outwardly simple line families. Hammer and Sobczyk in Planar line families. I and II give geometric and analytic characterizations of outwardly simple line families. We restate a few of these results here in slightly modified form. Pictorially, an outwardly simple (o.s.) line family is generated as follows.

Two runners x and y run on a circular track according to the following They start at the same time at diametrically opposite points and run continuously without stopping in the same sense and at the end of one hour x is to be at the starting position of y and y is to be at the starting position of x. If x and y carry a line between them then an outwardly simple line line. Conversely every outwardly simple line family is generated by family may be so obtained. This picture may be extended to a closed convex curve with stops of runners permitted. rules.

5.1

Let F be a family of lines with pedal function p.

Then

necessary and sufficient conditions that F be outwardly simple are that (1)

+ it) = —p(O)

for all 6, and

(2) p be uniformly Lipschitzian.

REMARKS. The first condition assures that infinite points (directions) are simply covered, whereas the second gives both a continuity condition and boundedness of the set of intersection points of line pairs from F. Note that the derivative p'(O) exists almost everywhere and is uniformly bounded. DEFINITION. The set of all points v(O) = P(O)u'(O) —

for which p has a derivative p'(O) is called the virtual envelope of F. The

following theorem proved in detail by T. J. Smith in his thesis is useful later. 5.2 THEOREM (SMITH). Let F be an o.s. line family. Let X be the set of all intersection points of line pairs from F. Then the closed convex hull of X is identical wi/h the closed convex hull of the virtual envelope of F. The set of pedal functions for all o.s. line families form a linear space which is complete

but inseparable with the norm

liP

II

=

sup v'(P(O)P

+ [p'(O)J'.

VI. Analytic representation of curves of constant Mlnkowskl breadth. 6.1 Let p(O)u(O) be the symmetrical convex unit circle for a Minkowski metric. Let p be the pedal function of an outwardly simple line

family F. Then there is a constant number q such that the curve r = r(q) given by

P. C. HAMMER

298

x(O) = p(O)u'(O) +

PP

+

with F as a representing is a convex curve of constant breadth relative to family of diametral lines. Furthermore every convex curve of constant breadth relative to may be so represented (by proPer choice of F and q). PRoOF. First we show that if F is a family of diametral lines representing a convex curve T of constant breadth relative to then r is representable in form (1).

Let r be represented in form

x(8) = p(O)u'(O) + f(8)p(O)u(t9)

where f(O) + f(8 +

a positive constant. Now for almost all 0 there is

is

a nonnegative number g(0) such that x'(O) = g(0)[p(0)u(0)]'.

Substituting from (2) in (3) and omitting the arguments, we find, on setting coefficients of u'(O) and u(0) equal to zero, + fp = gp, —P +f'p +fp' = gp'.

Eliminating g and solving for f' we have

Pt'

f1

P

Hence for some constant q t.e

"t'

Jo

P

P

But from (4) it follows that necessarily

4+ Hence

P(°)

any convex curve

of constant breadth relative to p is representable

by (1).

Now, however, suppose F is a given outwardly simple line family with be the minimum real number q such that

pedal function p. Let qo

p'(0) p(0)

for all 0.

Now let q >

o

P

and consider the curve r represented by (1). Then

is a simple closed curve containing the virtual envelope of F since fp> —p' and hence r is simply covered by F (Smith's theorem). Moreover x(0) — x(0 + 2r)

"I' dcti]p(e)

= + and hence the length of chords of r on F is Constant relative to 46. Moreover

r is convex since there is g(6) > 0 and

x'(O) = g(0)[p(0)u(0)J'

CONVEX CURVES OF CONSTANT MINKOWSKI BREADTH

299

for almost all 0 and hence r is locally convex and hence convex. The value of q may now range down to q0 and to + to obtain all convex curves represented by F and of constant breadth relative to P. In the case that F are symmetrical and r(qo) reduces to a single point. is a pencil the curves Q.E.D. NOTES.

Letting p(0)

1 we obtain all constant breadth curves in the form

we previously gave: x(0)

=

+ (q +

0

In the above proof we used the facts that

+

= —p(0) and p(0 + ir) =

The generality of representation was implemented by direct definition of the outwardly simple line families and by their representation using pedal functions. Other attempts to obtain the constant breadth curves by starting with the envelope curve failed to achieve generality since the specifications for this curve were too stringent to allow all outwardly simple line families. It is conceivable that the representation of Meissner could, in view of later development of theory of Fourier series, be shown to represent the curves of relatively constant breadth. If we write p(O).

=

cos (2j —

1)0

+ Eb, sin (2] —

1)0

and if these series are chosen so that P(0) is Lipschitzian one may obtain many outwardly simple line families. However, for applications to kinematics it may be that using finite Fourier series or other representations of p will be adequate. It will be observed that (1) will still give closed curves intimately related to a symmetrical curve P which is not necessarily convex. Since the transformation from a convex curve 5 to a symmetroid curve is theoretically simple we are in a position to obtain curves r equivalent in breadth to a given curve 5. In doing this, of course, the outwardly simple line family F cannot be chosen arbitrarily. VII. Arc length function of equivalent breadth curves. Let C be a convex body and let F be an o.s. family of diametral lines of C which represents C. Define the function s(0) as the sum of lengths of the (two) arcs of r between rn(O) and m(0). Then s(0) is called the arc length function of r. Note that s(,t) is the circumference of C. 7.1 THEOREM. Two convex bodies C and D are equivalent in breadth if and only if they have identical arc length functions. have the same arc length PROOF. We show that C and its symmetroid function regardless of the representing family F used. With p as the pedal function of F we represent r by x(0) = P(0)u'(0) + f(0)p(0)u(0)

where p(O)u(O)

is the vector function of

and hence f(0) +f(0 + ir) =

2.

P. C. HAMMER

300

Now s(8)

=

II x'(8)

I

Jo

÷ I x'(O + it) j}dO = I I x'(8) — x'(O + it) I dO Jo

since x'(O) and x'(O + it) are parallel the derivative vectors exist. But I x'(O) — x'(O + it)

I

in opposite sense unless one is zero when

= Ix(0)

—

x(8 + it))' I =

2 I [p(8)u(0)J' I

since f(0) + 1(8 + it) = 2. Hence s(O) = 2S1 [p(O)u(O)]' do which is the arc for the pencil of diametral lines through 0. Hence if length function of C and D are of equivalent breadth they have identical arc length functions. and Now suppose C and D have the same arc length function s(8). Let 5* be the boundaries of the respective symmetroids of C and D. Then since

have the same circumference and are centered on 0 they must and intersect in at least four points, two of which make a central angle with 0 5* there must be a pair of points (say at 01 < 0,, less than it. Hence if say, is
convex

body C then

VIII. Finite constructlona of relatively conatant breadth curves. In this section we give the geometrical construction of curves of constant breadth based on o.s. line families composed of a finite number of sectors of pencils. The description is simple, the equations appear more formidable: We give the description first. Let m0, m, ..., mk-l be a set of k lines, no two of which are parallel, and let us suppose that these lines have respectively directions

0=00<01<

as

n rn,. the intersection F consisting of m0,

Take the family of lines such that m(0) for 8, <8 <

Then and

=

v• also.

is called a subpencil and its virtual envelope is the set of points v0, -, Vk—l. Now let any symmetric convex curve : p(0)u(O) be given. We construct a curve r comprised of arcs similar to those on by starting at a point x, 0

on ,n0 and taking v0 as center of similarity describe an arc from x0 to x1 on m,

which is similar to the arc on ft from to 8, - From x, on m1 and with v, as center of similarity an arc similar to that on ft between 8, and is constructed meeting m2 in x2. This process is continued until we have returned to the starting point x0! We always return after constructing at most 2k

CONVEX CURVES OF CONSTANT MINKOWSKI BREADTH

301

arcs. We will always have a convex curve r of constant breadth with respect to p if we choose x, sufficiently remote from v0 and the family F will be comprised of diametral lines of r. We must now examine the construction more closely to determine exact conditions for convexity and to prove that we do indeed return to the same point x0. Let subscripts in u1 correspond to angles 0.. We define 01÷, = 05 + 2T. We define as follows: top.u. =

— v0,

=

— V.,

= =

—

t0p1u1 t,p1u1

x(O)

0 01,,

= v0 + top(0)u(0),

x(8) = v1 + 11p(0)u(0),

8

—

tjpjuj = Xj — — =

v,

x(0)

,

=

v + t,p(0)u(0)

,

0

0,

Now assume I. is fixed. Then eliminating x1, x1,, ••, we find , 1—10+ (v0—v1).u1

=

(vi—v2)•u2

(v0—v1)•u1

Pt

Pt —

+

+

Pt

v)

PS

1

It is clear that if we are concerned only with a finite set of values t. (we shall have 2k values), then t0 may be taken large enough so that t 0 for t + t1+1 which will establish that r is of conall j. We now prove stant breadth relative to We have 1)

j

+±(vJ_I_vI).uJ

P1

1

1

k+i

P

P

But

= _±(vJ_l_v,).uJ since Vh÷, = Vi,

Pi+i

=

P1

and Uj+k = —U,. t$

Hence

+ t$+1 = to +

= to from I = k and r is a closed curve. Now choosing t0 so that Hence all tj 0 gives us 4 + 4 is the relative breadth of r and r is convex. To determine a minimal value to° to give a convex curve r, we calculate — for i = 1, 2, .. ., k. Then the maximum value of the set .

of these k summands and zero is the smallest choice of 4 to insure that

P. C. HAMMER

r is convex and C contains all of to, we may take

Now if t' +

2 for this minimal choice

and then t0 + 4 = 2 so that r is equivalent in breadth to r. If + > 2 then it is not generally possible to obtain r equivalent in breadth to with the line family F. 8.1 TaEOREM. Let P be a symmetrical convex polygon with 2k sides. Then all convex curves of constant breadth with regard to may be obtained by taking parallel to lines through only subpencils of lines based on lines m0, opposite vertex pairs of P. In particular, if and only if a symmetrical convex curve is a parallelogram are all curves r of constant breadth relative to

similar to a translate of P. A convex polygon has at most as many sides as its symmetroid. Raii&aics. In forming a convex curve r of constant breadth relative to a the line family F of our representation symmetrical convex curve

is best taken so that it contains a sector of a pencil for directions corresponding to a pair of parallel opposite line segments in P. This assures that the line family F will be the essential diametral line family of the curve r and that the representation will give a minimal convex curve equivalent in breadth to IX. Concluding remarks. We have presented here the representation of relatively constant breadth convex curves promised. The outwardly simple line families which were used also appear in other connections. For example, the lines dividing a convex curve in half (the circumference), and the lines dividing the area of a convex body in half each form outwardly simple line families. The line families for which the pedal function is Lipschitzian are slated to be used in discussing differential equivalence as well as generalized envelope curves. Extension of the representation given here to higher dimension is an unsolved problem.

We have omitted significant and interesting work by Ohmann and by Grlinbaum, for example, on Minkowski plane geometry. The references directly quoted should be supplemented from the bibliographies of Bonnesen and Fenchel, Hadwiger, and Busemann. We are indebted to T. J. Smith for several discussions concerning convexity and families of lines. He provided most of the entries for the bibliography. BIBUOGRAPHY

1. W. Barthel, Zuin Inhaltabegriff in der Minkowskischen Geometrie, Math. Z. 58 (1953), 358-375.

2. , Zur Minkowaki-Geometrie, begründet auf dem FiéAcheninhattsbegriff, Monatsh. Math. 63 (1959), 317-343. 3. P. Belgodère, Stir ies surfaces minima en giometrie de Minlcowski, C. R. Acad. Sd. Paris 24.0 (1955), 1504-1505.

CONVEX CURVES OF CONSTANT MINKOWSKI BREADTH

303

4. F. Bohnenblust, Convex regions and projections in Minkoweki SpaCe8, Ann. of Math. (2) 39 (1938), 301-308. 5. T. Bonnesen and W. Fenchel, Theorie der konvexen Korper, Ergebnisse Math. 3, part 1, Berlin, 1934.

6. H. Busemann, The foundations of Minkowakian geometry, Comment. Math. Helv.

24

(1950), 156-187. , The

7. 8. 9.

geometry of geodesws, Academic Press, New York, 1955. Intrinsic area, Proc. Nat. Acad. Sci. U.S.A. 32 (1946), 5-8. , The isoperirnetric problem for Minkowski area, Amer. J. Math. 71 ,

(1949), 743-769.

10. U.S.A. 27 11.

, Metric condition8 for zymmetrw Finsler spaces, Proc. Nat. Acad. Sci. (1941), 533-535.

, Motions with maximal displacements, Comment. Math. Helv. 28 (1954),

1-8.

12. , Two dimensional geometi-ics with elementary areas, Bull. Amer. Math. Soc. 53 (1947), 402-407. 13. H. Busemann and C. M. Petty, Problems on convex bodies, Math. Scand. 4 (1956), 88-94.

14. N. David, A characterization of Minkowgkian geometry, Bull. Amer. Math. Soc.

53

(1947), 196-200.

15. A. Dinghas, Zum I8operirnetrischen Problem für die nichteuklidean Geometrien,

Math. Ann. 118

(1943), 636-686.

16. H. G. Eggleston, Notes on Minkoweki geometry. 1, Relations between the

radius, diameter, inradius and minimal width of a convex set, J. London Math. Soc. 33 (1958), 76-81. 17. H. Gericke, Zur Relativ-Geomet,-ie ebener Kurven, Math. Z. 47 (1941), 215-228. 18. B. Grunbaum, Borsulc's partition conjecture in Minkoweki planes, Bull. Res.

Council Israel Sect. F 7F (1957-58), 25-30. 19. , On some covering and intersection properties in Minkowski spaces, Pacific J. Math. 9 (1959), 487-494. 20. P. C. Hammer, The centroid of a convex body, Proc. Amer. Math. Soc. 2 (1951), 522-525.

21.

, Convex bodies associated with a convex body, Proc. Amer. Math. Soc. 2

(1951), 781-793.

22.

4

P. C. Hammer (1953), 226-233.

and A. Sobczyk, Planar line families. I, Proc. Amer. Math. Soc.

23. , Planar line families. 11, Proc. Amer. Math. Soc. 4 (1953), 341-349. 24. P. C. Hammer, Diameters of convex bodies, Proc. Amer. Math. Soc. 5 (1954), 304-306.

25. 26.

, Maximal convex sets, Duke Math. J. 22 (1955), 103-106. , Constant breadth curves in the plane, Proc. Amer. Math. Soc. 6 (1955),

333-334.

27. P. J. Kelly, On Minkowski bodies of constant width, Bull. Amer. Math. Soc. 55 (1949), 1147-1150.

, A property of Minkowekian circles, Amer. Math. Monthly 57

28.

(1950),

677-678.

29.

D.

Laugwitz, Konveze Mittetpunkt8bereiche und normierte Raume, Math. Z. 61

(1954), 235-244.

30. K. Leichtweiss, Zwei Extremaiprobleme der (1955), 37-49.

Math. Z. 62

304

P. C. HAMMER

31. H. Lippmann, Zur Winkeitheorie in zweidimensionalen Minkowski und Raumen, Nederl. Akad. Wetensch. Proc. Ser. A. 60 Indag. Math. 19 (1957), 162-170. 32. L. A. MacCoil, Elementary Lrs paces, Trans. New York Acad. Sd. (2) 14 (1951). 35-39.

H. Minkowski, Gesammelte Abhandlungen, Vol. II, Teubner, Leipzig, 1911. 34. V. Nasu, On the normality in Minkowekian spaces, Kumamoto J. Sd. Ser. A.

33.

2 (1954), no. 1, 11-17. 35. D. Ohmann, Extremaiprobteme fur konvexe Berewhe der euklidischen Ebene, Math. Z. 55 (1952), 346-352. 36. C. M. Petty, On the geometry of the Minkowaki plane, Riv. Mat. Univ. Parma 6 (1955), 269-292.

37. Yu. G. An extremal problem from the theory of convex curves. Uspehi Mat. Nauk 8 (1953), no. 6 (58), 125-126. (Russian) 38. F. A. Valentine, A characteristw property of the circle in the Minkoweki plane, Amer. Math. Monthly 58 (1951), 484-487. UNIVERSITY OF WiSCONSIN

SEMISPACES AND THE TOPOLOGY OF CONVEXITY BY

PRESTON C. HAMMER

Introduction. In the year 1951 we discovered some of the properties of the convex sets we have named semispaces. A semispace in a linear space M is a maximal convex set which excludes some point p M. At the same time, due to the awareness of the properties of semispaces we asked the question, "Is there a system which fruitfully includes convex hull closure and ordinary closure?" In our case as it seems to have occurred to other mathematicians the study of convexity thus led us into a field seemingly remote from things convex. Professor T. S. Motzkin had also defined semispace for finite dimensioned spaces at about the same time as we did. We have been unable to obtain any material concerning his work. In 1955 we published a paper on semispaces [21 and that same year we published a papet [3] introducing our version of extended topology. Subse. quently Klee [1] published a paper concerning the structure of semispaces. In 1959 S. T. Rio completed his thesis [10] giving new results based on the topological system of our first paper. In 1959-60 the first opportunity arose for us to spend uninterrupted time in further consideration of extended topologies. We have been able to obtain enough results to feel that the extension of topology may survive our blundering efforts.

It was only in January of this year that we finally obtained one of the principal topological theorems which apply nicely to convexity theory. One important theorem may be loosely stated as follows: fn any extended topology in which the minimal convergent sets are finite there exists a unique minimal base of neighborhoods for the class of all neighborhoods of each point.

The ramifications of this theorem and closely related ones are by no means completed. Applications of this theorem give the existence of semispaces and

hyperplanes in linear spaces for example. Also included are maximal subgroups in a group, maximal subsemigroups of a semigroup and so on. In this paper we review some of the basic properties and applications of semispaces in Chapter 1. In Chapter 11 we present a few of the basic concepts and theorems of extended topology relevant to the topology of convexity, and in Chapter III identification and application of these concepts to convexity are made. The reader may find that alternation between Chapters II and III will

relieve the tedium of reading the necessarily long list of definitions in Chapter II. The only reason we can offer for the late appearance of semispaces in the theory of convexity is that most applications of convexity and virtually all the theory until recently has dealt with closed convex sets. However, the 305

P.

306

C. HAMMER

study of convexity, even closed convexity, cannot be properly achieved without an awareness at least of the "prime numbers" among the convex sets— namely, the semispaces. Extreme points are virtually dual to semispaces and extreme faces can be described by use of convex sets interpolated between open halfspaces and semispaces. Among other applications, the semispaces induce simple orderings of linear spaces, they may be used to extend certain partial orderings to simple orderings. The intersection of a semispace at 0 in with the unit spherical surface provides a representing set for n-dimensional projective space—i.e., a semispace "is" a projective space. However, we should not like the reader to feel that current applications are the sole conceivable purpose for grasping the underlying theory of pure convexity. CHAPTER I.

BASIC PROPERTIES OF SEMISPACES

We assume that the real linear space M with null set N and origin 0 is at least one-dimensional. We will use p + X = {p + q : q e X) for the translation of X by p. Also X ± V = (p ± q : pc X, q e Y}. If X and Y are convex sets then X + V and X — Y are also convex sets. The minimal convex set containing a set X is designated hX and is called the convex hull or convex closure of X. Only in referring to open or closed half-spaces do we consider a standard topology of M but this is generated by the topology of the line which we do use.

The function h maps each subset X of M into another subset hX. It has the properties: 1. hX X (enlarging). 2. X V implies hX h V (isotonicity). 3. h(hX) hX (idempotence). These are the properties we require in our

extended topology for a closure function. In this instance we also have hN = N, h{p) = (the null set N and singletons {p) are convex). Moreover, h is domain finite—i.e., letting lxii be the cardinal number of X then hX U {hY: V X, II Yti finite). That is, the convex hull of X is the union of the convex hulls of its finite subsets. This is discussed more fully in Chapter III and it is a reason for the existence of semispaces. A semispace at p e M is a maximal convex set which does not include P. Such a semispace may be designated and then S, — p is a semispace at 0. The existence of semispaces at each p M is a corollary of Theorem 2.3. 1.1 THEOREM. Let X be a convex subset of M so that there is a point P X. Then there exists a semispace

which includes X.

Note. This theorem is a corollary of Theorem 2.4, part 4. 1.2 COROLLARY. The union set of any nonemPty class of convex sets which is simply ordered with respect to inclusion is a convex set. 1.3 THEOREM. If S,, is a semis pace at p then the intersection of S, with each

line through p is an open half-line with origin p. Hence

is a maximal

convex cone which excludes its vertex p. PROOF. If E1 is a line through p then S,, n E, cannot have a point in both rays from p in E1 or else S,, which is convex. Suppose X is a ray from

SEMISPACES AND THE TOPOLOGY OF CONVEXITY

p in

E1 such

that (2p — X) n

307

= N. Then if qeX but q C S9 contrary to

u {q)) is a convex cone with conclusion we observe that the convex hull {q}) and S9 is not a maximal convex h(S9 u vertex q and "base" S,. Hence p Hence X set excluding p. S9. 1.4 THEOREM. 1. Let S be a semisPace at 0. Then if p * 0, p + S either

pro perlv contains S or is properly contained in S. 2. Let S1, S2 be semispaces at 0. Then p1 ± S1 p2. S1 = S2 and pt 3. Let S be a semispace at 0.

p2 + S2 if and only if

The relation < defined by p < q Provided

q — p £ S provides a simple ordering of M. 4. Let < be a simple ordering of M such that p1 O} is a convex set. Then C is a semispace at 0. S. Then if q = p + /1 such that P, eS we have If, on the other hand, p€(—S)\{0) = then is a semispace at 0 and hence p + S S. 2. This is a consequence of (1) since (p, — p2) + S, = S2 gives (p — P2) + S PRooF.

p+p,€S

1.

Suppose

or p +

S.

as a semispace at 0 and also at p, — p2 and hence p1 = p2 and then S, = S2. 3. Since any convex cone with vertex 0 partially orders M in the manner defined we need only verify that if P q then either p 0, > p > 0, etc. Now if Pc C then P/2 e C since either p12 > 0 or —p/2 > 0 and if —p/2 > 0 then 0€ I pp —p121 C which is impossible. Hence C contains an open ray from O through each of its points and C is a convex cone. Now if C is not a semispace at 0 then there is a line E1 through 0 which does not intersect C.

hence p
a simple order. But then —p + p < —p or () < —p and —p€C which is a contradiction. Hence C is a semispace at 0. Q.E.D. REMARK. This theorem shows that the semispaces are intimately connected with simple ordering of spaces. 1.5 THEOREM. The class comprised of M and alT semis paces at all Points of M is the ,nini,nal intersection basis for the class of all convex subsets of M.

{}

PROOF. Let X be a convex subset of M. If X = M then is the subbe a seniispace at P containing class in point. If X M let p X and let p X. Then X = fl X}. On the other hand if S,, is properly contained in any convex set C then pe C and hence S,, is not the intersection of any class of convex sets which does not contain Q.E.D. EXAMPLES. In finite dimensional spaces M the seinispaces, as Klee Ill proved, are isomorphic to one and they are easily generated as shown in 121. Thus on the line a semispace is an open half-line. In the plane a semispace is the union of an open half-plane and a bordering relatively open In a semispace at 0 is the union of relatively open halfspaces Ii) of dimension

P. C. HAMMER

308

•, n such that H1 is in the boundary of H1+1 and 0 is the boundary 1, 2, point of H1. The proof of the following theorem will be omitted. Let S = be a basis set of vectors for 1.6 THEOREM. Let p1 , p2, -, Then S is > 0, the first s1 not to vanish is positive). {sipt + •-• + at 0. Conversely to each semispace S at 0 there is a basis set of a points so that S is thus representable. REMARKS. It wa' noted in [2) that a similar method generates semispaces in infinite dimensional spaces. However, the work of Klee [1) should be consulted for a study of such semispaces since he has shown that there are

j=

-

nonisomorphic semispaces in infinite dimensional spaces and given the cardinal

number of their types. 1.7 THEOREM. The semispaces at 0 form the minimal intersection basis for all convex cones with excluded vertex 0. 1.8 THEOREM. Let B and C be disjoint convex sets. Then there exists a con-

vex set H such that L\H is convex and H B and L\H C. PROOF. The set B — C is convex and 0 B — C. Hence let S be a semispace atO includingB— C. Then H= fl{p+ S:PeC} Band Hisconvex and H n C = N. Moreover since L\H = U {P — S : p C) is convex and contains C we have the representation required. Q.E.D. REMARKS. The above two theorems are given in [2) and Theorem 1.8 is

attributed to Kakutani. A point p is an extreme point of a set X provided PG X but P h(X\{pJ)—Le., p in X but not in the convex hull of X\{p). In topological terms p is an isolated point of X. The complement of a semispace has the vertex as its single extreme point. Hence we have the following; 1.9 THEoREM.

1.

A Point PG X is an extreme Point of X if and only if

there is a semisPace

S,

A set X is the convex hull of the set E of its extreme Points if and only if for each p e X the complement of each semispace at p has a point in E. PROOF. 1. Since the complement of a semispace S, is a minimal neighborhood of p this may be stated: a point P X is an isolated point of X provided there exists a neighborhood Y of P such that Y r'i (X\{p)) — N. The proof of this is trivial but the association between semispace and extreme point is important. Neighborhoods are defined in Chapter II but in this case are represented as the complements of convex sets. 2. To prove necessity suppose there is a semispace S at a point pe X and that E S S. Then hE S S and hence X * hE. For sufficiency let p e X. If p E hE then let S be a semispace at p containing hE. But then (M\S) n E = N since S hE E. Hence Pc hE and X = hE. Q.E.D. REMARKS. This theorem is proved in [2]. A more general theorem, stated in terms of neighborhoods, is the following where a minimal neighborhood of p is simply the complement of a semispace at p. 2.

1.10 THEOREM. A point pe hX

and only if every minimal neighborhood of p

SEMISPACES AND THE TOPOLOGY OF CONVEXITY

contains a point of X.

We now leave the direct discussion of semispaces and related maximal convex sets until the extended topology has been defined and we return to them in Chapter III. It should now be clear that the semispaces cannot be ignored in any reasonable introduction to convexity. We have not taken the trouble to track down the numerous papers on extremal elements of convex sets to determine the applicability of semispaces. However, it may be assumed that the use of semispaces wilt simplify materially much of this work. CIu1'rBR H. EXTENDED Topoi.ooy

We have now stated several particular results concerning semispaces and related convex sets. In this chapter we present a short introduction to our system of extended topology which arose from the attempt to consider certain processes and concepts associated with convexity as topological in character. We first present the definitions of the basic functions of extended topology in complete generality. We then show how an arbitrary set-valued set-function generates an extended topology. Then we state the basic theorems which later we show imbed certain of the results concerning convexity in a much more general framework. Let be the class of all subsets of the space M which has null set N. into itself. Let , Let F be the family of all functions mapping M, JNX N, eX X, and cX M\X. c be the functions in F defined by f,X generate the corresponding The set algebra and inclusion relations in algebra and order relations in F. Thus we define (fl

fl(f0X) We treat composition and we take S to mean f1X f2X for all Xe as a product and indicate it by juxtaposition. We observe that F is a semigroup with respect to composition with identity element e. The following subfamiljes of F are among those we have found useful in extended topology. 1. The idempotent or projective functions P = (f: f = f} 2.

The isotonic or inclusion preserving functions F0 = {f: X Y implies JX S fY)

={f:f(Xu —{f:f(Xn Y)SfXrifY).

= {f:f C). The limit functions The Primitive functions F' = {f:JX U {fY\Y: Y 5. The expansive functions F. = {f: f e, fe F'0). 6. The closure function F. = P n F2. 3.

X}) c

F0.

functions F-. = (f:! S e,fe F.). functions F-. = P n F_.. additive functions F'4 = .(f:f(X U Y) JX u fY} c F.. F.-4 (f:f(X n Y) IX n JY) c additive functions dual

F..

4.

7. 8. 9. 10.

The contractive The interior

The

The

P. c. HAMMER

310

In this paper we will not be concerned with topological spaces in the usual sense. However, a Kuratowski closure function f is an additive closure function

such that fN = N. The antitonic functions f such that fc is isotonic are represented here only by the complement function c which is also involutory: =

e.

The dual of f is cfc. Note that duals of idempotent, isotonic, expansive, closure, or additive functions are respectively idempotent, isotonic, contractive, interior, or dual additive functions. The duals-of limit-functions and primitive

functions are not used here. An extended topology is specified by the pair M, where q is an expansive function. An arbitrary function fe F generates an extended topological system of functions in the following manner: ii. The f-limit function c is the maximal limit function contained

in!.

12. 13.

The f-primitive function f' is the minimal isotonic function containing It may be represented by:f'X U Y: Y X} U {fY\Y: Y X}. The f-expansive function g is the minimal expansive function containing f

(or!' orf1). It is representable by g=euf' or X}. U{ftY:

U{fY:

The f-contractive function r = cgc. Then r is the maximal contractive function contained in cf'c. 15. The f-closure function h is the minimal closure function containing f (or equivalently, f1 ,f', g). There exists a unique minimal ordinal number 0 such that g(qA0) = gXO and then h = g"O Operands for transfinite composition of a well-ordering of. expansive functions are formed by unions of preceding compositions at limit ordinals. rA0. Then i is the maximal interior 16. The f-interior function i = chc function contained in r. 17. The internal f-primitive function, u = e n f', is the maximal contractive function contained in 1'. 18. The primary f-self-dense function v is the u-interior function. If vX = Y then Y is the maximal subset of X such that 1' Y 2 Y. Two functions! and w are equivalent provided they have the same expansive (or primitive) function. The reader may verify that the f-prhnitive function f' is a primitive and it is also the g-primitive function. Moreover, if we write t'f = and t2f g, the f-expansive function, then = U and = U(t2fa). A point p is a Primary f-limit Point of X A point is an f-limit point of X provided P€f'hX—i-.e., P is a primary f-limit point of hX the f-closure of X. A set X is f-closed provided 9X = X (or hX = X). The complement of an f-closed set is f-open. 14.

f'

-

Let g be an expansive iunction and let r =

cgc.

Then a set X is an r-

neighborhood (or g-neighborhood) of p provided p rX. The class of all r-neighborhoods of p is empty if and only if p e crM = gN (i.e., p is a primary

g-limit point of N). The class member sets. A subclass

of

contains all supersets of each of its is called a base for provided to

SEMISPACES AND THE TOPOLOGY OF CONVEXITY such that Y X. If ..A is empty then there exists Xe necessarily empty. The following theorem is proved in [91.

Ye is

2.1 THEOREM.

Let g be an expansive function, let r =

cgc,

let h be the g-closure

function and let i = chc. 1. Pc g 'X is equivalent to p e g(X\{p)). 2. peg'X if and only if Yn (X\{p}) * N/or each r-neighborhood Y of p. = 3. PG g 'hX (i.e., P is a g-limit point) is equivalent to p e g (hX\{ h(hX\{ 4. peg'hX if and only if Yn (hX\{p}) * Nfor every r-neighborhood Yof p. 5. peg 'hX if and only if Y ri (hX\{p}) * N/or every i-neighborhood Y of p. REMARKS. This theorem is stated to indicate that the neighborhoods as defined are sensibly connected with the limit point concept. It is simple to use a neighborhood system {..-4: Pc Z M) to define a contractive function r.

I

DEFINITIONS. A function f is domain finite provided fX U {fY: X, II Y finite) where II Y II is the cardinal number of Y. A function f is domain I

bounded with domain bound n 0 provided there exists a minimal integer n such that fX U UY: Y X, II Ill n}. The domain bounded and domain finite functions are important in discussing closing and closure when the sets analogous to convergent sequences are finite. The following theorem is taken from [8) and it gives a few salient features of domain finite functions. 2.2 THEOREM.

1.

The union function of any family of domain finite functions

is a domain finite function. 2. The intersection function of any finite family of domain finite functions is a domain finite function. 3. The composition function of any well-ordering of domain finite functions is a domain finite function provided operands at limit ordinals are taken as unions of all Preceding compositions. 4. The closure function h of a domain finite expansive function g is domain finite and either h = for some minimal n 0 or h = = U {g': n <w}. 5. To each domain finite expansive function g there corresponds a unique minimal limit function u such that g is the u-expansive function. Then uX * N = {X: uX * N) is a class of finite sets (the implies II XII is finite. Hence minimal convergence sets for g). REMARKS. Examples of domain finite expansive functions include all algebraic

closures, and convex hull and linear hull closures in linear spaces. Not included are the topological closures requiring infinite convergent sets. 2.3 LEMMA. Let f be a domain finite function and let be a nonempty sub• class of ...4' such that X1 , e 'it' implies there is an X0 such that X0 X1 U (In lattice terminology, contains an upper bound for each of its finite subclasses.) Then f{IJ X: Xe = U {fX: Xe PROOF. Since every domain finite function is isotonic, f(U X) U (fX). Hence we show that if p €f(U X) then there is X0 e such that p €fX0. But if p ef(U X) then there is a finite subset Y of U X such that p efY. If Y = N

P. C. HAMMER

312

and since is not empty we are done. be taken so that peX1, Otherwise suppose Y={Pi,•-.,Pe} and let i = 1, .--, k. Then p €/ Y f{U X1} fX. where X0 contains U

then pefY JX for each Xe Q.E.D.

REMARKS. This lemma or its dual is the principal observation used in proving several theorems Including the following ones. In [81 we give a systematic treatment of several other related results including the converse of this lemma. of ...4 satisfies the condition of the lemma Note that a nonempty subclass

if it is closed with respect to finite union and in particular then if simply ordered with respect to inclusion. The lemma is trivial if

is has a

maximal set. The next theorem is fundamental for its applications to a variety of mathematical systems. It is proved in [81. 2.4 THEOREM. Let g be a domain finite exPansive function; let r — cgc; let h be the g-closure function and let i = chc. The following statements hold: 1. There exists a unique minimal neighborhood base 91, for the class then no proper subset of X of r.neighborhoods of p for each j)E M. If X€

is in .91,. of sets X such that 2. Dually, to each p e M there exists a unique class p e cgX but zf Y is a proper superset of X then p€ gi'. The class is the class of all complements of sets in the minimal r-neighborhood base .91,. of all i-neigh3. There exists a unique minimal base 91 of the class borhoods of p in each p e M. If Xe .91 then X is g-open (iX = X). of maximal g-closed 4. Dually, to each p e M there exists a unique class

sets I such that

e cX = chX.

The class

is comprised of the complements

of sets in .91. Let h be a domain finite closure function and let i = chc. There exists a unique minimal intersection basis for the class of all closed sets. This class is comprised of M and each maximal closed set X which excludes some Pc M. 2. There exists a unique minimal union basis for the class of all open sets. This class is comprised of N and all minimal oPen sets X which contain some 2.5 COROLLARY.

1.

P e M. 3. There exists for each set Z M a unique minimal intersection basis for the class of all closed sets X such that X Z. This class is comprised of all maximal closed sets contained in Z and all maximal closed sets I such that each peZ. 4. There exists for each Z M a unique minimal union basis for the class of all oPen sets containing Z. 2.6 COROLLARY. There exist minimal intersection bases for each c/the classes

of sets indicated below. 1. Let M be a group and let the closed sets be the subgroups in M. 2. Let M be a and let the closed sets be subsemigroups. 3. Let M be a ring and let the closed sets be the ideals (or the subrings). 4.

Let M be a linear space and let the closed sets be linear manifolds (not

SEMISPACES AND THE TOPOLOGY OF CONVEXITY

313

necessarily topologically closed). 5.

Let M be a linear space and let the closed sets be convex subsets.

REMARKS. The number of examples available is rather embarrassing.

More-

over, the reader will observe that we have also demonstrated existence of certain sets. Thus the semispaces in M exist from our general theorem, the maximal linear manifolds (hyperplanes) in M exist and maximal subgroups excluding a point exist. See, for example, Theorem 1.8 (3) in [12]. ing theorem is proved in [8). 2.7 THEOREM.

Let I be any domain finite function.

1.

To each

The followthere

be the class exists at least one maximal set 1, such that p e ci Y,. Let comprised of all such 1', for pe M and of M. Then to each Xe .4' there implies Y X and such that Y corresponds a subclass of is minimal. With respect to this Property fX = fl {fY: Ye 2.

fl {Y: Y X, Ye where is the class of maximal sets Then v is a closure function and fv = f. In case f is expane and if f is a closure function then v =f.

Define vX =

described above.

sive then f2 v

CHAPTER Iii.

TRK

OF CONVEXITY

Now that we have given some of the basic definitions and results of extended topology we are in a position to illustrate them with applications in convexity. We identify the functions named in some detail since it seems important that the strangeness of the concepts be justified by their relevance in various areas. Again, M is a linear space and 0 is its origin. The usual expansive function g of convexity is defined by

U{(p0,p1):P.,P1eX) where (P0. p1) is the open line segment between po and p1. 3.1 THEoREM. 1. The expansive function g of convexity is domain bounded with domain bound 2.

The closure function h of g is equal to if M is n-dimensional and 2 [11]. For M infinite dimensional h = The function h is domain finite in all cases. 3. The minimal limit function w which generates h has the following 2.

Properties:

(a) wX * N implies that X is the set of m + 1 vertices of an rn-dimensional simplex for some m 1. If M is n-dimensional then m n. (b) If wX * N then wX is the relative interior of the rn-dimensional simplex determined by X. (c) If Y c X then wY n wX = N. 4. The minimal limit function w which generates g has the following properties:

(a) wX * N implies X is a two-point set. (b) If wX N then wX is the interior of the 1-dimensional simplex determined

by X. REMARKS.

Part

3

is essentially Carathéodory's theorem.

Now with

P. C. HAMMER

314

r = cgc, h the g-closure function, i = chc, we have the following description of minimal neighborhoods and maximal sets. 1. The minimal base 3.2 for the r-neighborhood class ..4 of

p is comprised of all sets X which intersect each line through p in a closed half-line with origin P. of maximal sets X such that p gX is the class ofcomple2. The class ments of sets in if X intersects everj' line through p in an open halfline with origin P then Xe Cl,. The class Cl, contains the semispaces. Each

value 9Y M of g is the intersection set of a class of subsets of U Cl,. 3. The minimal base for the i-neighborhood system A" of p is coinprised of all complements of at P. 4. The class of all maximal convex sets (= closed sets) which exclude P is the class of all semispaces at p. 5. Every g-open set X * N is the union of a class of complements of semispaces. But no such complement is the union of open sets contained properly in it. 6. Every convex (= g-closed) set X * M is the intersection set of a class of semispaces. No semispace is the intersection set of a class of convex sets which proPerly contain ii.

Apart from the details of verifying the neighborhood of p for this application everything in this theorem is identified with the general theory of domain finite topologies. The g-primitive function g' associates with each set X the union of the open line segments between each pair of points of X. These are the primary glimit points of X. A set X is said to be Primarily self-dense provided g'X X. 3.3 THEOREM.

1. A convex set X is primarily self-dense if and only if X

has no extreme points. Such a set is perfect. 2.

Let u = e n g' and v =

where

is the projecting order of u. Then

vhs the unique maximal subset Vol Xsuch that

Y. Siswevisan

interior function, the union of any class of primarily self-dense sets is a primarily self-dense set. Problem 1. Determine the projecting order of u. Problem 2. Describe the r*neighborhood bases in n-dimensional space. 3.4 THEOREM. Let M a unique class Cl. of maximal convex sets Y excluding M*. To

each Ye qC

there corresponds a semispace S at 0 such that Y=fl{q+S: such that X n (M* U (kb)) = Nfor

Each maximal convex set Xe

is either Cl or X is the intersection of a set Ye such that p e Y with a semispace at P. The minimal intersection basis for the class of all convex subsets of cM* is Cl = Cl. U Cli. some PG

PROOF.

This theorem is an application of Theorem 2.5 to an important

special case in convexity. Since there exists for each Ye a semispace S at 0 containing Y — M*, we have V S 1) {q + S: q e M*} = Z But Z is a = Nand Z V. Hence Z Y. Such a semispace S may convex set, Zn

SEMISPACES AND THE TOPOLOGY OF CONVEXITY

315

be called parallel to M5.

If X is a maximal convex set such that Xn (M5 u {p}) = N for pecM5 containing X. The set V necessarily then there exists a maximal set Ye containsP or Y=X. If PCY then Yn(M*u{p))=N and since S, n Y, but Y = X. Let S,, be any semispace at p containing X. Then X and hence X = S, n Y. Q.E.D. S0 n Y is a convex set excluding M5 U Comments. This theorem gives the existence of the maximal convex sets "between" open halfspaces and semispaces. Thus if M* is n-dimensional and then Y is a maximal convex set excluding M* and cY is a minimal Ye y-open set including M*. These sets are important for describing extremal faces of convex sets. Continuity. We have defined continuity as follows: Let I: M1 —, M2 and let Then it is (u, v)it, v respectively be expansive functions on and continuous provided tu vi where I now also refers to the induced mapping Let now M1 be a linear space, let h be the convex from /1 into . hull closure and let M2 be the real numbers with h0 its usual closure. Then .

/: M, —b M2

is (h, h,)-continuous implies either I is constant or t has two distinct

values say a,

such that r'a,

are complementary convex subsets of M1..

the transformations I which are (h, h)-continuous Problem 3 (a). where 1: —' M2 and M1 and M2 are linear spaces and h is used as the convex

hull closure in each M, and M2. (b) Do the same when h is the closed convex hull closure for M and M2 finite-dimensional. (c)

is

If g is the convex expansive function then 1 (g, y)-continuous implies I

h)-continuous. Are there examples where the converse is not true? Proble,n 4. If M = E2 show that there is an expansive function g with the

convex hull closure Jz which is not domain finite. Note: If h is any closure consider the class

with h as closure. Let

A0 = sup

of all expansive functions

= M} where Aa is minimal. The root order bouizd for h. If there exists an = Ii but ?' * Ii for a < A, then (/ is called

ordinal A, may be called the expansive function such that an ultimate root of h0. For convex hull closure does such an ultimate root exist? Observe that if is not a limit ordinal then an ultimate root of h necessarily exists. Concluding remarks. We have just begun to study the applications of the extended topology in other fields as well. Since we have developed theories

of continuity, convergence, separation and connectedness as well as other topological extensions the reader may refer to published papers to obtain a more complete picture of the extent of the system. However, we are particularly pleased that the system of extended topology is not futile in application to the area which prompted it—convexity. REFERENCES

1. V. L. KIee, Jr.. The structure of semispaces. Math. Scand. 4 (1956). 54-64. 2. P. C. Hammer. Maximal convex sets, Duke Math. J. 22 (1955) 103-106.

p. c.

316

HAMMER

3. General topology, symmetry and convexity. Trans. Wisconsin Acad. Sci. Arts Lett. 44 (1955), 221-255. 4. , Kuratowski's closure theorem, Nieuw Arch. Wisk. (3) 8 (1960), 74-80. 5. , Extended Reductwn of limit functions, Nieuw Arch. Wisk.

(3) 9 (1961), 16-24.

6.

.

Extended

topology: Domain finite expansive functions, Nieuw Arch.

Wisk. (3) 9 (1961), 25 33.

7.

,

Extended topology: The Wallace functions of a separation, Nieuw Arch.

Wisk. (3) 9 (1961), 74-86.

8. 9. -

Extended .topology: Domain finiteness, Manuscript available August 1961. , Extended topology: Set-valued set-functions, Nieuw Arch. Wisk. (3) 10 ,

(1962), 55-77.

10.

S.

T. Rio, On the Hammer topological system. Ph. D. Thesis, Oregon State College,

1959.

11. 'F. Bonnesen and W. Fenchel, Konvexe Koerper, Springer, Berlin, 1934, p. 10. 12. Marshall Hall, The theory of groups, Macmillan, New York, 1959, p. i8. UNIVERSITY OF WISCONSIN

ON SIMPLE LINEAR PROGRAMMING PROBLEMS BY

A. 3. HOFFMAN 1. Introduction. In the considerable research that has been devoted to linear programming since the subject was first formulated in 1947 by George

Dantzig, there have been a number of occasions when it has been noticed that some particular classes of problems were amenable to "obvious" solution.

For the most part, the source of the obvious solution has been insight into the physical or economic meaning of the problem. The purpose of this talk is to point out that almost all of the classes of problems which the author currently knows to be amenable to simple solutions, and which have the further property that the particular answers are integral when the particular data are integral, can be shown to be special cases of one simple observation. In a certain sense, therefore, this simple observation provides a unified mathematical insight as a substitute for the physical and economic insights. Even more remarkable is the fact that the essential idea behind the observation was first noticed by G. Monge in 1781 [4]! Monge remarked that if unit quantities are to be transported from locations X and Y to Z and W (not necessarily respectively) in such a way as to minimize the total distance traveled, then the route from X and the route from Y must not intersect; and it is this idea which shall be exploited. We first define a special class of transportation problems which can easily be solved by inspection. Next, we will take up in detail the "warehouse problem" of A. S. Cahn, which has been shown by several authors to be amenable to solution by inspection. This will be demonstrated afresh, by a succession of two transformations which result in a restatement of the problem in such a form that the Monge idea applies. While much more cumbersome than other methods of solution, our procedure has the virtue of exhibiting a variety of devices used in the trade. We close with remarks on simplifying devices and suggestions for future research. The tranaportatlon problem and the Monge eequence.

2.

Let a1, -

a,.,,

given non-negative integers such that L = b. Let C = by n matrix of real numbers. The transportation problem is to discover among all non-negative m by n matrices X = such that b1,

-

. -, b,, be be an m

Z=

L

= a and b a matrix which minimizes For the task of actually calculating answers, several efficient iterative algorithms are

known. Our concern here is to show that if the coefficients

(ci,) satisfy

certain special conditions, then the solution can be obtained by inspection. To that end, we introduce the following definitions: A Monge sequence is a rearrangement 317

A. J. HOFFMAN

318

•.., (Iip,,t,Juiui)

(i1,j1), of the pairs of indices (i, j).

A Monge sequence (2.1) is said to be consonant with a matrix C =

if,

wherever

(i) p
(iii) (Iq,jp) we have

(Iir,fr),

+ Cjqjp + C;qjq Consider now the following procedure for choosing a matrix

(2.2)

=

(i,,).

Step 1. Set k= 1. Step 2. Set X;klk mm (alk, b,k). Replace — Step 3. Replace a$k by by bik — Step 4. If k = mn, stop. If k <mn, replace k by k + 1 and go to Step 2. . ., b,, be given non-negative integers such •, THEOREM 1. Let a1, If the Monge sequence that La, = (ia) produced by the algorithm (2.1) is consonant with C, then the matrix X = If (2.1) is not consonant with C, then (2.3) solves the transportation problem. such there exist non-negative integers a1, •, a,., b1, •--, b,, with L a, = that the matrix X = (ii,) produced by (2.3) does not solve the transportation

(2.3)

-

problem.

PROOF. Assume (2.1) consonant with C. We shall apply induction on m + n.

The result obviously holds if m + n = 2, and assume it holds for all smaller values of m + than the current one. Among all solutions of the transportation problem, let Y = (y,,) be a solution with the largest value of x,1,1 (obvi< ous continuity arguments establish the existence of Y). Assume that y,151 < b1. It follows that there exist indices r * 1 and s * 1 such that

> 0 and Let e =

, y,,1).

> 0.

Consider now the matrix Z = + e = Yr. + £

— Z,j1 =

£ ,

—e,

y,j for all other pairs of indices (i, i). Clearly Z satisfies the boundary conditions of the transportation problem, and by (2.2), L c•;z,, is not larger than L E, Hence Z is a solu-

tion to the transportation problem with a larger value of

than

This contradiction establishes that, among all solutions to the transportation problem, there is one in which =

ON SIMPLE LINEAR PROGRAMMING PROBLEMS

319

By (2.3), the new value of at least one of a11 , b1 is 0. For the sake of definiteness, assume the new value of b1 is 0. The algorithm (2.3) will then to be zero. In fact, it is clear that our problem is compel all i* by deleting now reduced to an n ii,,,. we consider the order in the changing a,, to — If and column j', Monge sequence obtained from (2.1) by deleting those entries corresponding to column .11, it is clear that the conditions (2.2) are hereditary. Hence the induction hypothesis applies. This completes the proof that the consonance of the Monge sequence with C justifies (2.3). Suppose (2.2) is not satisfied, i.e., we have +

(2.4)

Let (2.3) Cipiq

+

>

= aiq = b1, = bsq = 1, all other a, and b, at zero. Then the algorithm But this is bigger than will produce + cj,;, for L + Ciqip. This completes the proof of the second part of the theorem.

L

As a simple illustration of the application of this theorem, consider the following problem [15]. An individual has n jobs to perform. Job j takes t ... d,,, assume the 1., and the hours and is due d, hours from now, d,

d, are all positive integers, and the individual works in hourly units. How should he schedule his work in order to minimize the maximum tardiness? It is well known that an optimal procedure is to perform the jobs in the order of their due times. Let us now prove this as a special case of our theorem. Let m = L t,. Consider the m by n transportation problem in which b = a, = 1, and the are defined as follows: = 0 for i d,, c.s,+1..a = Me, where M is a very large number. It is clear that, for M large, the objective function is dominated by the largest r such that Xdj+r.J = 1 for some j. Hence the solution of our transportation problem will schedule the work so as to miaimize the maximum tardiness. Construct a Monge sequence (1, 1), (2,1),...,

..., (m, ,z).

It is easy to verify that (2.2) holds. 3. The warehouse problem: first transformation. Let Pi, Pu, c1, ., be given positive numbers, 0 A B, a pair of constants. The problem is to choose x,, and y,, ...,y,, subject to (m, 1), (1, 2), . . •, (m, 2),

(3.1) (3.2)

(x, +x2)—(y, (x, +

...

+

x,,_1) —

+

...

(3.3) y2

A + x1 —

+Xu+i)—(yi+

B—A, B

—

A,

A. J. HOFFMAN

320

in order to minimize c,x, —

(3.4)

p,y,.

Let K be the convex set of all points satisfying (3.1)-(3.3). K is not empty,

since x = y1 = 0 satisfies all conditions. Further, K is bounded: (3.1) and

the first line of (3.3) show that y1 is bounded, so by the first line of (3.2) x1 is bounded, therefore, by the second line of (3.3) Yz is bounded, etc. We now show that, if for some j, P, c.,, then the problem can be split into two problems each of the same form as the original. Let p c-. Then there is a solution in which + ... + y, + ... + x, — = A+ (3.5) .. ., = 9, Otherwise, let . , 9,,) be a solution in which 9) = assumes its maximum value. If (3.5) does not occur, then . .

(3.6)

If we increase i, and 9, by a small amount, it follows from (3.6) that (3.1)(3.3) will not be violated, and since p, c1, we will not have increased (3.4). This contradicts the definition of (1,9), hence we may assume (3.5). If we substitute the value of y, from (3.5) in (3.2) and (3.3), we obtain

x1—y1_B—A,

(3.7)

and (3.8) X,

+

X,÷1 —

Yi+t B

+X,,-1) —(y,+, (3.9)

and (3.10)

Y,+i

y1, .. . , y), (3.8) Now (3.7) and (3.9) involve only the variables (x1, . . ., and (3.10) involve only the remaining variables. Hence our original linear programming problem has been broken into two parts. It is clear that (3.7) and (3.9) are in the same format as the original. We now work on (3.8) and and a new p}
ON SIMPLE LINEAR PROGRAMMING PROBLEMS

we may assume the problem so posed that with that stipulation. consider Consider now inequalities


and

321

we return to

(3.11)

—x, + y2

A + xi — Yi.

Let L be the convex set given by (3.1), (3.2) and (3.11). L is unbounded, and clearly Kc L. We shall show that: (1)

(ii)

(3.4) is bounded from below on L.

L has at least one vertex, and a minimum of (3.4) is attained at a

vertex. (iii) Every vertex of L is in K. It will follow at once that minimizing (3.4) on L is equivalent to minimizing

(3.4) on K.

Suppose (i) false, so that there exists a sequence of points in L on which (3.4) decreases (not necessarily monotonically) without bound. This is only is unbounded. Let k be the smallest index possible if at least one of the is unbounded on the sequence. It follows from the kth line of I such that (3.11) that there is some index i k such that is unbounded on the sequence. and let m k be the least such index. If m
positive for any vector in S, it follows from the definition of k(T) that k(S) > k(T). But it follows from the definition of T that k(S) contradiction completes the proof of (i).

k(T). This

To prove (ii),. observe that, since all variables are non-negative, L must contain a vertex, for it is a theorem that a closed convex set in finite dimensional space that contains no line must contain a vertex, and the first orthant contains no line. It is also a theorem that a concave function le.g., ç3.4)) bounded from below on a convex polyhedron attains its minimum, and at a vertex if the polyhedron has any vertices. (Proofs of these theorems are contained in L2].)

To prove (iii), observe that, for any k, a vertex of L cannot have both 'k and Ye positive. For we could change Xe and ye by ± E, leaving all other co-

ordinates unchanged, and exhibit the alleged vertex as the mid-point of a line and Ye is zero. We segment contained in L. Therefore, at least one of is satisfied. will use this to show that each inequality in

322

A. J. HOFFMAN

If k = 1, and > 0, then = 0, and the first inequality of (3.11) coincides with the first inequality of (3.3). If y' = 0, then the first inequality of (3.3) > 0, then Xk = 0, and the kth is satisfied since A 0. If 1
where (4.2)

with (4.3)

=

and

j=1,"-,n.

(4.4)

It is assumed that there exists at least one vector (x1, . . ., x,,) satisfying At this point, the meaning of our symbols has shifted, and we

(4.2)—(4.4).

are dealing with a generalization of the problem.

By virtue of the non-negativeness of all x,, it is no loss of generality to assume (4.5)

For each b; without changing the problem. Henceforth we assume (4.5). These preliminaries over (which amount to several trivial transformations of a generalization of the problem as it appeared at the end of § 3), we are

now ready to pose a transformation of the problem to the point where the Monge algorithm applies. First, introduce the non-negative variables y1, -.

. , y,,

by the equation

t=1,-•-,n.

(4.6)

The variables y, may be thought of as unused capacities, if the Xt are conceived as bounded production variables. Next, introduce the non-negative variables Xjj,X12, •..,X15 , ,

xl",

ON SIMPLE LINEAR PR(XRAMMING PROBLEMS

323

by the equations

+ x, = xlt ± X2—x22+...+X2n,

(4.7)

xl, = xflfl

tie production If (4.2)-(4.4) be considered as a production problem, with as that part of production in period i, then one may interpret the variable in period i which is used to satisfy demand in period j. Then it is natural (and will be later justified) to impose the following conditions on the new variables: (4.8)

0,

Xi.?

= b1

X13 +

x2 + x22 = + x33 =

b3 —

=

—

4-

+-

—

Note that, by (4.5), the right-hand sides are non-negative.

Similarly, introduce

the non-negative variables Yii ,Yiz ,

= Y22, ",Y2* , Yn,t

by the equations y'

(4.9)

=y' + =

+

+

,

but one might try thinking of it as that part of the unused capacity in period i "tagged" to period j. As a guide to the conditions analogous to (4.8), let us reconsider the rightIt is not easy to interpret the

hand inequalities of (4.2). Substituting by (4.6), we obtain

1= 1, Let us now define d1

+ •..

A. J. HOFFMAN

324

It is obvious that (4.12)

and that, because the

are non-negativc, the conditions

+ -•

(4.13)

+

are equivalent with (4.10) and hence with the right-hand inequalities of (4.2). We now show further that, if the inequalities (4.2)—(4.4) are consistent, then =

(4.14)

(recall

a1

+ -•- +

—

(4.3)).

To prove (4.14), observe first that its right-hand side is non-negative; otherwise (4.4) and (4.3) would be inconsistent. Next, assume there is some k < n such that

then (4.15)

but

x1 + •.- +

(4.16)

4,

Xk

-

(4.16) and invoking the last of the inequalities of

Adding the (4.2), we have

which

violates (4.15). Thus (4.14) holds.

Now, in analogy to (4.8), we imposer the following conditions on the variables (4.9):

y,, >

(4.17)

0

= (li

Yii

Y12 + Y22 = d2 — d1 Yia

+ )izs

+

Yin

+

=

+ynn

dn

Now consider the following problem: (4.18)

+ xi,,) +

p1(x11 + ...

where the variables x

,

-, x,,,,,

-

Minimize

+

y,

d2,

d3 —

-,

-. + v,,,,

X2n)

+

- -

-+

satisfy (4.8), (4.17) and

ON SIMPLE LINEAR PROGRAMMING PROBLEMS

325

(4.19)

+

+X2,,

+

+Yzn xe,,, +

=Oz,

y,.,, =

It can be shown that: satisfying (4.8), (4.17) and (4.19), the (i) given any variables x,, and variables x obtained from (4.7), (4.9) and (4.6) satisfy (4.2) and (4.4) and yield a value for (4.1) identical with the value of (4.18); (ii) conversely, given any variables x, satisfying (4.2) and (4.4), one can find variables x,5 and Yu satisfying (4.8), (4.17) and (4.19), such that (4.1) equals (4.18); (iii) the conditions (4.8), (4.17) and (4.19) are those of a transportation problem which can be solved by inspection because an appropriate Monge sequence can be identified. The proof of (iii) will occupy the next section. The proof of (ii) is somewhat long (see [3]) and will be omitted. The proof of (i) will now be given.

Observe that the content of (i) and (ii) jointly is that our transformed problem is equivalent to the original one. PROOF OF (i). It is clear that, using (4.7), (4.9) and (4.6), one obtains (4.4) and the equality of (4.1) and (4.18) immediately. What remains is (4.2). To prove the left side of (4.2) observe that-

=

+

(Xis

+ x22) + - - + -

+ -.

+

Xtt)

A similar discussion shows that

and ((4.11) and (4.13)) this implies (4.10) and hence the right side of (4.2).

This completes our construction of the transformed problem. Note that this construction required not only the notion of tagging production in any given period with the period whose requirements it would help satisfy, but also the notion of tagging unused capacity in any period with some period whose requirements it would not help satisfy.

The usefulness of this idea in the present problem will be apparent in the

seems such a strange thought that there may very well be sequel, but other opportunites for using it when its meaning has been absorbed. 5. ApplicatIon of the Monge sequence. To fix our ideas, consider the case n = 4. All the phenomena for general n are already illustrated in this case. Consider the four by eight transportation problem with cost coefficients, row sums and column sums given by the following tableau:

A. J. HOFFMAN

326 (5.1)

b3 — b, d3 —

b4 —

d4 — d3

d1

b2 — b1

d2 — d1

Pt

0

P'

0

Mt M' M6

M Mt

P2

0

0

0

M

0

0

M5

M'

M

0

b1

M3

pt

M2

0

Pt

b3

0

M is an arbitrarily large positive number. Notice first that this transportation problem has non-negative row and column sums, and satisfies the condition that the sum of the row sums equals the sum of the column sums, for the sum of the column sums is b4 + d4. By (4.3) and (4.14), this is

b4+aj+a2+a3+a4—b4=a1+-•.+a1, which is the sum of the row sums. The odd columns refer to variables the even columns refer to variables The large coefficients Mk compel certain variables to be zero. It is clear that this transportation problem is then identical with (4.18).

Next, arrange the 32 elements of the cost matrix in a sequence by the following rule: (i) list first the elements of the first column in ascending order of magnitude, (ii) list the elements of the second column in ascending order of magnitude, (iii) list the elements the third column in ascending order of magnitude, (iv) list the elements of the fourth column by first putting the zeroes with indices whose corresponding P's are in descending order of magnitude, then the powers of M in ascending order, (v) list the elements of the fifth column in ascending order of magnitude, (vi)

list the elements of the sixth column by first putting the zeroes with

indices whose corresponding p's are in descending order of magnitude, then M, (vii) list the elements of the seventh column in ascending order of magnitude, (viii) list the elements of the eighth column in any order. Then one sees

that the stipulations (2.2) have been satisfied for this sequence, so the algorithm of § 2 applies. It can he shown, of course, that if inequalities (4.2) and (4.4) are consistent, the algorithm will never choose a positive if is a power of M. 6. Remark8. The first transformation used above is a special case of a device which appears to have been used for the first time by W. Jacobs in [11]. The second transformation is based on an idea of Prager [14]. (Incidentally, the simple algorithm proposed by Beak [5] for the solution of Prager's

formulation of the caterer problem can be shown to be a special case of the Monge idea; so can the algorithms presented in parts of references Besides these transformations, other tricks are the use of the duality theorem [7] and various devices for standardizing the structure [1]. In the not too distant future, it should be possible to present a catalogue of devices

ON SIMPLE LINEAR PROGRAMMING PROBLEMS

usable in making linear programming problems "simple," whether or not the Monge idea applies; at present, they are too fragmentary to justify listing. By far, the most interesting direction of study is that initiated by Jacobs in [11J. In this instance, he gave an example of how one could minimize on a

set K by minimizing on L K, because it was possible to show that a minimum on L occurred at a point of K. A less ingenious instance of this was given in § 3 above. A comprehensive theory giving classes of cases where such transformations are possible would be very desirable. REFERENCES

1. A. J. Goldman and A. W. Tucker, Theory of linear progra,n,nsng, Linear inequalities and related systems, pp. 53-98, Annals of Mathematics Studies, No. 38, Princeton Univ.

Press, Princeton, N. J., 1956. 2. W. M. Hirsch and A. J. Hoffman, Extreme varieties, concave functions and the fixed charge problem, Comm. Pure Appi. Math. 14 (1961), 355-369. 3. A. J. Hoffman, Some recent appiwations of the theory of linear inequalities to external co,n6inator-ial analysis, Combinatorial analysis, pp. 95-112, Proc. Sympos. Appi. Math., Vol. X, Amer. Math. Soc., Providence, R. I., 1960. 4. G. Monge, Débiai et remblai, Mémoires de l'Académie des Sciences, 1781. 5. E. M. L. Beale, Letter to the editor, Management Sci. 4 (1957), 110. 6. E. M. L. Beale., G. Morton and A. H. Land, Solution of a purchase-storage programme, Operational Research Quarterly 9 (1958), 174-197. 7. A. Charnes and W. W. Cooper, Generalizations of the warehousing model, Operational Research Quarterly 6 (1955), 131-172. 8. C. Derman and M. Klein, Inventory depletion management, Management Sci. 4 (1958), 450-456.

9. M. Fréchet, Sur lee tableaux de correlation dont Lee marges sont donnéee, Ann. Univ. Lyon Sect. A (3) 14 (1951), 53-77. 10. J. W. Gaddum, A. J. Hoffman and D. Sokolowky, On the solution to the caterer problem, Naval Res. Logist. Quart. 1 (1954), 223-229. 11. W. Jacobs, The caterer problem, Naval Res. Logist. Quart. 1 (1954), 154-165. 12. S. M. Johnson, Sequential production planning over time at minimum cost, Management Sd. 3 (1957), 435-437. 13. H. Lighthall, Jr., Sch'duling problems for a multi-commodity production model. Tech. Rep. 2, 1959, Contract Nonr-562(15) for Logistics Branch of 051cc of Naval Research, Brown University, Providence, R. I. 14. W. Prager, On the caterer problem, Management Sci. 3 (1946), 15-23. 15.' W. E. Smith, Various optimizers fir single-stage production, Naval Res. Logist. Quart. 3 (1956), 59-66. INTERNATIONAL BUSINESS MACHINES CORPORATION

TOTAL POSITIVITY AND CONVEXITY PRESERVING TRANSFORMATIONS BY

SAMUEL KARL!N1 1. Let X and Y be real intervals and let K(x, y) be a bounded measurable function defined on the rectangle X ® Y. It is useful and interesting to find conditions on K which imply that the transformation

xeX,

9— Tf

sends bounded convex functions into convex functions. Our objective is to establish the relevance and utility of the concept of total positivity and related ordering- properties (see below) to this problem. More precisely, we shall show that the property of total positivity provides simple sufficient conditions

to insure that the transformation T is convexity preserving. We will not enter the question of determining to what extent the converse is true. In this connection, see [9; 31. The

theory of total positive functions and, more generally, sign regular

functions, has been widely applied in several domains of mathematics, statistics and mechanics [1; 2; 3; 13]. An extensive summary of the recent literature

which also contains some extensions of the theorems of this paper can be found in a forthcoming monograph by this author [3]. 1 and 2 we develop alternative definitions and preliminaries dealing In with the property of sign regularity. A number of -theorems are stated in. dicating the relative strength of the various definitions. Basic variation diminishing properties that the transformation (1) inherits when K is suitably sign regular are also stated.

§ 3 is devoted to a discussion of several important

examples which show the usefulness of alternative formulations.

In § 4 generalizations are given and various specific applications are indicated stemming from the original question of the convexity preserving nature of T.

2. Prellminsries. A function K(x, y) of two real variables ranging over sets on the real line X and Y, respectively, is said to be totally positive of order r (abbreviated Ti',) if for all m, 1 m r, and for all' x1 <x, < < ...
we have the inequalities I This paper was supported by National Science Foundation G-9669. I acknowledge indebtedness to A. Novikoff of Stanford Research Institute for several valuable discussions.

' Unless stated explicitly to the contrary, the determinants (2) shall always have the rows and columns arranged according to increasing values of x and y, respectively. 329

SAMUEL KARLIN

330

K(x1,y,)

K(x2,y1) K(x2,y2)."K(x8,y,1)

(2)

o. —

•- K(x,, , y.,)

K(x., ,

If strict inequality holds in (2) then we say that K is strictly totally positive of order r (STP,). Typically, X is an interval of the real line, or a countable set of discrete values on the real Fine such as the set of all integers or the set of non-negative integers; similarly for V. When X or Y is a set of integers, we may use the term "sequence" rather than "function." if X and Y are finite sets, then K reduces to a matrix.

A more general concept is that of sign regularity. A function K(x, y) is sign regular of order r (abbreviated SR,) if there exists a sequence of numbers each

and

either +1 or —1 such that for every <X,Yt
1 m

(x1eX;yje Y)

r x2.

\Y*,Y2,

..•,

o.

The notion of strict sign regularity of order r (SSL) corresponds to strict 2,,, then inequality in (3). In the case of a SR, function, the symbol denotes the sign of the mth order determinant. we say that K is "RR,," where the letters In the case i,,, = suggest the abbreviation of sign reverse rule and may be interpreted as asserting that (2) is positive, when the order of the rows are reversed. If a TP, function K(x, y) (x c X, ye Y) is a probability density in one of the variables, say x, with respect to a o-finite measure jz(x), for eath fixed value of y, then K(x, y) is said to be Pólya type of order r (PT,) The concepts of PT1 and PT2 densities are familiar ones. PT2 functions correspond to those possessing a monotone likelihood ratio (M.L.R.). The name M.L.R. refers to its statistical context (see [6]). An important specialization occurs if a TP, function may be written in the form K(x, y) = f(x — y) where x and y traverse the real line; f(u) is then said to be a Pôlya frequency density of order r (PF,). If x and y range through

the set of integers, f(u) is said to be a Pólya frequency sequence of order r which we abbreviate in the same way (PF,). Finally, if the subscript is omitted in any of the previous definitions, then the property in question will be understood to hold for all values of r. We record for ready reference two simple properties of SR kernels. (a) If K(x, y) is SR, and so(x), 4'(y) are nonzero functions, maintaining a constant sign for xc X and y Y, respectively, then L(x, y) = ço(x)çb(y)K(x, y)

is SR.

TOTAL POSITIVITY AND CONVEXITY PRESERVING TRANSFORMATIONS 331

define and v = (b) Let K(x,y) be SR, (xcX,ye Y). Let u = strictly increasing functions which transform X and Y into U and V, re-

spectively. Consider L(u, v) =

m = 1,2, •, r. If u(x) is strictly in. Then L(u, v) is SRr and e.(K) = and creasing while v(y) is strictly decreasing, then L(u, v) is

= We are motivated to introduce the definitions below by the following elementary observations. The monotonic character of a function h(I) of a real variable can be expressed in various forms. We say that h(t) is increasing if h(t1) provided h(11) 0 for all I in I. We now formulate, analogously, various definitions of sign regularity. It is necessary to provide a meaningful and useful interpretation for the determinant

\Yi,Ys,

.,y

where

Y)

in which several of the x's and/or y's are permitted to coincide. The asterisk sign on the K will always appear when this is the case and indicates that a special evaluation as described below is to be made. In writing (4) we shall

always assume that X and Y are open intervals of the real line and that K(x, y) is sufficiently continuously differentiable. Now when there occurs a block of equal x values in (4), the corresponding rows are determined by successive derivatives, i.e., K, 8K/Ox, ., For

example, in the case of two equal x values, the initial row associated

with these values is formed in the standard way and consists of the elements K(x, yj), K(x, ye), .. ., K(x, the following row consists of the elements OK

OK

y1), ..., OK

y.).

Blocks of equal y values are treated analogously, in this case partial derivatives in the y variable occur wherever necessary. As an illustration, let m = 6 and suppose x1 <x2 = x, <Xs = x.; Yi
____________

SAMUEL KARLIN

332

= K(xj,y1)

OK

K(xj,ys)

K(x11y.)

K(xiy2)

K(x2,y.)

8'K

OK

K(x4,y1)

K(x4y2)

K(x5

K(x5 ,y2)

OK

OK

-ç4-(x4,y2) ,

02K

Ya)

OK

84K

03K

,y,)

,

,

,

K(x4,y.)

, y2)

0'K

OK

An important special case is K' (6)

—'x 8y' ' '

OK

•,

K*(Xl X,

'

y)

82K

8x81y y)

9K(x, y)

y) -

This convention resembles the familiar procedure in dealing with sets of data

used for interpolation theory and in the theory of ordinary differential equations to cover the case of "coincident points." The next two theorems show the close relation between the sign regularity properties of determinants involving x and y values which are distinct on the one hand and coincident on the other hand. 1.

(a)

Let K be

and continuously differentiable m — I times1

r for x in X (X is an open interval); then

m (7)

X,1,

(b) (9)

Let K be SR, and

in the variables x and y; then

\Yi,Ys, ",Y'.'/

(xeX,y1€

TOTAL POSITWITY AND CONVEXITY PRESERVING TRANSFORMATIONS 333

where

and y• satisfy (5) with egualities allowed for the y's as well.

It should be noted that even if K is assumed to be SSRI, the case of equalities in (7) and (9) may happen. We already pointed this out for the case of monotone functions. However, there exists a converse theorem where the emphasis is on strict inequalities. THEOREM 2.

in X® I (X is an oPen interval). Let

Let K be

(a)

r for all xeX and arbitrary y1

of each order m

< ...

Y);

then Xj

>0

\y1,y2, for any selectzon of (b) Similarly,

andy1
sat:sfytng X1

\y,y,...,y / >0 •

for any selection of Xj satisfying x1<x1< ••• interval, then

for arbitrary x,,y, satisfying x1 <x2 <

y€

I

<x. and all yel=an open

<x,,y1

and I denote open intervals) of each order m r,

then

where

and y1 satisfy (5).

Actually, more general theorems can be asserted in which Theorems 1 and 2 are special cases which refer to arbitrary arrangements of coincidences. A complete discussion of these notions, their detailed proofs and extensions 'will be found in [3]. The result of Theorem 2 is essentially due to P6lya [13]. It is useful to ascertain to what extent the requirement of strict positivity is dispensible in the hypothesis of Theorem 2.

The following result resembles a familiar criteria for verifying that a quadratic form is positive definite or semi.definite.

SAMUEL KARLIN

334

the hypotheses of Theorem 2, Part (a), hold when m r —

COROLLARY.

for all x X, y1
(b) (c)

is allowed

...xr)

(12)

for

1

\Y1,Y!,

•,YT

A symmetrical statement to (a) is valid reversing the roles of x and y. the hypothesis of Theorem 2, part (c), holds when m r — 1 and

If

(13)

X, ye Y of order r, then (14)

for all Xf

y < 1.

x,; x, < X; Yi

X2

The following simple example shows in the assertion of Corollary 1 the need for strictness in the lower order determinants.

Let Y={1,2},X=[—oo,oo),K(x,1)=f(x)=x'; f—2x—1,

K(x,2)=g(x)=1 Note that K(x, 2)

0,

—1<x<0,

0 but that it vanishes on an interval. Then

0,

x<—1,

But

<0 whenever x1 <0< A discrete version of Theorem 2, part (C), is known as Fekete's theorem [2, Chapter 5].

With the statements of Theorems I and 2 firmly in mind we are now ready to formalize the relevant definitions. The previously introduced notion of strongly increasing motivates the ing: DEFU4rn0N 1.

Let K(x,y) be

in XØ Y in which X and Y are each

open intervals. The function K is said to be extended sign regular of order r in the x variable (abbreviated ESR,(x)) zf

TOTAL POSITIVITY AND CONVEXITY PRESERVING TRANSFORMATIONS 335

\Y1,Y2,

-

= 1 we say that < •-
>0 for all xe X and y Y where the size of the determinant is m

r. = 1 for all m then K is said to be extended totally positive of order r in both variables As in the notion of strongly increasing, the emphasis in Definition 1 is on strict inequality. Using this language, Theorem 2 may be stated as follows: When

THEOREM 3.

Cases (a) and (b): If K is either ETP,(x) OT

then K is

sTPY. Case (C): If K is ETPS. then K is ETP,.(x) and ETP?(y).

The converse need not be true in general. However, in some special important cases the properties of STP,. and ETP, are essentially equivalent. 4. Suppose 1(x) is r — 1 times continuously differentiable and K(x,y) = f(x — y) is and STP,., then K(x, y) is Here x and y

assumes a particularly simple form: traverse the real line. The case r = namely for translation kernels STP implies ETP. Correspondingly for both variables, suppose f(x) is 2(r — 1) times continuously Then K(x, y) is ETP,. differentiable and K(x, y) = f(x — y) is TP,+1 and

The proof of the theorem will appear in [3]. The assertions of Theorem 4 may be extended to kernels L(x, y) derivable from a translation kernel by a strongly increasing change of variable. Explicitly, consider 77) =

çt'(,j)) =

—

(a <
TP,+1 and STP,. We obtain if X= Y=(0,oo) and ço=çb=exp. COROLLARY.

K(x,y) is

Let K(x, y) = g(x/y) (0 < x, y < co), and suppose g(.) is

If

and STP, then K(x,y) is ETP,..

We can express the ETPY property in a form which more resembles the concept of monotonicity. To this end we introduce a sequence of functions defined recursively in terms of the parameters , . . , y,, as follows:

SAMUEL KARLIN h1(x)

K(x,yj,

h2(x)

h,(x)

...

h,(x)

where

yeY, xeX.

yl
are (m = 2, •, p) are well defined provided h1, .-•, The functions all nonzero and K is m fold continuously differentiable with respect to x e X. Then Kis ESR,.(x) if and only strictly $sitive for all y and x satisfying (18) for suitable signs , •••, e,. Variation Dlmlnl8hing Property. Each of the concepts of SR carries with it a corresponding form of the basic variation diminishing property.

In order to state these results, we introduce the following notation. Let f(t) be defined in T where T is an ordered set of the real line. Denote by

V[f(t)]

sup

•

< •-•
ing multiplicity, the last being defined in the usual way if f is suitably

differentiable.

Let K(x, y) be defined in X x Y measurable and assume for simplicity that the integral S1K(x, y) dp(y) exists absolutely for every x in X. Let f be bounded on Y and consider the transformation (21)

g(x) = Tf(x) =

K is SR avd satisfies the requirements above, then

V1Tf) In the case that K is TP, then if V-f = Vg, the values of the functions f

(22)

and g exhibit the same sequence of signs when their respective arguments traverse the domain of definition from left to right.

TOTAL POSITIVITY AND CONVEXITY PRESERVING TRANSFORMATIONS 337 (b)

Suppose

that K is SSR, then we have

V(f) provided f is not zero (a.e. /1). (c) Suppose K is ETP(x) and that differentiation any number of times under the integral sign in (21) is permissible, then

Z*(g) V(f) provided I is not zero, a.e. 4a.

A converse of Theorem 5 can be shown to be correct, but we shall not formulate it precisely here. 3. Examplea of totally positive functiona. In this section we discuss a series of examples of some interest. The composition formula (24) below plays an essential role in the construction and analysis of these examples. Let K, L and M be functions of two variables satisfying =

(23)

C)L(C, v)de(C)

where the integral is assumed to converge absolutely. Here traverses X, C ranges through Y and varies over Z; each domain is a linearly ordered set on the real line, ti(C) denotes a (1-finite measure defined in Y. When Y consists of a discrete set then, of course, the integral sign is interpreted as a sum. If the formula (23) is viewed as a continuous version of a matrix product, subdeterminants then the extension of the multiplication rule which of M in terms of those of K and L becomes \71,,'iz, (24)

= J

•..

K

L

.1

\liz,7)2, x da(C1)dc(C2)

... dti(C,.).

For the proof we refer to Pólya-Szegö [10, p. 48, Problem 68]. The formula (24) frequently applies even if we allow equalities amongst the and ,j values; in that case the interpretation of the determinant conforms to that of (4). The relation (24) is certainly justified whenever it is permissible to differentiate (23) under the integral sign. Henceforth we assume that it is permissible to carry out an of differentiation and integration operations. The relevance and utility of (24) will be abundantly clear. 1.

The function e',

<x,y <

is

easily seen to be ETP [13]. Asan

immediate corollary, we obtain that K(x, y) is TP (STP) on X® Y provided ç(x) and çb(y) are increasing (strictly) on X and Y, respectively (see remark (b) of § 2). If X and Y are open intervals and çô and 4' are C and strongly increasing, then is ETP. More complicated examples of SR functions are constructed based on the example K(x, y) = with the aid of (24) (see Examples 2 and 3 below).

SAMUEL KARLIN

338

The function

(1, a<xy
K(x,y)—_10

(25)

a

direct calculation shows that 10, otherwise.

'\y1,y2,

Combining with remark (a) of § 2, we see that

()26 is

K"x \

—

fp(x)q(y), a < x

y

b,

TP provided P(x) and q( y) are positive functions on (a 2.

x, y b).

Many classical examples of TP kernels admit the following repre-

sentation

K(x, y) =

(27)

.=0

where u(x) and v(y) (x e X, yE Y) are increasing positive functions and a,, > 0,

n = 0, 1,2, •••. Of course, the series is assumed to converge for all xc X and ye Y. In order to prove that (27) is TP we write it as a Stieltjes integral: (28)

K(x, y)

=

exp [E log u(x)] exp

log v(y)1 dgl'(e)

where c& is a discrete measure with jumps a,, located at n = 0, 1, 2, . As noted above, exp log u(x)J is TP for real and XE X. It remains to apply

(24), thus concluding that K(x, y) is TP, strict when u and v are strictly increasing and ETP(x) when X is an interval, u and ,i(x) > 0. As a concrete illustration, we mention K(x, y) = I.(x y), where denotes the standard Bessel function of imaginary argument. 3. It is frequently simpler to verify that a function K(x, y) is ETP rather

than TP (just as one frequently proves that a function is increasing by proving the stronger fact that it is strongly increasing). As an example, we establish that the important noncentral t-density occurring in statistical theory is ETP an'd a fortiori TP. This density function is of the form A)

=

exp

[—

—

where c is a normalizing constant and a represents a fixed positive parameter. We now prove that p(t, A) is ETP for — oo
of no direct way to establish the TP property except by proving the stronger result of ETP. Consider first t > 0, let v = V'wI2a 1. Then after obvious simplifications p(t, A) =

erA exp (—a(v2/t2)ldci(v)

TOTAL POSITIVITY AND CONVEXITY PRESERVING TRANSFORMATIONS 339

where f and g are strictly positive functions and o(v) is a positive measure whose explicit form we need not write. Applying (24), we obtain that >0

for 1>0 and —oo
where f, L(v, A)

exp

p(f, A) =

(30)

=

I; then

and a satisfy the same conditions as!, g and ev?t

above. The function

satisfies

>0 where m is the order of the determinant. The function M(v, 1) satisfies

exp [—a(v'/15)J

>0 since

df 1\

2

<0.

By (24), we again obtain that

pa (1,

>0

for 1 <0 and —00 < A < 00• For the case when 1 0, a direct calculation yields a Vandermonde determinant and it follows that

>0, <2 < oo The three cases I > 0,1 <0 and t = 0 in Conjunction, verify that the noncentral f-density is ETP. By the same methods we can prove that the density function of the noncentral correlation coefficient, the noncentral Chi square and numerous other important statistical distributions are ETP. 4. An extremely rich source of examples of TP functions arises by considering the Green's functions of Sturm-Liouville differential equations. The totally positive functions associated with eigenvalue problems of differential equations are not extended totally positive, but the determinants are strictly positive in an intermediate sense explained below. denote the operator Let —oo

L(çc) =

+

and consider the eigenvalue problem L(çc') under the boundary conditions

SAMUEL KARLIN

340

=0, The Green's function has the form

K(X,Y

14,(y)z(x),

where 4' and Z satisfy the boundary conditions at a and Li, respectively. Gantmacher and Krein [2] have shown that (32) is TP provided 4'(x)x(x) > 0 for all

a x

6. We can prove the folldwing sharper statement

if and only if (5) holds and

i=1,2,".,n—1. This allows certain equalities between the x's and y's. The Green's function K(x, y) of the eigenvalue problem (ry")" = 2y under the boundary conditions

=0 = 0 , ry"

+ a0y

(ry")' + where

cc is

—

=0 =0

,

—

TP. (See [2).) We can prove in this case the

sharper statement that x1,

(35)

•

•,

",YR/

>o

if and only if (5) holds and < Yi+i

(36)

,

<

,

I = 1, 2, •.., n — 2

In comparison with (34) the restrictions (36) allow certain additional equalities

amongst the x's and y's. The generalizations to the case of higher order differential operators are evident. The proofs of (34) and (36) and their extensions will appear in 131. We close this section with another application of (24). SpecifIcally we investigate the sign regular nature of the function c(t + s) where (37)

c(t)

=

x)f(x)dx,

t > 0.

We assume that f(x + y) is sign regular (for example, either or RR,) and x) is TP, for t > 0 and x > 0 and obeys the semigroup property that

TOTAL POSITIVITY AND CONVEXITY PRESERVING TRANSFORMATiONS 341

p(t+s,x)=

(38)

Subject to these conditions we prove that c(t + s) is TI', (RR,) according as f(x + y) is TP, (RR,). To this end, we examine the expression c(i + 5)

(39)

+ s, x)f(x)dx

=

u)f(u + E)dude.

=

The last identity results by applying (38) and then performing obvious manipulations. With the help of the convolution property (24) we obtain that the function ço(s,u)f(u +e)du

sb(s,e)=

is sign regular of the same type as f(u + Now, (39) can be written in the form: c(t + s) =

since o(s, u) was assumed TP,.

e)sl'(s, E)dE.

Another application of the convolution property (24) yields that c(t + s) is appropriately sign regular. The above arguments hold virtually without change in the case that the variable t traverses over the set of positive integers. We summarize this discussion in the statement of the following theorem. THEOREM 6. Let f(x + y) be Ti',. (RR,) for x> 0 and y > 0. Suppose ço(t, s)

is TP,for x>O and t >0 (or t = 1,2,3, defined by (37). discrete case).

and satisfies (38).

Then c(t + s) iS TP, (RRr) for t, s > 0 (t, $ = 1,2,

Let c(t) be

in the

The hypotheses of the above theorem are satisfied in each of the following circumstances:

Let f(x) be a PF, density function of a non-negative random variable. where (a) Let ço(n, be the density function of the sum X1 + + -- + are positive identically and independently distributed random variables whose density function is PF,. - The fact that ç'(n, E) is TP, for n 1 and E > 0 is proved in [7]. The semigroup character of so(n, is clear. (b)

Let

=

1(1) '

Relation (30) and the TI', property may be verified by direct calculation. Let denote the density function of a stable process X(t) with non(C) negative drift whose Laplace transform is exp s > 0 (k = a fixed positive integer rt is shown in (4J that is for t, > 0.

SAMUEL KARLIN

342

Property (38) emerges from the fact that X(t) is a homogeneous process with non-negative drift. The above theorem also enables us to deduce various moment inequalities for PF, densities associated with non-negative random variables. Inueed, let t > 0,

c(t)

where f(E) is a PF density on the positive ray. As observed above in example (b),

ço(i,e)= f'(t)

'

0, satisfies the condition of Theorem 6. Hence c(l + s) is SRR for I, s > inequality of the second order determinant asserts 2a(s + t)

(41)

a(t) + a(2s + I)

0.

s,

,

The I>

0

where a(u) = log c(u) + 1. This shows that a(u) is concave (since a is conIn particular,

tinuous).

a(t) and since a(O) = (42)

+

+ t)

(1 is a density), we have

0

'I'

1

'I'

1

(1(1 + 1)

+

1)

for 0 I <s. This derives further interest by comparison with the classical moment inequality

e'f(e)de)'

I <s which also emerges by total positivity arguments in which the relevant second order determinant is of the opposite sign.. For another den: for 0

vation of (42), see 181. 4. Convexity preserving properties. The significance of the variation diminishing properties of sign regular functions described in Theorem 5 is best emphasized by drawing some of its consequences. These results are of interest in themselves and useful in applications. We consider the transformation (21) under the same conditions stated earlier. PRoPosITIoN 1.

Suppose ,.K (x, y)dp(y)

1 for all x X.

If K is SR2 and f(y) is monotone, then (43)

g(x) =

y)f(y)dp(y)

TOTAL POSITIVITY AND CONVEXITY PRESERVING TRANSFORMATIONS 343 is monotone. PRooF.

(a)

Let a be any real number and consider the relation g(x) — a

=

K(x,y)[f(y) — a]d1z(y),

x€X.

For any a, according to the hypothesis, f(y) — a changes sign at most once. The variation diminishing character of K implies that g(x) — a enjoys the same property. This is tantamount to the monotonicity of g(x). PRoPOsITIoN 2. In addition to the hypothesis of Proposition 1, we have: then y(x) (i) If K is SSR2 and f(y) is monotone and not constant (a.'. is strictly monotone in the same sense as f. (ii) Suppose X is an open interval and K is E.SR2(x). Assume that differenti. ation under the integral sign in (21) is justified. If f(x) is monotone and not constant (a.e. p), then g'(x) never vanishes. In the following proposition, the hypothesis is strengthened and now involves sign regularity of order 3. PROPOSITION 3. Let X be an open interval of the real line or an interval of integers (i.e., a consecutive set of integers). Similarly for Y. Assume that

xcX, a > 0.

y)dp(y) = ax + b,

Let K(x, y) be TP3. If f(y) is convex (concave) then g(x) is convex (concave).

PROOF. The proof is based on the identity (45)

g(x) — c1x— c2

=

K(x,y)(f(y)

—

c1y/a

— c2

+ cibla]dIL(y)

for any real c1 , cz. The function in brackets under the integral sign changes sign at most twice, since f is convex (concave). Hence, by Theorem 5, part (a), g(x) — c1x

— c2

changes sign at most twice and if twice in the same

direction as the function of the integral in (45). Thus in particular every line intersects g at most twice. Whenf is convex and f(y) — c1y/a — ca + c1b/a displays the arrangement of signs +, —, + (which is the only possibility of two sign changes) and if g(x) — c1x c2 also shows two sign changes then they occur necessarily in the sequence +, —, +. This implies that g is convex. Had we assumed in Proposition 3 that K is SR3, then we could have concluded

only that g is either convex or concave and which it is requires further discussion.

If we strengthen the hypothesis, the conclusions are correspondingly stronger. Thus PROPOSITION 4.

Assume the conditions of Proposition 3 hold.

(1) Let K(x, y) be STP8. If f(y) is convex (concave) and not linear (a.e. ia),

SAMUEL KARLIN

344

then g(x) is strictly convex (concave). (ii) Let X be an open interval;

K(x, y) is ETPS and differentiation under the integral sign in (21) is justified. if f(y) is convex (concave) and not linear (a.e. /2), then

g"(x) >

0 (g"(x) < 0).

A generalization of these assertions can be achieved by introducing the concept of convexity of order k. Explicitly a function 1(y) defined on an open interval Y is said to be convex of order k if, for every polynomial Ph_i(y) of

degree k —

1

with positive leading coefficient, f(y) —

displays at most

k sign

changes as y traverses Y.

k

changes, they occur in an arrangement ending with a + sign. In

sign

In

the circumstance that there are exactly

particular, first order convexity is synonomous with increasing and second order convexity coincides with the usual concept of convexity, etc. If I is k times

continuously differentiable, then it is elementary that f is convex of order k 0. if and only if PRoPosrrIoN 5.

Let X and Y be- open

Pj*(x),

K(x,

(46)

for

intervals.

that

X

1,.. •,k 1, and where P1(y) is an arbitrary real polynomial of degree I whose highest coefficient is positive then Pj*(x) is of the same type. Let K(x, y) be TPk; if f(y) is convex of order k, then g(x) defined in (21) is every I =

exact

convex

of

order k.

The proof

consists of an obvious adaptation of the method of Proposition 4.

Other stronger versions of this corollary hold under the condition that K is STPf, or alternately when K is ETPr(X). There also exists a version of Theorem 7 and its corollaries for the case that X and Y are sets of consecutive integers. The following examples occasionally make use of this version. We now present three applications of these propositions. 1. Let = K(x, y) denote a density function with respect to a sigmafinite measure where x serves as a real parameter. Let denote the random variable associated with p1(y). Then Pr{Y1 A) is a decreasing function (strictly) of x for fixed real A whenever K is TP1 (STP2) and /g(—Oo, Al > 0 and 00) > 0. The proof is immediate in view of Proposition 1 and the representation

A)

=

where

_(1,

y>A.

Examples fulfilling the above conditions include all the common statistical densities.

TOTAL POSITIVITY AND CONVEXITY PRESERVING TRANSFORMATIONS 345

2. We apply the method of Proposition 4 to the study of a function which arises in certain economic and queueing models. Consider the densit-y fl—I'

(n—rn)!

K,(n,rn)=

+!

+1!+ ... 0,

otherwise

(a is a fixed positive constant). Clearly, K is a density in rn = 0, 1,2. -- for rn) is TP for each n = 0, 1, 2, - •. A trivial calculation shows that

n,rn=0,1,2,•". Weprovethat

Cl'

1!

is a decreasing convex sequence of n

2!

+1.

0.

The function g(n) possesses economic interpretation; it expresses the marginal

loss in sales when n represents the capacity available in supplying demand. The fact that it is convex is of interest for applications. Let f(n) = 1 for n = 0, 0 for n > 0; then f is clearly decreasing and convex on Y = {O, 1,2, •--}. We observe that g(n) =

(49)

m)f(rn)dp(m)

where p is the counting measure on Y and g(n) is defined in (48). A straightforward computation produces the formula rn)dp(m) = n + ag(n) — a.

(50)

Therefore (51)

(c + aa)g(n) + an — aa + b =

m)[cf(m) + am + b]d,u(m),

a > 0.

We consider only the case a > 0. (The case a <0 is simpler to discuss since and by Proposition 1, g also is decreasing.) Now f(m) + am + b displays at most two sign changes as m traverses the non-negative integers and if two in the order +, —, + then, by virtue of Theorem 5, part (c), the left-hand side of (51) displays at most two sign changes and if two, then in the arrangement —, +. This implies that g(m) is convex.

I

3.

Let

(52)

(k=O,1,2,•••;O<x<1)

where n is a fixed integer. It is easily verifiable that (52) is ETP(x) and satisfies (44), viz.,

kK(k, x) = nx. The nth Bernstein polynomial approxi-

SAMUEL KARLIN

346

mation to a continuous function on the unit interval is defined as x) =

(53)

k=O

x).

'2

x) is convex whenever f is convex. Moreover, considering a tangent line to f at any point x (0 < x < 1) shows

By Proposition 4 we conclude that

x) f(x) (strict if f is not linear). Indeed, let ax + b define a tangent that x) > ax + b line to f(x) at a fixed but arbitrary point xo. Then, trivially with equality only possible if f is linear. Taking x = x0 we conclude that f(XQ) as asserted. x0) Next we observe that if x is fixed, then

L(n,k)=

—

,

is TP. (See also [7] or [4).) Using this fact since approaches f(x), we also obtain (54)

x)

n,k = 0,1,2,

x) f(x)

and

x).

Thus the Bernstein polynomial approximation to continuous convex functions occurs monotonically from above (cf. 112]).

The reader may quickly discern that these results generalize appropriately to kth order convex functions and with other summation methods. We mention a second example. Associate with any continuous function f(x), x 0, which grows to infinity not faster than an exponential the sequence of functions: x) =

We assert that if 1(x) is convex then x) is convex and converges to 1(x) monotonely from above. The proof of this fact is the same as that for the Bernstein polynomials. We can develop a generalization of these examples in the following manner.

Let 1(x) be the density function of a positive random variable, i.e., 1(x) = 0 for x <0. Suppose also that f(x) is a PF3 function. It is proved in [7] that the n-fold convolution

is TPa with respect to the variables n = 1,2,

and x> 0. Assume for simplicity that all moments of I exist and specifically that = 1, or otherwise properly rescale the x axis. We form for any continuous function g which grows to infinity not faster than a polynomial, the sequence A) =

(56)

It is easy to prove that Since

A>

0.

A) converges uniformly to g on any finite interval

TOTAL POSITIVITY AND CONVEXITY PRESERVING TRANSFORMATIONS 347

the methods used above for the Bernstein case apply and we deduce that if g is convex and grows no faster than a polynomial at oo then and converges to g monotonely from above.

A) is convex

The case (53) arises when f is the single trial binomial density and (55) corresponds to the Poisson density both of these being limiting discrete versions of (56).

Finally, we remark that the transformations (56) also preserve generalized k-convex functions provided 1(x) is PFk+I. REFERENCES

1. 1. 1. Hirschman and D. V. Widder, The convolution transform, Princeton Univ. Press, Princeton, N. J., 1955. 2. F. Gantmacher and M. Krein, Oscillation matrices and vibrations of mechanical systems, GITTL, Moscow, 1950. (Russian) 3. S. Karlin, Total positivity and applications to probability and 8tatistics, Stanford Univ. Press, to appear in 1963. 4. Total positivity absorption probabilities and applications, Bull. Anier. Math. Soc. 67 (1961), 105-108. 5. , Decision theory for Pólya type distributions, case of two actions. 1, Proc. Third Berkeley Sympos. on Probability .and Statistics, Vol. 1, pp. 115-129, Univ. of California Press, Berkeley, Calif., 1956. 6. S. Karlin and II. Rubin, The theory of decision procedures for distributions with monotone likelihood ratio, Ann. Math. Statist. 27 (1956), 272-299. 7. S. Karlin and F. Proschan, POlya type distributions of convolutions, Ann. Math. Statist. 31 (1960), 721-736. 8. S. Karlin, F. Proschan and R. E. Barlow, Moment inequalities of Polya frequency functions, Pacific J. Math. 11 (1961). 9. Jan Krzyz, On monotonity-pre8ermng transformations, Ann. Univ. Mariae CurieSklodowska Sect. A 6 (1952), 91-111 (1954). 10. G. Pólya and G. Szego, Aufgaben und Lehrasatze aus der Analysis, Vols. I, II, ,

Berlin, 1925. 11. I. J. Schoenberg, Smoothing operations and their generating functions, Bull.

Amer. Math. Soc. 59 (1953), 199-230. 12. , On variation diminishing approximation methods, On numerical ap. proximation, pp. 249-275, Univ. of Wisconsin Press, Madison, Wis., 1958. 13. G. Pólya, On the mean value theorem corresponding to a given linear homogeneous

differential equation, Trans. Amer. Math. Soc. 24 STANFORD UNIVERSITY

(1922), 312-324.

INFINITE-DIMENSIONAL INTERSECTION THEOREMS BY

VICTOR KLEE 0. Introduction. The report of Danzer-Grunbaum-Klee [1] (DGK) discussed two dual (or polar) aspects of finite-dimensional convexity: intersection proper-

ties of convex sets and the representation of convex hulls. It was natural to link the two aspects since their duality leads to an illuminating interplay (Sandgren (1], Valentine [1]). The duality persists in the infinite-dimensional situation as described, for example, by Bourbaki [1]: If the linear spaces E and F are dual with respect to a bilinear form < , > and {Ma : a E A) is a family of convex sets in E, each including the origin and each closed for the topology a(E, F), then (flGeA M0)° = cl cony M, where the closure is for the topology a(F, E) and where M° = {y e F: SUPX€I <x, y> 1) for each M c E. A more sensitive duality has been presented by Fenchel [1] for the finitedimensional case and can be extended. Even though the duality persists in the infinite-dimensional situation, it has not yet been applied in any significant way to derive infinite-dimensional results of Helly type from those of Carathéodory type or vice-versa. Clearly the infinite-dimensional duality theory merits further study, but at present it would seem artificial to couple the infinite-dimensional intersection theorems with representation theorems for closed convex hulls. (The latter topic will be treated elsewhere.) The present report serves as an infinite-dimensional

supplement to DGK; its aim, like that of DGK, is to supply a summary of known results and a guide to the literature. In addition to infinite-dimensional results, the present report describes some finite-dimensional intersection theorems which deal with infinite families of noncompact convex sets. However, some of the material included- here has little in common with Helly's theorem except in the broad sense that they both deal with intersection properties of convex sets. Perhaps this is an inevitable concomitant of the shift to infinite-dimensional spaces, where the purely combinatorial aspects of the theory of convexity are often submerged in topological considerations. 1. Some Intersection theorems for infinite families. Let us begin with the finite-dimensional discussion promised in § 4 of DGK, viewing Helly's theorem as a statement about the structure of certain families of convex sets in R' such that is the intersection of all members = 0 (where of We know of no satisfactory generalization which applies to all such families but will discuss some partial results. The following extension of Helly's theorem is due to Sandgren [11, the proof below to Grunbaum.

1.1.

Suppose

is a family of closed convex sets in R", 349

is nonempty

VICTOR KLEE

350

and compact for some

9c51 with card

c

and

= 0. Then r5 = 0 for some

— ,r..9' has a countable dense subset and of is an open covering of Z, there is a countable subset be an enumeration of 9 and T2, = ir.V. Let Then some set A, must be bounded. Indeed, suppose the contrary and for each m let Ym€Dm with !lYmIl > ,n. Assume without loss of generality that 0€ ir,V. Clearly the sequence y,, admits a subsequence such that the sequence y,,,./Ily,,,011 converges to some point qeR" with = liqil = 1. Since [O,yM1] c I?, for all i mj, it follows that [0, oo[q 7r.Y, contradicting the fact that ,r.V is bounded. Now with Dm bounded, suppose ,r5 0 whenever 9c. and card 9

PROOF.

Since the set Z =

S: S .Y" such that let I),, =

n + 1. Helly's theorem implies that . has the finite intersection property, Then whence the same is true of the family of compact sets (K n D,., : Ke * 0 and the proof is complete. of course ir.

In a study of the structure of families of convex sets with empty intersection, the families of open halfspaces are of special interest, both intrinsically and because of the following fact (Gale-KIee [1]): 1.2.

Suppose (K0: a€

is a family of convex sets in R' such that

K0

=

0. Suppose further that each set K0 is open, or each is closed and admits neither asymptote nor boundary ray. Then there is a family of open halfspaces (J0: a A} such that K0. J0 = 0 and always

We next state Klee's extension (31 of Helly's theorem to compact families of open convex sets. (A related compactness condition appears in a covering theorem of Karlin-Shapley [11.) 1.3. Suppose is a family of open convex sets in such that mt Ce = whenever C is the limit of a convergent sequence of members of If n + 1. 0, then ir5 = 0 for some .9c .5t with

A theorem of Ramsey [1] asserts that if all r-membered subsets of an infinite set A' be divided into a finite number of classes, then there is an infinite subset Y of X such that all r-membered subsets of V belong to the same class. Combining this result with Helly's theorem, R. Rado (11 has proved: 1.4. If is an infinite family of convex sets in there is an infinite subfamily .3/' of .2" which has one of the following two properties: .3/ has the finite intersection property;

for some 5 n it is true that each j members of .3/' have

inter-

section but each j + I members have empty intersection.

In any discussion of intersection properties of convex sets, the semispaces should be mentioned. When P is a point of a linear space E (of dimension a maximal convex subset of E — p. When S is such 1), a a semispace, its reflection 5' in p is also a semispace and E—. {p} = S u S'. The semispaces were introduced by Motzkin [1) and Hammer [1], who showed

INFINITE-DIMENSIONAL INTERSECTION THEOREMS

351

that in the following sense they form the unique minimal intersection base

for the class of all convex sets; 1.5. Every convex set is an intersection of semispaces. If a semispace S is the intersection of a family of convex sets, then Se .X.

A detailed discussion of semispaces is given by Hammer [1; 2] and by Klee [6), who proves the following intersection theorem: 1.6. For a convex set C in a linear space of countable dimension, the following two assertions are equivalent: C is the intersection of a countable family of semispaces;.

whenever .2' is a family of convex sets for which n.5r = C, then ,r.9' = for some countable .9' c

C

From 1.6 it follows (in a linear space of countable dimension) that if a family of convex sets has empty intersection, then so has some countable subfamily. While this property is characteristic for spaces of countable

dimension, the same conclusion holds in more general spaces when the convex

sets are subjected to suitable topological restrictions. A result in this direction was established by Klee [6] (1.8 below), and Corson has recently formulated the underlying topological conditions. For a family .9' of subsets of a topological space, let = ci n.5'. For a family let = countable .9' c Then Corson's result is as follows: 1.7. Suppose X is a topological space and W is a family of subsets of X such that X %'° has the Lindelof properly and ,r.9' is different from 7cW for each countable 3.' c Then W° is a proper superset of ,r4' and U n W° is dense in for each Uc W. PRooF.

Since {X— ..V*: countable .9' c W} is an open covering of X— fe'°,

there is a sequence , 9', , --• of by the Lindelof property of X— countable subsets of such that X— ,.V). With = = we have

U

= ci ir.5r = cI

c

c

=

=

jf = then = ci whence = Since is a countable subfamily of this contradicts the hypothesis and shows that *° is indeed a proper superset of For each whence

U€W,

(Un 'l'i°)=cl(Uri whence ci (U n 1.8.

=

and

the proof is complete.

Suppose E is a complete separable metric linear space and W is a

family of convex subsets of E such that is closed. Suppose further that each member of W is closed, finite-dimensional, or open relative to the smallest closed fiat containing it. Then ,r.9' = irW for some countable .9' c Yl. PROOF. Suppose the conclusion is false, whence 1.7 implies the existence

352

VICTOR KLEE

For each z e ?I°, let D, denote the union of all = open line segments in W° which pass through z (and D, = lx) when is a complete separable metric convex set, it is known that {x}). Since such that A is dense in ?I° (see Klee (5]). Since there exists z and it is easy to that D, is is closed we may choose ye Jx, z( is Now consider an arbitrary Ue %', and note that Un dense in When by 1.7. When U is closed, then of course dense in U is finite-dimensional, then '//° is finite-dimensional and y must be a relawhence y U. Suppose, is dense in tively interior point of W° (since finally, that U is open relative to the smallest closed flat F containing U. If y U, then y lies in a closed hyperplane H relative to F such that H misses U. Since U n is dense in by 1.7, W° cannot lie in H and hence must intersect one of the halfspaces (in F) which is bounded by H. Since is dense in must intersect the other open halfspace also, whence U intersects both open halfspaces. But then U intersects H and the contradiction implies that yE U. We have now proved in each of the three cases that y e U, whence y and the contradiction completes the proof of the theorem. Theorem 1.8 actually holds for a somewhat more general class of sets, as described by Klee (6]. While separability of the containing space plays an important role by way of 'the Lindelöf property, it is unclear whether completeness is essential. In particular, the following problem appears to be open: If is a family of open convex subsets of a separable normed linear is closed, must there exist a countable subfamily ..V of W for space and = When (or, a fortiori, has nonempty interior, an which affirmative answer follows by the above method; when = 0, the answer

of a point xe

is apparently unknown. 2. Intersection theorems involving the weak topology. Among the infinitedimensional relatives of Helly's theorem, those of the most significance for

functional analysis deal with intersection properties of closed convex sets. These results are strongly topological in flavor, but convexity enters the picture because of the fact that a closed convex subset of a locally convex topological linear space must also be weakly closed (closed under the weak topology). It is known, for example, that a normed linear space E is reflexive

if and only if its unit cell lx: t

1} is weakly compact, and hence it is x II evident that E is reflexive if and only if 0 whenever .5/ is a family of bounded weakly closed subsets of E which has the finite intersection property. Theorems of Smulian [1] and Eberlein [1] show that reflexivity and weak compactness can be characterized in terms of intersection properties of sequences of convex sets. As refiaed by Dieudonné [1] and Day [1], the result may be stated as follows: I

2.1. A weakly closed subset X of a complete locally convex space E is weakly

compact if and only tf X has the following Property: whenever K. is a decreasing sequence of closed convex sets in E and each K1 meets X, then meets X.

K,

INFINITE-DIMENSIONAL INTERSECTION THEOREMS

353

This result has interesting implications for the geometry of convex sets (Klee [1; 2], Dieudonné [1], James (1; 2], Köthe [1]) and in topological studies (Klee [4]). For the latter purpose, the foUowing intersection theorem was proved by Klee [7] and applied by him and by Corson-Klee (1]: 2.2. In every infinite-dimensional normed linear space there is a decreasing sequence of unbounded but linearly bounded closed convex sets whose intersection is empty.

For convex sets, the following improvement of 2.1 is due to Floyd-Klee [1] and Pták [1]: 2.3. A bounded closed convex subset K of a complete locally convex sPace E is weakly compact if and only if K has the following Property: whenever H. in E and each finite intersection 11' is a sequence of closed H, meets K. meets K, then

(1] shows that if U is the unit cell of a separable nonreflexive then some continuous linear functional fails to attain its maximum on U and thus there exists a decreasing sequence f, of closed halfspaces such that hr misses U even though each set J, meets U. (The same result for the nonseparable case is stated by James 12].) Klee [8] conjectures (at Banach

least in the separable case) that this property is characteristic for weakly closed sets which are not weakly compact, and proves an intersection theorem which seems to support the conjecture. Let us say that a family of sets has the countable intersection property In line with 1.8 and 2.1, provided * 0 for each countable ..V c it is natural to wonder which Banach spaces E have the following property: 10 whenever a family of closed convex subsets of E has the countable intersection Property, then ,r.2' 0. This and related properties of E are studied by Corson [2] in a deep investigation of topological properties of spaces E., (a Banach space E in its weak topology). He considers the following properhas the Lindelof proPerty; 1110 the space E is the closed linear extension of a weakly compact set; IV° the space is normal; V° for each natural number n, the product space E is normal; V!° the space El is fraraties: 110 the space

compact. It is well-known or obvious that 110._la, II°=.VI°=-4V°, and V°—'IV°.

Corson proves that VI°=II° and conjectures in an oral communication that the conditions II°—VI° are all equivalent to each other but are not equivalent to 10. In fact, let D denote the Banach space of all bounded real valued functions

f on [0, 1] which at each point of [0, 1] are continuous from the right and have a limit from the left, with 11111 = sup (Ifx : x [0, 1]). Then Corson can show that the space Dl, lacks the Lindelof property and he suspects that D has property 1°. (By a theorem of Rudin-Klee [1] and Michael [1), E is a separable metrizable locally convex space.)

Corson's paper [2] contains other interesting results and examples concerning intersection properties of convex sets in certain Bánach spaces. For example, the space (m) lacks property and the space C.[O, 2[ actually contains

354

VICTOR KLEE

a family of closed metric cells (homothetic images of the unit cell) which has the countable intersection property but empty intersection. If C is a compact

group and L1G the space of all Haar integrable functions on C, then the indicates the topology space (L1G),,. has the Lindelöf property, where a(L1G, CC) obtained by regarding the members of L1G as continuous linear functionals on CC. Corson's most complete results concern spaces C,X, where this is the Banach space of all continuous complex-valued functions which vanish at infinity on the locally compact Hausdorif space X. 2.4. If X is met rizable, then (C0X),, has the Lindelof property hence C0X has Property 10. When X is a topological group, then of X, the Lindelof ProPerty for and normality of are all equivalent. Another interesting paper by Corson [11 contains an intersection theorem which is related to results of Berge [1] and Ghouila-Rouri [1] discussed in § 4 of DGK. For comparison, we repeat the result of Ghouila-Houri: Suppose C,, - . ., C,,, are closed convex sets in a topological linear space and each k of the sets have a common Point, where 1 k < m. If C is convex, then some k + 1 of the sets have a common Point. Corson's result is as follows: 2.5.

If W is a family of bounded convex sets covering an infinite-dimensional

reflexive Banach space E, there is a point xe E such that every neighborhood of x meets infinitely many nwmbers of of an infinite-dimensional normed Now consider instead an open covering let ö V denote the diameter of V. When linear space S, and for each V ,.oV < a simple dimension-theoretic argument shows that for each have a common point. The reasoning of finite m, some m members of Corson [1] shows that if each member of is bounded and convex, and is a subfamily of {cl V: Ve which has the finite * 0 whenever intersection property, then some finite-dimensional cube in S meets infinitely This applies in particular when S is a conjugate space many members of and F is a covering of S by open metric cells. Even when is such a special covering of Hubert space, it is unknown whether some point must

lie in infinitely many members of F. propertlea of metric cells. Completeness of a metric space 3. is characterized by the condition that if a family of nonempty closed subsets is directed by inclusion and has arbitrarily small members, then it has nonempty intersection. Without any restriction on the size of the sets, the condition is characteristic of compactness. However, there are spaces in which closed sets of a certain form always have this intersection property, irrespective of their size and compactness. The following result in that direction is due to Harrop-Weston [1]: 3.1. Suppose B is a bounded sequentially closed subset of a locally convex space E, and the closed symmetric convex hull of B is sequentially complete. Let {&}AE.f be a directed system of sets of the form BA = XA + rAB with EE

INFINITE-DIMENSIONAL INTERSECTION THEOREMS 0, where

and

A

< A' (in the ordering of A) implies BA

355

Then flAE4 BA

is a set x + rB with x = tim XA and r = urn rA. Theorem 3.1 applies to metric cells in a Banach space, where by (closed) metric cell we mean a set of the form V(z, r) = {x€M: p(z,x) r), M being a metric space with distance-function p, z M, and r 0. (Although such sets are often called "spheres", we feel that such terminology is misleading. Our spheres are sets of the form {x M: p(x, z) = r}.) An example of Sierpinski

[1] shows that the metric cells of a complete metric space may lack the intersection property expressed by 3.1. More recently,

[1] observed that

the real line can be given an invariant metric with respect to which it is complete but contains a decreasing sequence of metric cel!s with empty intersection. (Let p(x, y) = x — y I when I x — y I

when

Ix—yl>2.)

I

2 and p(x, y) =

1

+ (I x — y

I—

intersection properties of metric cells were discussed in 6—9 of the treatment there was mainly restricted to convex cells in Similar ideas have been discussed in a more general setting by several authors, Various DGK, but

some for metric spaces (Aronsza;n-Panitchpakdi [11, Gehér [1], Grünbaum [2)) and others for Banach spaces of arbitrary dimension (Nachbin [1, 2], Lindenstrauss [1]).

The expansion constant of a metric space M is the greatest lower bound EN of numbers p such that for each family { V(z,, r.) a 6 A) of pairwise intersecting cells in M, the intersection fl,6A V(z1 , is nonempty; if always V(z,, 0, the constant is said to be exact. The Jung constants

are defined similarly with respect to families of pairwise intersecting metric cells which are all of the same radius r, = r. (Of course

=

when

M is a normed linear space, but this equality fails in the general case. Compare with the definition of J(p, r) in § 6 of DGK.) Clearly EM for all r, and inequality is possible even for a two-dimensional Minkowski space (GrUnbaum [1]). On the other hand, GrUnbaum [2) describes a Banach space

M for which fit = E1 though the first constant is exact and the second is not.

A metric space M is said (by Griinbaum 12)) to have the finite intersection property (f.i.p.) provided * 0 whenever is a family of metric cells in M such that every finite subfamily of has nonempty intersection. This property is possessed by every space which is boundedly compact, and also by every Banach which is conjugate to another. Grünbaum [2] conjectures that f.i.p. is characteristic for conjugate spaces. Clearly the constants EN and are exact whenever M has f.i.p., but this is not necessary for exactness of the constants. He notes that = 2 and the constant is exact, while (c0) lacks f.i.p. even for families of cells of the same radius. The following result of GrUnbaum [2] extends a theorem of Bohnenblust [1] on projections (cf. 3.4 below): 3.2.

E1,

2 whenever M is complete.

A metric space M is called totally convex if for each pair x and z of points of M and each A €10, 1[ there exists y e M such that p(x, y) =

Ap(x, z)

356

VICTOR KLEE

and p(y, z) = (1 — A)p(x, z). The retraction constant rM of M is the greatest lower bound of numbers p which have the following property: whenever Y

is a metric space which Consists of M and one additional point, there is a transformation r of Y onto M such that rx = x for all x M, and p(ry, ry') pp(v, y') for all y, y' e Y. If there always exists such an r with p(ry, ry') rMp(y, y'), then rM is called exact.

Griinbaum [21 proves 3.3. For each metric space M, rE, while = rN when M is totally

If either constant is exact, so is the other. From this he deduces a result of Aronszajn-Panitchpakdi [1) on the extension of uniformly continuous transformations. When X is a normed linear space, GrUnbaum [2] defines the projection constant Px as the greatest lower bound of numbers p having the following property: for any normed linear space Y which contains X as a subspace of deficiency one, ,there exists a continuous linear projection P of Y onto X with p; exactness is defined in the obvious way. He proves II P11 3.4. For each normed linear space X, if either constant is exact, so is the other. convex.

He discusses the connection of this result with the extension of linear transformations and deduces Nachbin's result [1] to the effect that a normed linear space has the Hahn-Banach extension property if and only if the family of all its metric cells has the binary intersection property (cf. 7.2 of DGK). Another projection constant, is defined like above but with respect to all spaces V having X as a subspace (no restriction on deficiency). Obviously Px Px. The constant Px has been extensively studied, and in particular, GrUnbaum [31 has evaluated Px for a number of finite-dimensional spaces. Except for those involving the condition = 1 (ci. 3.6 and 3.7), the many interesting results on Px seem to have little connection with intersection properties, so will not be discussed here. The interested reader may consult the report of Nachbin [21, pp. 94—96 of Day [1], and other papers listed by them; for some more recent results and references, see Isbell-Semadeni 111. For a cardinal r 3, a metric space M is called r-hyperconvex if for each family rd,) : a A) of metric cells in M with card A < r and r. + the intersection is nonempty A V(ZG, + e) * 0 for each e > 0>. The space M is hyperconvex pro. These and some related notions are studied extensively by AronszajnPanitchpakdi [1], who observe that 3-hyperconvexity is equivalent to total convexity, and (in different language): 3.5. For a metric space M, the following three statements are equivalent: M is hyperconvex; M is complete and almost hyperconvex; M is totally convex; its expansion constant is equal to 1 and is exact.

A consequence of 3.5 (noted by Griinbaum [2)) is that the expansion constant of a bounded totally convex space must be exact if it is equal to 1.

INFINITE-DIMENSIONAL INTERSECTION THEOREMS

For all Aronszajn and Panitchpakdi supply a topological characterization of those compact Hausdorff spaces H for. which C(H) (the space of all continuous real functions on H) is r-hyperconvex. (Two of their problems in they describe this connection are solved by Henriksen [1).) For each r an H such that C(H) is rhyierconvex but not r'-hyperconvex for any r' > for r-hyperconvex spaces which are not (r + 1)r hyperconvex. From results of Hanner [11 (7.1 of DGK) it follows that a 3dimensional 11-space (regular octahedron as unit cell) is 4-hyperconvex but not 5-hyperconvex, and that each finite-dimensional Banach space which is • The latter statement is 5-hyperconvex must in fact be proved by Lindenstrauss [1] for an arbitrary Banach space.

In connection with 3.5, Aronszajn and Panitchpakdi ask whether, for there exist complete metric spaces which are almost r-hyPerThey produce such an example for r — 3, convex but not and show that in general, almost r-hyperconvexity implies — 1)-hyper3< r <

convexity. Lindenstrauss [1] proves that for Banach spaces, almost rhyperconvexity implies r-hyperconvexity.

Aronszajn and Panitchpakdi prove that each compact hyperconvex space is an absolute retract and ask whether every compact absolute retract can be hyperconvexly metrized. A result related to this question was obtained by Plunkett [1]. So far in this section, Banach spaces have been mentioned only incidentally.

Let us now summarize some of the most important results on intersecting cells in Banach spaces. Combining results of several authors (AronszajnPanitchpakdi [1J, Goodner [IJ, Griinbaum [2J, Kelley [1], Nachbin [1]), we note: 3.6. For a real Banach space X, the following eight statements are equivalent: E1 = 1; rx = 1 Pz 1; P1 = 1; X is almost hyperconvex; X is hyper-

convex; X has the Hahn-Banach extension property; X is equivalent to the space C(H) of all continuous real functions on a ext remally disconnected Hausdorff space H.

And the following (as well as other equivalences) is due to Grothendieck [1] and Lindenstrauss [11:

For a real Banach space X, the following statements are equivalent: (i.e., the second conjugate space satisfies the conditions of 3.6); X is X is weakly almost 5-hyperconvex (i.e., for each family { i 4) of four 1): 1 intersecting unit cells in X, and for each e > 0, the intersection V(z1, 1 + is nonemPty; whenever 3.7.

Px.. =

1

Y is a closed linear subspace of a Banach space, T is a compact linear operator

from Yb X, and that } I TI I

(1

+

> 0, I

T admits a compact extension T from Z to X such

I TI I; the preceding condition holds for dim (Z/ Y)

1.

Lindenstrauss also shows that if the unit cell of X has an extreme point, then the above conditions are equivalent to X's being isometric with a certain type of subspace of a space C(H). Nachbin [1; 2J asked whether an Banach space whose unit

VICTOR KLEE

358

cell has an extreme point must be equivalent to a space C(H). Lindenstrauss [11 supplies a counterexample, which also disproves a conjecture of Grothendieck [11. (His example is the space of all real sequences x = (x1, x2, ...) for with I Ix I = sup,, I x,, I.) On the other hand, which urn,,..,. x,, = (x1 + Lindenstrauss proves 3.8. A Banach space X is eguivalent to a space C(H) (H compact Hausdorff) the set of all extreme points of the unit if and only tf X is cell of X is nonempty, and the set of all extreme points of the unit cell of X5 is w5-closed. (None of these three conditions is implied by the other two.) Fullerton [1] defines a CL-space as a Banach space X with unit cell U such thai for each maximal convex subset F of the boundary of U, U cony (F u — F). Hanner [1J showed that every finite-dimensional 4-hyperconvex Banach space is a CL-space (but not conversely). Fullerton (11 characterizes those conjugate CL-spaces which are L-spaces or C(1f)-spaces, and Lindenstrauss [1] obtains the following characterization in terms of intersection properties:

3.9. A C

CL-space X is a C(H)-sftace if and only if are four metric cells in X such that each three have a

common point. BIBLIOGRAPHY

N. Aronszajn and P. Panitchpakdi 1. Extension of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 8 (1956), 405-439. C. Berge 1. Sur une propriété combinatoire des ensembles convexes, C. R. Acad. Sci. Paris 248 (1959), 2698-2699.

H. F. Bohnenblust 1. Convex regions and projections in Minkowski spaces, Ann. of Math. (2) 39

(1938),

301-308.

N. Bourbaki Espaces vectoriels topoiogiques, Hermann, Paris, 1955; Chapters 111-V. H. H. Corson 1.

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of convex sets which cover a Banach space, Fund. Math. 49

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2. The weak topology of a Banach space, Trans. Amer. Math. Soc. 101 (1961), 1-15. 14. Corson and V. Klee 1. Topological classification of convex sets, these Proceedings, pp. 37-51. L. Danzer, B. GrUnbaum and V. Klee 1. Helly's theorem and its relatives, these Proceedings, pp. 101-180. M. M. Day 1. Normed linear spaces, Springer, Berlin, 1958. J. Dieudonné 1.

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W. A. Eberlein 1. Weak compactness in Banach spaces, Proc. Nat. Acad. Sci. U.S.A. 33(1947), 51-53.

INFINITE-DIMENSIONAL INTERSECTION THEOREMS

359

W. Fenchel

1. A remark on

convex sets

and polarity, Medd. Lunds Univ. Math. Sem. Tome

Supplémentaire (1952), 82-89.

E. E. Floyd and V. L. Klee 1. A characterszatton of reflexivity by the lattwo of closed subapaces, Proc. Amer. Math. Soc. 5 (1954), 655-661.

R. E. Fullerton 1. Geometrical characterizations of certain function spaces, Proceedings of the International Symposium on Linear Spaces, Jerusalem, 1960, pp. 227-236, Academic Press, Jerusalem, 1961. D. Gale and V. Klee 1. Continuous convex sets, Math. Scand. 7 (1959), 379-391. L. Gehér 1. Ober Fortsetzungs- ned Approximations problem. fur stetige Abbildun gem von ,netrischen Raumen, Acta Sci. Math. Szeged 20 (1959), 48-66. A. Ghouila-Houri 1. Sur l'étud.e combinatoire des families de convexes, C. R. Acad. Sci. Paris 252 (1961), 494-496.

D. S. Goodner 1. Projections in normed linear spaces, Trans. Amer. Math. Soc. 69 (1950), 89-108. A. Grothendieck 1. fine caractérisation vectorislie-mitrique des espacea V. Canad. j. Math. 7 (1955), 552-561.

B. Grünbaum 1. On some covering and intersection properties in Minkowski spaces, Pacific J. Math. 9 (1969), 487-494. 2. Some applications of expansion constants, Pacific J. Math. 10 (1960), 193-201. 3. Projection constants, Trans. Amer. Math. Soc. 95 (1960), 451-465. P. C. Hammer 1. Maximal convex sets, Duke Math. J. 22 (1955), 103-106. 2. Semispaces and the topology of convexity, these Proceedings, pp. 305-316.

0. Hanner 1. Intersections of translates of convex bodies, Math. Scand. 4 (1956), 65-87. R. Harrop and J. D. Weston 1.

An intersection property in locally convex spaces, Proc. Amer. Math. Soc. 7 (1956), 535-538.

M. Henriksen 1.

Seine remarks on a paper of Aronszajn and Panitchpakdi, Pacific J. Math. 7 (1957), 1619-1621.

J. Isbell and Z. Semadeni 1. Projection constants and spaces of continuous functions, Trans. Amer. Math. Soc. (to appear). R. C. James 1. Reflexivity and the supremum of linear functionals, Math. Ann. 66 (1957), 159-169. 2. Characterizations of reflexive Banack spaces, Mimeographed abstract (2 pages), Conference on functional analysis, Warsaw, 1960.

M.Jüza 1. Remark on complete metric spaces, Mat.-Fyz. Casopis. Slovensk. Akad. Vied. 6 (1956), 143-148. (Czechoslovakian. Russian summary)

S. Karlin and L. S. Shapley 1. Some applications of a theorem on convex functions, Ann. of Math. (2) 52 (1950), 148-153.

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J. L. Kelley 1.

Banach 323-326.

spaces with the extension property, Trans. Amer. Math. Soc. 72 (1952),

V. Klee 1. Some characterizations of reflexivity, Rev. Ci. Lima. 52 (1950), 15-23. 875-883. 2. Convex sets in linear spaces. II, Duke Math. J. 18 3. The critical set of a convex body, Amer. J. Math. 75 (1953), 178-188. space, Trans. Amer. Math. homeomorphisms in 4. Convex bodies and Soc. 74 (1953), 5. Separation properties of convex cones, Proc. Amer. Math. Soc. 6 (1955), 313-318. 6. The structure of semi8 paces, Math. Scand. 4 (1956), 54-64.

7. A note on topological properties of normed linear spaces, Proc. Amer. Math. Soc. 7 (1956), 673-674.

8. A conjecture on weak compactness, Trans. Amer. Math. Soc. 104 (1962), 398-402. G. Kbthe Topologische lineare Räume. I, Springer, Berlin, 1960. J. Lindenstrauss On the extension property for compact operators, Bull. Amer. Math. Soc. 68 (1962), 484-487.

E. Michael 1. On a theorem of Rudin and Klee, Proc. Amer. Math. Soc. 12 (1961), 921 T. S. Motzkin 1. Linear inequalities, Mimeographed lecture notes, Univ. of California, Los Angeles, 1951.

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2. Some problems in extending and lifting continuous Linear transformations, Proceedings of the International Symposium on Linear Spaces, Jerusalem, 1960, pp. 340-350, Academic Press, Jerusalem, 1961. R. L. Plunkett 1. Concerning two types of convexity for metrics, Arch. Math. 10 (1959), 42-45. V. Pták 1. remarks on weak compactness, Czechoslovak Math. J. 5 (80) (1955), 532-545. R. Rado 1. A theorem on sequences of convex sets, Quart. J. Math. Oxford Ser. (2) 3 (1952), 183-186.

F. P. Ramsey 1. On a problem of formal logic, Proc. London Math. Soc. (2) 30 (1930), 264-286. M. E. Rudin and V. L. Klee 1. A note on certain function spaces, Arch. Math. 7 (1956), 469-470. L. Sandgren 1. On convex cones, Math. Scand. 2 (1954), 19-28. W. Sierpinski 1. Sur une definition des espaces complete, Ganita 1 (1950), 13-16. V. Smulian 1.

On the principle of inclusion in the spaces of the type (B), Mat. Sb. (N. S.) 5

(47) (1939), 317-328. (Russian. English summary) F. A. Valentine 1. The dual cone and Helly type theorems, these Proceedings, pp. 473-493. UNIVERSITY OF WASHINGTON

ENDO VECTORS' BY

T. S. MOTZKIN 1. IntroductIon. In a vector space V over the field R0 of real numbers, a convex set may be defined as a subset S of V containing 21s1 + A2s2 for all S, s2 S, and all

=1 Strengthening the requirement by deleting the restriction (1), or (2), or both, on the two-dimensional coefficient vector A = (2k, 22) E R0, we obtain, instead of the convex sets, the family of flats (linear subspaces) in case (1) is deleted, of 0-flats (flats through the origin 0 of V, or empty) if (1) and (2) are deleted, and of convex O-halfcones if (2) is deleted. Thus each of these four families 2

is defined, with respect to the "center" 0, by certain characteristic "endovectors" A, where A is an ordered pair of coefficients of a linear combination under which every member of the family is closed. The following study of sets of endovectors defining families of point sets falls into a general theory (Chapter 1) and into investigations of the translation invariant case (II), of sets of one-dimensional endovectors (ffi—V) and of other special endovector sets (VI). The results are partly of an arithmetic (in a wide sense of the word), partly of a geometric nature. Among those of the first type we note Theorems 20— 22 on the structure of translative endovector sets with two-dimensional integral generators, 76—79 on certain nontranslative endovector sets with a single two-dimensional integral generator and on proper endothety, 36-37 on centers of symmetry and autothety, 29-32 on semigroups of one-dimensional endovectors, and 27 on proper dimensions of endovector sets; of the second type, 50-51 on the structure of ambiconvex sets and 49 and 54-73 on overstar and inverse overstar centers. The present study has connections with several topics of current interest

(theory of structures, affine analysis of convex sets) but presupposes only standard material, as shown by the absence of references. Numerous points will be noticeable where the investigations have been, more or less arbitrarily, broken off and which are awaiting further research. CHAPTER I.

2.

GENERAL THEORY

FamIlies. A set S is said to include every

S

(element of S) but to

contain every c S (subset of S). This paper includes results presented to the American Mathematical Society on October28, 1961. 361

T. S. MOTZKIN

A family F, i.e., a set of sets, is called a classification of its union U F into classes if it consists of disjoint nonempty sets. Every classification of U F corresponds to an equivalence relation between the elements of U F. A family F is called intersectional if F includes the intersection of every subfamily of F. In particular, F includes the intersection of the empty family;

this intersection is the "largest" F (i.e., it contains every F) and hence is U F. Every Sc U F determines a "smallest" e F containing it, viz., the intersection of all F containing S; this smallest F is said to be spanned by S and called the hull H(S). The hull of a union of sets is their join. A family F is called unional if F includes the union of every cF. In particular, F includes the union of the empty family; this union (whether or not stipulated as empty) is the smallest e F and hence is the intersection fl F. Every S fl F determines a largest e F contained in it, viz., the union of all F contained in S; this largest F is the core of S. A family F of subsets of a set V is called complemental if F includes the complement V — S of each member SE F. A complemental family is unional if and only if it is intersectional. We shall use THEOREM 1. Every intersectional, unional and complemental family with nonempty union consists of the unions of the classes in a classification of its union.

These classes are the hulls of the singletons contained in the union. 3. Notation. We denote in the sequel by R a, not necessarily commutative, ring with unit 1, by R(1) the subring spanned by 1, and by x (the "characteristic" of R) the cardinal (number of elements) of R(1). Then Z = 2, 3, 4,-.., or

By V we denote a vector space (left module) over R: the operation v1 + v2 for points (elements) of V is an abelian group, and the operation p V, p E R, V E V. is distributive in p and in v and associative in p, with 1.v = v. The point 0 is called origin; we assume that V includes at least one point besides 0. A linear transformation T of V into itself or another vector space over R is a mapping v —* T(v) such that T(p1v1 + p2v2) =

T(v1) + Pt T(v2). A homo-

thety is a mapping v pv; it need not be a linear transformation. A translation is a mapping v —' v + v0. If S is a subset of V then the image of S under these mappings is denoted, respectively, by T(S), pS, S + v0. If the image is S itself the mapping is, respectively, a linear automorphism, an autothety, an autotranslation (period).

The particular vector space called k-space over R and denoted by R" (k =

Consists of all k-tuples A = (A,,, . . ., with coordinates R, with the usual definitions of addition and scalar premultiplication. By Uk we denote the set of the k unit vectors (the rows of the k by k 1,

2,

•..,

unit matrix) by ik their sum (1, ..., We also consider unions of disjoint vector spaces over R. The union R1 U R2 is denoted by R, and the set U,, U U2 Li-- c R by U. 4. Endovectors. DEFINITION 1. For a given subset S of V, the vector A =

ENDOVECTORS •, •,

(As,

s1,

363

is called a k-O-ENDovEcToa or S if S includes

ASj for all

S is said to be ENno-A or to have A-CENTER 0.

Similarly if S is a subset of a union U V1 of disjoint vector spaces we call A an endovector if S includes A1s1 for all s1, •, e S for which this sum has a meaning. For any ye V, O-endovectors of S are called v-endovectors of the translated set S + v. The vector A = (A1, •-, Ak) is a v-endovector of S if and only if v + A'(s, — v) E S for all s1, •, 5. If v becomes infinity (topologically or projectively) the condition becomes w 1s S, A = 1; for k = we call w an endotranslation (hemiperiod). 1

A

THEORFM 2.

The set of 2-centers (A =

ifSisendo.A*, A*=(Al,...,Ak,

(As,

..., A'))

of S contains S if and only

For either condition amounts to saying that S includes s +

—

s)

for

seS.

all

For any set 4cR,2 a set Sc V (or c U V1) such that every AcA is an 0endovector of S is said to be endo-A or to have A-center 0. The set of A-centers of S contains S if and only if S is endo.4*, where A* (defined as in Theorem 2) 4* if A e A. The empty set 0 is endo-A. The set of A-centers of 0 is V. Immediately verifiable are: 3. if S is endo-4 so are pS, if pA, = A,p for every coordinate 2 of every A €4, and T(S) for every linear transformation T. (For S + v see § 12.) THEOREM 4. If S1 and S2 are endo•A so is their sum + S2, and, in particular, their direct Product S , 0) + (0, S2). = The sum S1 +S2 of the sets S1 and S2isthesetof ails1 +s21s1eS1,s2€S2. 5. A-hulls. Obviously the family a(4) of all endo-A sets in V (or in U V1) is intersectional. The hull of S c V (or c U V1) in this family is called the

We have SEa(4) if and only if H4(5) = S; if and only if A•S c S, where 4-S is the set of all A,s5, s, ES, A €4; and if and only if S' = S, where S' = S'(A) = S U 4-S. A-hull H4(S).

THEOREM 5.

(1) (2) (3)

H4(S)=SUS'tJS"U.-.; S'(HA(U1)) U S'(H4(U2)) U = S'(J-fA(U)); HA(S) = Us1 H4(S1), where S1 runs over all finite subsets of S. HA(S)

Indeed, H4(S) consists of all 2L1L2- --•

•L1 s with A

4 U U, L1, -

, L matri-

ces whose rows are A U U with interspersed zeros, s a column with elements in S. Subdivision according to j or to the number of elements of s yields (1) or (2), respectively. From (2) follows (3) in view of H4(S1) = SI'(H4(Uk)) for any S1 with at most k points. Instead, the theory can be developed for ,1 c R'. the set of sequences over R with tinitely many nonzeros.

T. S. MOTZKIN

can , If repetition of elements in s is permitted then the matrices L,, if A In addition, be required to have only one nonzero in each column.

includes a vector with sum unity and not a unit vector, then the rows of L,, •, L can be required not to be unit vectors. On the other hand, to obtain each e H4(S) but not with minimal j it is sufficient to use matrices all of whose rows except one are unit vectors. 6. Complete endovector gets. DEFINITION 2. A set A c R is called COMPLETE if there exists a set S such that A = A(S), where A(S) denotes the set of all endovectors of S. THEOREM 6.

The statements:

(1) A is complete; (2) A=J-b(U);

(3) A=A(A), are equivalent.

Indeed, (1) (2) expresses that a set is complete only if it includes all L, defined as before, classified according to the number of columns

of L. Further, if (2) holds, then every AL is an €A (i.e., Ac A(A)) and,

since A contains U, an endovector of A must lie in A, whence (3). Finally is trivial. We note specifically these properties of complete sets: (1) Every complete endovector set contains the set of all unit vectors. (2) Every complete endovector set is permutalionally symmetric (for every matrix of A permutation ir,, •, of 1, . . k we have A = (A,, . .., e A. permuted unit vectors = (Art, ..., O)e A. A is equivalent to (A,, .. , A —o (A, + A,, (4) The elements can be compressed, i.e., (A,, . .., Ak) e A. (To every classification of (1, . . ., k) belongs a compression of A.)

(3) Zeros are irrelevant, i.e., (A,, .. . ,

.

If A is complete and includes 0 (in some Rk), then (A,, . .., A,,÷,) e A

(A,, ...,Ak)EA ("suppression").

If A is complete then H4{1} is the set of sums Z

A e A, while HA{0, 1)

is the set of coordinates A, , A E A. If Ak(S) means A(S)flRk then the sets Ak(S) can be characterized by = Ak. The set Ak determines 4k-1, but not conversely. The k by k matrices L with rows in Ilk form a multiplicative semigroup, there correspond different semigroups. In particular, to different sets A, is a multiplicative subsemigroup of R. Also the w by w matrices whose rows are infinite sequences consisting of an e A and zeros form a multiplicative semigroup M(A). Denoting by M_ the set of all w by w matrices whose rows consist of one 1 and zeros and by M.,. the set of all w by w matrices with rows R" we have: THEOREM 7.

A set M of w by w matrices is an M(A) for some complete

endovector set A if and only if M is a multiplicative semigroup and M. c Mc M+. 7.

Intersection of complete endovector sets.

THEOREM 8.

The intersection

ENDOVECTORS

of complete endovector sets is complete.

contains U and AL, where For if the sets 4( are complete then A = and therefore in A. A and every row of L belong to A, is in every The complete endovector sets form thus an intersectional family, and every set A c ,k has a complete hull H(A). THEOREM 9.

We have H(A) =

114(U); hence 114(S) =

S'(H(A)).

For HA('U) consists of all ALLL L, and is therefore complete; hence H(A). On the other hand, by (2) in Theorem 4, H(A) = Hfl(4}UD !14(U).

H4(U)

If A = H(Ak) for some k, the smallest of these k is called the basic diménsion dim A of A; if no such k exists we write dim A = For an example with k = 3 see § 14. 8.

Complete families.

DEFINITION 3.

A family a of sets S c R is called

COMPLETE if there exists a set A c R such that a = THEOREM 10. There is a 1-1 correspondence between complete endovector sets and complete families, given by a = a(A) or equivalently by A = fla A(S).

For firstly, if a = c(A) for some A c R then A c A(S) for every Sea, hence also H(4) c A(S), hence a = a(H(A)). Secondly, A c fl,A(S) and if A is complete then A(A) = A, hence Ac a(A), hence one A(S) (viz., A(A)) A, hence A = flaA(S).

The correspondence has the duality properties of being antimonotone if a increases, A decreases, and of associating intersections with joins. It goes without saying that also the class of complete families is intersectional, and thus every family a has a complete hull H(a). Clearly 11(a) = u(A), where A = flgA(S). 9. The four main families. To U and the smallest and largest complete endovector set, correspond c( U) and a(R), the largest and smallest complete family; a(U) is the family of all sets s c R, while a(R) is the family of 0-fiats. Note that U = H(Ø), a(U) = a(Ø), while R = H(Rt), c(R) = c(R2). In this statement, R2 can be replaced by the set of all (A1, 1) or by R' U {(l, 1)) (or, if R is a division ring, by R' U {A}, with any A with more than one nonzero element). The 0-flats in R1 are the (left) ideals. The third important complete endovector set, A(UI),' consists of all vectors with sum unity; a(A(Uj)) is the family of flats. The straight line through two points v1, (v1 * v2) means their flat hull v2}. If R is a division ring other than {0, 1) then A(U1) = H(A2(U1)), a(A(U1)) = a(A2(Uj)); for R = {0, 1), replace A2(U1) by {(1, 1, 1)). Finally, U U (0) is complete (where (0) is the set {0 in R', 0 in R', . . and a(U U (0)) = u(0) is the family of 0-sets (sets empty or containing 0). As (1,1) is in R but not in U, UiJ (0) or A(U1), these three differ from R and hence (cf. proof of Theorem 12) from each other; the latter follows also by noting that 0 is in UU (0) but not in A(U1), and that (1, 1, —1) is in A(U1) but not in U. .

T. S. MOTZKIN

366

THEOREM 11.

For R =

(0,

1) the sets, fiats, 0-sets and 0-flats are the only

complete families.

For by permutational symmetry and irrelevance of zeros a complete ii in R, for R (0, 1), is given by stating which vectors

endovector set

(1, •••, 1) Rk, = 0 e Rt, belong to it. But ltk+1 4, k 1, implies lzk+I + (1,0, . . ., 0) + (2k — 1)(0, 1, ., 0)€ 4, hence 12*_I €4, and similarly Likewise '2k A, k 1, implies l2k + (2k — 1)(1, 0, - -,0)e A, hence l2k+s A. e A, and similarly A; also 121+1 + (2k — 1)(1, 0, . •, 0) A, hence h, A. '2k—i or , k 0, or of L and Thus A consists either of all or of all Another proof goes via Theorem 79. only of Q.e.d. lk

•

10. Complete classes. DEFINITIoN 4. A class C of complete endcwector sets, and the corresponding class of complete families, are called COMPLETE if C includes the intersection and the join of every subclass. The totality of complete classes is intersectional, and every class of complete

endovector sets or families has a complete hull. THEOREM 12.

The families of sets, flats, 0-sets and 0-flats form a complete class.

For obviously it(U1) fl (U U (0)) = U; on the other hand H(4(U1) U (U U (0)) = 11(4(U1) U (0)) = since every 2 can be lengthened to be an E 4(U1). Hence the join and intersection of the families of flats and 0-sets are the and 0-flats. 11. Gapless classes; maximal and minimal families, DEFINITIoN 5. A class C of complete endoveclor sets, and the corresponding class of complete families, are called GAPLESS if 4' C, A" e C, A' c 4 c A", A complete imply A C. The totality of gapless classes is intersectional, and every class of complete

endovector sets or families has a gapless hull. DEFINITIoN 6. A complete endovector set 4 * U is called MINIMAL if (A, U) is gapless; a complete endovector set A * R is called MAXIMAL if (.4, R) is gapless. The corresponding complete families a(A) are called MAXIMAL and MINIMAL,

respectively. THEOREM 13. The family of 0-sets is If and only if R is a division ring the family of flats is minimal. Indeed, UU (0) is minimal. Further, if the complete endovector set A contains .4(U1) and includes some A not in 4(U1) then, if R is a division ring (other than (0, 1), in which case see Theorem 11), A contains the straight lines through A and points of .4(U1), hence (repeating this argument once) all points of R. On the other hand, if R is not a division ring then R contains an ideal R' different from R, (0) and 0, and the set A of all 2 with E 1 while 4(U1) cA cR, A * 4(U1), A R. (modR') is complete (cf.

CHAPTER II.

12.

Translatlve families.

TRANSLATIVITY

A vector v

in

V such that S€u(A) implies

ENDO VECTORS

S + V 6 a(A) is a period of u(4). DEFINFrION 7. A family u(A) in V is called V'-TEANSLATIVE if V' gs the set of periods of o(A).

This is equivalent to saying that, for 2 A, S c V, if 2 is a v-endovector of S for some v, then A is a (v + v')-endovector of S for all v' V'. THEOREM 14,

The statements:

(1) o(4) is V'-translative; (2) (3) {v'}€c(A) for every

VI,

are equivalent.

eS for 2€ 4, s, ES implies v' + A(s, — v') S For if (2) holds then S, v' V', i.e., (1). Every family e(A) includes the set {O), hence (1) implies (3). If (3) then A V-translative family g(4) is called translative; this is equivalent to saying for A e A, s2

that, for every S c V. the set of A-centers is either 0 or V, or that Se e(A) implies that also every affine transform T(S) + v o(4), where v V and T is a linear transformation. 13. Translatlve endovector sets. In particular, a(A) is translative if Ac 4(U1). A family a(A) in I? is translative if and only if A c 4(U1), i.e., if c(A) includes the flats. Subsets of A( U1) will therefore be called translative endovector sets. The complete translative endovector sets form the gapless hull of { U, 4(U1)). For complete translative A, the sets U, U U (0), 4 and A' = H(A U (0)) form a complete class. The family a(A') consists of the 0-sets in o(4); conversely, given a complete endovector set A' (0), the family a(4), where A = A' fl A( LI1),

is the complete hull of the translates of the sets in a(4). The translates themselves form a complete family if and only if A' = H((4'

fl

U (0));

this is not always the case. In fact, the H(4 U (0)), 4 complete translative, are only for R = {O, 1) the gapless hull of {U. U (0), R}, as shown, e.g., by A' = H(R1), the set of all A with at most one nonzero (cf. § 21): here A' fl 4(U1) = U, H(UU (0)) = UU (0) * A'(R * {0, 1)). 14. Gilds. For every subring R' c R with 1€ R', the set of = Uk vectors A with all e R' and the set 4(U1) fl are complete; the corresponding

complete familes are the 0-R'-flats and the R'-flats. Exactly as in the proof of Theorem 11 we see that The 0-R'-flats are fl 4(U1)) U (0)) = thus the 0-sets in the translative family of R'-flats. For the smallest subring, viz. R(1), we obtain the 0-grids (additive groups or 0) and grids (0 or cosets of additive groups). THEOREM 15. The complete endovector sets corresponding to the family of 0grids and to that of grids are H{(1, 1)) and H{(1, 1, —1)), respectively. = H{(1, 1)). Further if For obviously R(1) c H{(1, 1)) whence easily

let

A'=

T. S. MOTZKIN

368

= (1, 0, •, 0); then A' e A, u1 A, A ÷ is1 — A' A, hence 0, 1 — Ak, Ak), A =(2 + is, — A') + A' — is, H{(1,1, —1)) if (1 —Ak,Ak) and A + is1 — A' are e H{(1, 1, —1)). But A + is1 — A' belongs to a smaller k, and we end up with (0,

•

Now (2,—1)e H{(1,1,—1)}; hence also (3, —2)=2(2,----1)—1(1,0)€

1, —1)),

H{(1, the

etc.

to

Another proof translates an endo-(1, 1, —1) set

0;

then

set becomes endo-(1, 1).

Note that H{(2, —1)) = H{(1, 1, —1)} only for odd X (see § 17). The autotranslations of an arbitrary set S c V form an 0-grid ST. S is a grid then is a translate of S.

If S is

an 0-grid then

The

periods

of an arbitrary complete family o(4) form an 0-grid.

15. Convex sets. For every subserniring (subset closed under addition and multiplication) R' c R with 0 c R', 1 R', the set A of vectors 2 with all A eR' A, = 1 is complete, and A = H(A fl Rt); c(A) is the (translative) family and of R'.convex sets, and HA(S) is the R'-convex hull of S. The family of R'convex 0-sets is o(A'), where A' = H(A U (0)) consists of all A with A1 R', 2, R' (these are the A that can be lengthened to be an e A; the com1 —

pleteness If R'

is easily verified).

is a ring the R'-convex sets and 0-sets are the R'-fiats and

For the smallest subsemiring, R'(1), we obtain for finite R(1) again (since = R(1) in this case) the 0-grids and grids, for infinite R(1) (when A = U)

R'(l)

the 0-sets and sets. 16. If S is R'-convex so is Indeed, if s€S s + v,eS and plies s + X21v, = +

-

if

A,eR' and

=

then seS

1

im

co,nplete hull H({,a}), p = 1, has the coordinates 1), and of all vectors obtained from the by Permutation, compression and insertion of zeros. Allowing suppression we obtain H({p} U {0)). 16. SIngle generator. THEoREM 17.

consists

of all p"', 1

0, 1,2,

-

The

-, where

This follows from Theorem 9; (1), § 5; and the next to last remark in § 5. Denoting by H1 the set consisting of p" and of the vectors derived from

we have suppression

THEOREM 18.

elements,

H,+,,

H({p))

H0 U H1 U - -. -

Similarly for

H,'.

where

is allowed.

+p = 1,

The set R2

but no more; for R1 fl

Indeed, p" has

H

coordinates

may have

the maximal number is

() coordinates

-

-. (note

that

ENDO VECTORS

= pp,) and each A R2 n H. is given by its first coordinate A,, which may be any partial sum of the coordinates of p", viz. A, = + •.+ where

e0=0,1;e, =0,

Similarly AER°flHj' is given by

(1) o,e, + €, , A, = (2) = , Thus R fl H, has 2 elements (1,0), (0, 1) for 1 = 0, at most 2 others , p,) for 1 = 1, at most 8, 52,636, 16724, 1036272, --• others for I = (P, , ps), 2, 3,4,5,6, •••. The corresponding numbers for R2 fl H.' are 3, 6,45,846, 55800, for 1 = 0, 1, 2, 3, 4, 5,The maximal numbers for Rb fl H, and 1? fl H!, p = (p,. - - -, Pb) with comk)+ k) where k)+k — 1) and muting Pb, are similarly seen to be 17232264,

the products are extended over all k-nomial coefficients (I; k) = 1

=

For k

to n" and (,i + 17.

3

and noncommuting Pt the exact bounds are increased

1)14.

SIngle Integral generator. THEOREM 19. For p = (—1,2), H({p)) consists For E R(2) for every j * R(1) for one = 1,

of all A with E

H({.p}U(0)), omit ZAi=1. Firstly the condition is necessary since, mod 2, H = H({p)) is U. Secondly, (2, ,1

—A)cH,(A,—1,2—A,)eH—.2(A,,1—A,)—(A, —1,2—A,) =(A, +1,—A,)eH;

R(2) ( = 2, -• •, n; hence (2,, 1 — A,) e H for all A, R(1). Finally if 2, R(1), n 3), and if every such 2 for n — 1 belongs to H, then (A,, •-•, 2,,) = 2(0,- -,0,1 —A,,/2, 2,,/2)—(—A,,-- -, —A,,_2,2—2,,—A,,_,,0)EH. The second con-

tention follows by suppression and completeness check. If is (finite and) odd then R(2) = R(1). H,, THEOREM 20. For p = (ji,, 1 — p,), R(1) infinite, p,e R'(l), p, * 0, 1, (A, , Furthermore, or HI, with maximallA,l +12,1, we have IA,! +1211 =12p, — 1!'. = 1,2, 0, 1,p, or 1 —ps (A,, 22)6 H({p)) if and only if A, ER(1), 23€R(1), A, + + 2, = 1 and writting A,, A,, (mod p,(1 — p,)); for H({p} U (0)), omitting A, + A, instead of A, gives a necessary condition.

The first statement follows from (1), § 16 by selecting = 1, = 0, e, = = 0, --•; (2) yields nothing larger. Secondly, the set of (at most) four (h), residues 0, 1, p,, 1 — p, (mod — p,) is easily seen, by 16 trials, to be endo-p. Further if 4(0, 1) includes p,, 1 — p, , p,(1 — p,) and p,(1 — p,) + 1 then (as seen by substituting the latter two or 1,0 for 0, 1 and repeating) it includes Ap,(1 — p,) + A', 2' = 0, 1,p,, 1— p,, for every AER(1). Clearly p,, 1— p, and 4(0, 1). We shall now show that p, — + 1 is of p,-0 + (1 — p,)-p, are the form (1), § 16, with 1 = p,. Write 1 as E — p,)t and — as Hence — p,)1; the coefficients are then too large for (1). = subtract from terms k and k + 1 equal and opposite amounts, viz.

T. S. MOTZKIN

370 —

1),4"'(l —

and YkPi

Pt.

k =

1,3,

and

the set

5,

=0 of

•,

—

where'

= 2,

This

•

, p1

works for all

four; for Pt

1)])

—

]

2 for odd p1 and k = 1, 3, 5,

for all other k.

nine (for

—

([(lit — 2)/a

=1+ for

(1

— 3,

p1 — 2 for even Finally

1) e R'(l).

= 3, twelve)

residue pairs complying

the last condition of the theorem is easily seen (by checking the residues A,, + A, separately) to be endo-p and endo-0; but Theorem 19 shows of that the condition is not sufficient. 18. Other Integral cases. THEoREM 21. For R = R(1), a set A c A,(1) is a 4, if and only if the set A' of first coordinates of elements of A is the union to

multiplicative semigroup F of idempotent residue classes e R(l)IR(') of a containing 1 — r with r. onto R(1), finite, preserves hySince a homomorphism of R(1), x = and conclusion, we may assume x = Because of the translativity of A and by Theorem 20 it follows that the set of A such that A A', A + I €4' R'(I), and A' is the union of a set F of consists of all multiples of some idempotent residue classes R(1)/R(v). For A = A,(4,) it is necessary and sufficient that 0€ F, 1€? and that F. F implies + (1 — i',, pothesis

Setting Ti=O,Ti=l we have 1—r,EF; setting we have rireef'. On the other hand if F is a nonempty multiplicative semigroup of idempotent residue classes containing I — r with r then i(1 — r) = 0€ F, 1 — (1 — (1 —

—

— r.(l

+ Ts

Ti))

= T.Ti +(1

0

= 1 e F, and

T*Ts)(l + TiTs — —

22. For x = 1' for the number of such sets A and given v is 1 for ii = 0 (4 = U2) and i' = 1 (4' = R(1)); for v (decomposition into primes), it is the number c, of classifications of a set of k = k(M) elements. Every A belongs to only one M 0, except that if A belongs to an odd v it belongs also to For finite X, the total number of sets A is

where c0 = 1.

Indeed, r(l — r) = 0 shows that if those prime factors

of r for which j €1., c (1,

of

divide

every

the other pj divide every element of 1 — r and determines r uniquely. Since = f1.1 U I.,,, F corresponds to a complemental and unional family of subsets of {1, .., k) and therefore by Theorem 1 to a classification of (1, •, k), whire every classification of {1, •, k} leads to a family of subsets and to a semigroup F with the required properties. The cardinal of F is even and divides 2*. If 4 belongs to more than one R'(l) then t'0, the smallest v of A, is the smallest A(*0) R'(l) such that 2 £ A, A + 1 A; and any other v of A is a multiple pu,, p 2. Since — a',,) 0 (mod pv0), we have 1 — t'o 0 (mod p). But also 0 (modpv,), ml 0 (modp); hence p element

*

means largest integer.

•, k)

371

ENDOVECTORS

(mod 2). Every A belonging to an odd v. belongs also to 2v0, since 2(1 — Q.e.d. (mod implies 2(1 — A) 0 (mod

2)

0

We have Ck = E C*.J, where 4.j is the number of subdivisions of {1, - -•, k} = into j classes; Ck,J are the Stirling numbers given by c*., = 4.k = 1, •-• begins 1,2,5, 15,52,203, 877,. sequence c1 The Ck-lJ_1 + JCI-L.J.

begins 1,2,2,2,3,2,3,3,3,2,6,2,3,5,4,2, •••. The sequence 19. Real eases. For R = R,, the field of real numbers, we restrict ourselves of nonnegative numbers, to closed A,. Letting R' in § 15 be the semiring is a flat. we obtain the convex sets and convex 0-sets. For convex S, TNEOREM23. ForR=R, completeAc 4(1) and closed A,,A <2, * 0, * 1, o(4) is either the family of flats or of convex sets. —1 These two families form a gapless class. For if 0 < < 1 then I?' 11 H({2}) is dense on the segment C, connecting (1, 0) and (0, 1); for A, divides the interval from 0 to 1 into intervals of lengths not exceeding p max (As, 1 — A,) which are in turn subdivided into intervals is between + (1 — whose length is at most p', etc. If 1 <'2k <2 then <0. And if the complete endovector set 0 and 1; similarly for —1 A c 4(1) contains U (the convex hull of U) and some 2 not in U then A = for A has at least one negative coordinate 2,, and compressing the others we obtain (Aj, 1 — A,) A, hence also ((1 —

1 — (1 —

A, hence A, = I?' fl 4(1),

4 = A(1). Probably, for R = R0, there exist no other closed complete sets A c 4(1) than A(1), U, and those contained in R?1S. Among the latter, those of dimension 2 were discussed in § 18. 20. Complex cases.

THEoREM 24.

If R is the comPlex field, then a translative

family c(A) that includes a bounded set with more than one point includes the convex sets, and for closed H(A) no others. For if A e A and not A e U then A has at least one coordinate Aj that is not and compressing the others we obtain (A,, 1 — A,) H(A), hence also e H(A). Every set in o(A) with more (1 — than one point will be unbounded if the 27 are unbounded, i.e., if 121> 1, and similarly if 1 — I> 1, which must hold for some k = 1,2, if without For AcUsee Theorem 23 and its proof. (2', 1 — A) and ((1 — A.)*, 1 — I

CiurrEa III. UNIONAL FAJIIUES 21. Cones. The sets H(R') and H(R' — {O)) consist of all A with at most one or, respectively, exactly one nonzero; dr(R1) and o(R' — {O)) are the families of strong 0-cones and of 0-cones. They differ from each other and from the four families in § 9 if R is not {0, 1}; in the latter case the strong 0-cones and 0-cones coincide with the 0-sets and sets. According to the parenthetic statement in § 9 the family of strong 0-cones is minimal if R is a division ring. 25. For R = (0, 1, —1) the sets, 0-sets, 0-cones, strong 0-cones, flats

T. S. MOTZKIN

372

and 0-fiats are the only complete families.

Here a complete endovector set A is given by enumerating the vectors lk,1 (k ones, 1 minus ones; 10.. — 0) in A. By adding k — 1 and subtracting l

1, for k—lEO (mod3):

units at a 1, a —1 or outside, we get from and Ikd÷l; for k —

1*—i.,, 1

l

—1 we get lh—ia+1,

and subtracting k—i units to

we subtract (k

1), 'k—i.l+* (k

1) e A implies

k

we get for k—la —1:

for k — 1 0: ik—1,l+t, ik.l—a, For k — 1 1, 1 1, from k positive and L — 1 negative units and repeat, obtaining

all Ii,, with h — I

2), ik+1,1—l,

ik—i.l-I ,

Since ik (k 4,

C A, it follows that + 2 (at a 1) + (k — 2)(outside) = 1, except imply each other and Since 1.,, (1 3,

0) implies 1 and —11.0 —2 (at a — 1)— it follows that every (1—3) (outside) = with k — I 0 or —1, except 1, and 10,1, implies all 'k, The contention follows now easily. Another proof 1

goes via Theorem 79. 22.

Endovector sets of basic dimensIon 1. THEoRBII 26. For a complete

endovector set A, these four properties are equivalent:

(I) dimA=i;

(2) a(A) in R includes the strong 0-cones; (3) a(A) in R is unional; (4) the proper dimension of A A is at most 1. The Proper di,nension of a vector (As, •--, is the number of its nonzero coordinates. The equivalence of (1), (2) and (4) is immediate. If (4) holds then HA(S) = Uses HA({sJ) (cf. (3), § 4), hence (3). If (3) holds then R' e c(4) implies Ha(Uk)Eo(A), hence Ak C HR(Uk), i.e., (4).

The proof shows that, for a set A of basic dimension 1, o(A) in any V is unional; and, of course, o(A) in any V includes the strong 0-cones. The sets of basic dimension 1 form the gapless hull of { U, H(R')J. Among them, only U is translative. Proper dimensions of endoveetors. Condition (4'): the proper dimension of A e A is bounded, is not always equivalent to (4), as shown by A = H(2, 2), 23.

It is if R is a ring without zero divisors: Tuzoanu 21. If R is a ring without zero divisors then the set of proper

X = 4.

di,nensions of vectors A

A, where A is complete, is one of

the last onlyforx=2,A={11,i.,---). For if (A,, A, 2,2. * 0, then (At, 2122, 22) C A; similarly •we see that A includes vectors of arbitrary positive proper dimension. On the other hand (2k, '',Ak)€A, ff2, * 0, k 3, implies (A,,..., + 21)EA which is of proper dimension k — 1 except if Ak_i + Ak = 0. If the latter holds for any two coordinates of A then X = 2, 2 = and 2,1.-, A. Hence either , e A

ENDOVECTORS

e 4 which implies (At, (see before) or (As, that A includes a vector of length 2 except if for every implies 24. Endothetle aeta.

it follows + , + = 1 and not 0 €4. Finally

The sets 4 of basic dImension 1 correspond biuniquely

to the multiplicative subsemigroups G' c R with 1 eG', via G'= A fl R, 4= H(G'); u(G') = o(4) is the family of 0-G'-endothetw sets (en4othetic = into (self )homothetic).

For the smallest subsemigroups G' except (1), viz. G'(A) (the subsemigroup 1.2, A', -.. spanned by A), we obtain the 0-2-endothetic sets. (To be exact,

the minimal G' 0-sets.

1 may be only some of the G'(A)) For A = 0 these are the

If A has an inverse A", then S c V is 0-A-endothetic if and only if the compl(ment V — S is 0-A1-endothe*. sets. THEOREM 25. The comPlete endovector set 4 gives rise

to a untonal and complemental family a(A) in R if and only if A = H(G) for multiplicative group G c R with leG. For only if the subsemigroup G' c R, 1 e G', is a multiplicative group does A'€H0.(A) imply

If G c R is a multiplicative group with 1 e G then e(G) in any V, the family of 0-G-autothetic sets, V is unional and complemental and consists therefore

by Theorem 1 of the unions of classes in a classification of V; one of the classes is {O}.

In pirticular, for G = G(2), 2 invertible (the group (--•,t1, 1,2, •} spanned by A), we obtain the 0-A-autothetk sets. For S€ o(G(A)) = c(2, 4') we have' AS = S 26. Central ymetry. The sets H({—1}) = UU — U and H((O, —1}) = U U — U U (0) are complete; c({ —1)) is the family of 0- — lautothetic or 0-

symmetric sets (sets symmetric with respect to 0), e({O, —1)) that of 0-{O, —1)-

endothetic sets: these are the 0-symmetric 0-sets.

If r =2 then all sets are 0-symmetric. If R = {O, 1, —1) then the strong 0-cones coincide with the 0-symmetric 0-sets, and the 0-cones with the 0symmetric sets. In all other cases the sets, 0-sets, flats, 0-flats, 0-cones, strong 0-cones, 0-symmetric sets and 0-symmetric 0-sets in Rare eight distinct families; indeed, the corresponding complete endovector sets intersect differ-

ently the set of four points {O,—1,p,(p,1—p)}, where peR,p*O,1,—1. 27. Nonnegative cases. THEOREM 29. For R = R0 all closed multiplicative subsemigroups C' c (the set of nonnegative numbers) with 1 e C' can be distributed into these 13 mutually disjoint classes:

la. G'=l.

2a. G' nowhere dense, mm (G' — {1)) = p> 1. 3a. mm (G' — {1)) = p > 1, + A c C' (A being the smallest number for which

this holds). 4a.

G' ={A',n = -.-,—1,O,1, •-.;O},A >1.

T. S. MOTZKIN

6a0.

G'

any G' in 2a,3a,4a, the set of the recipn*cals of its elements and of zero.

+1) has 1 as limit point then G' is dense in 14+1; since G' is closed, G' =14+1. If C' fl (14+1) contains an interval then 2 c C' for some 2. If G' * (1), C' n (14 + 1) = (1) then the set of reIndeed, if G' fl

ciprocals of elements of G' — {O} is a closed aemigroup. Finally, if 2€ G',

peG' and log 2/log p is negative and irrational then C' is dense in 14 and thus C' =14. 28. Real cases Tasouu 30. For R = R. all closed multiplicative subsemigroups G' c R with 1 G' can be distributed into these 36 mutually disjoint classes:

The 13 classes of Theorem 29.

lb, ..•,4b'O. For any C' in la, •••,4a'O, the set of its elements and of their negatives. 2c. 3c.

G' nowhere dense, min(G' fl14—{1})=p> 1,max(G'fl <—1. —1,14+2cC', —14--2c

G' (2 being the smallest number for which this holds). 4c. 5c0. 2c0, 3c0, 4c0,

4c'O.

Indeed, if —leG' then G' = (C' n 14) U —(G' n 14). Verification of the ccases rests on a simple examination of the possibilities for G' with given TasoasM 31. For R = R0 all connected multiplicative subsemsg,vups G' c R with 1 G' con be distributed into these 11 mutually disjoint classes 10, 40, 6b0;

401,

4b'O, 4C10, 6a0;

6a (the G' in the

four classes without their lefi

endpoint).

Proof immediate. 29. The corresponding families u(G') are, for 16 selected semigroups, or classes of semigroups (some of which were formerly mentioned only after

union with {0}): la: all sets; lb: the 0-symmetric sets; 4a: the inverse 0-stars; 40_I:

the 0-stars; 4a'O: the strong 0-stars; 4b': the 0-symmetric 0-stars; 4b10: the 0-symmetric strong 0-stars; 4c: the inverse 0-2-overstars;

ENDOVECTORS

4c1: the 0-2overstars;

4c'O: the strong

0the the the

the strong 0-cones. The complements of sets 4c, 4c', 5a, Sc, 6a, 6b la, lb. 4a, are (disregarding the status of 0, and in two cases with instead of 2) sets la, lb. 4a', 4a, 4c', 4c, 54, 5c, 6a, 6b. 30. Complex cues. By examining the possibilities for G' with given I G' I (set of absolute values) we find: TlaEoazi 32. If R is the complex field then all closed multiplicative subG' c R with leG' can be distributed into these 72 mutually disjoint classes:

The 36 classes of Theorem 30.

ld, . - -, multiplied by •

- ., 4e'O.

For any C' in la,

•, 4a'O,

the set of its elements, each

--.,i? = 1, where k 3.

For any G' in la,

•,

the set of ill elements, each

multiplied by all numbers of absolute value 1. (omitting 6a0 and laO), certain For any G' in 2a, - - -, 21, -• •, nonreal subsemigroups not in d and e whose elements have C' as set of absolute values.

The 18 classes whose notation includes 1,4 or 6 and a, b or e have a single member; all others have infinitely many members. CHAPTER IV.

CENTERS OP ENDOTHETY

31. Endothety centers of grids. TRE0aEM 33. For given 2, the set Sa of centers of 2-endothety of a given grid S c V is empty or a grid that contains S. We may assume OESA. The point VESA if andonlyif (1—2)v-(-lseS for

every seS. If S is a grid and OeS, this is equivalent to 2seS, (1 — 2)veS. This condition on v is fulfilled by all points of S, and by v1 — Vg fulfilled by v1 and v,.

The set of 1-s-endovectors, se S, of a grid S does thus not depend on s; this set is a ring, called the endoring of S. The preceding proof shows that contains the normalizer of A in the endoring of S. the endoring of 32. Centers of symmetry. THR0ERM 34. For any set S c V the set S is a grid and includes with v also v + h, for every h with 2h =0. = H(1, 1, —1). Now if and only if For by Theorem 15, A(Uj n

T. S. MOTZKIN

376

I = 1,2,3, then SE S implies 2v. — se S, hence also — s))eS. Q.e4. — (2v. — 2(vi + — va) — s = 35.

+ h) — se S

and

Every nonemj'ty grid that includes with v also v + h, for every

h with 2h =0, is the set of centers of symmetry of some set. For assume (after a translation) the grid includes 0. Let S be the set of all 2vwherevisagridpoint. Then every v+hES1, and if then OeS

implies 2s€S, hence 2s=2v, s=v+h with 2h=0. For z = 2 the grid of centers of symmetry of any set is 1', and never

empty. On the other hand: THEonEM 36. If Z * 2 then in every R4 there is a set without center of symmetry,

for h = 1 and R = {O, 1, —1) or R = {O, 1, —1,2, —2).

For k 2 the set 0 U U4 has no center of symmetry; for if v were such a center then 2v €0 U U4, and for s s * 2v, also 2v — SE (14, which is impossible (2v — s has one coordinate —1). For k = 1, if the set {O, 1, s) has a center of symmetry such that 0 corresponds to 0 then s = —1; if 1 corresponds to 1 then s = 2; if s corresponds to S then 2s = 1. The latter equation has at most one solution, for 2h = 0, 2s = 1 implies h = 2hs = 0. Hence a set {0, 1, s} without center of symmetry exists if R has more than 5 elements. If x is even then {O, I) has no center of symmetry. There remain the two cases mentioned in the theorem, in which every set does have a center of symmetry. 33. Centers of autothety. We denote by S0 the set of centers of Cautothety, and by S, the grid of autotranslations 14) of a set S c V. S,. THEoREM 37. The set S6, tf nonempty, is a grid whose 0-translate is We may assume OeS9. The point v€S0 if and only if (1—A)v+.As€S for every s E S and every A C. As s e S is equivalent to As S. v S0 if and only if every (1 — A)v S,; hence S0 is a grid. Since every autotranslation of S is an autotranslation of S0 and 0€ S0 we have S, c S0. The condition for v e S,0, together with = implies Sir9 = Sg (1 — A)v The proof shows also that the endoring of S0 contains the normalizer of G in the endoring of S,. also, if every 1 — A(A e G — {1)) has an inverse, for We may have *

the latter may not be in the endoring of S,: for R the field of rationals, S = S, the ring of rationals whose denominator is a power of 2, and G the set of powers of -—2 with integral exponents, we have 1/3 e S,0. (1 — A)v

Indeed, has to be checked only for A = —2, or, in general, for a set spanning

G (as a multiplicative group), since (1— Ap)v = (1— 2)v + A(1 — p)ve 34.

Cone centers.

THEOREM 38.

(1 — A)v e

S, and

(1 — p)v

S, imply

= — A)veG. If R is a division ring and if the set S, of

and (1 —

cone centers of Sc Vis not empty then/or every AER,.A*O,A*1, the set SA = Se.

For let r SA and Si E S,,s, * r. If s S is not on the straight line through r and then S contains s,s —. {s,}, and by A-endothety a parallel line less its

ENDOVECTORS — hence the plane intersection with then an usual). If {r} c S. Q.e.d. c S. In either case From the proof we see, by a remark in § 9:

THEoREM 39.

(parallel and plane defined hence

rs1—(s1}cS, hence

If R is a division ringotherthan {O,1) then the setS, of cone

centers of a set Sc V is aflat; the set

sfrang cone centers of

S is the sante flat, or empty; and for the complement V —

of S we have

S

(but=St ifS=Ø orS= V).

THEOREM 40.

nonempty then S is a flat.

ii s is convex and

For 35.

ilalfeone centers. THFtoarni 41. The set Si, of halfcone centers of a set

Sc V(over R0) is a flat; the set St =Si,flS of strong halfcone centers of S is the same flat, or empty; and (V — S)i, = S1. (V — S)t = S1 —

St (but = St

if S = 0 or S = V). It is enough and easy to verify this in a plane through two halfcone centers v1and

A set with no cone center and at least one halfcone center v will be called halfcone; if v is a strong halfcone center, strong halfcone. THEOREM 42. A halfcone has no center of A-endothety for A <0.

Reasoning similarly as in the proof of Theorem 41 one shows that a center of A•endothety, A <0, of a halfcone would imply a cone center. Likewise we obtain: THEOREM 43.

Every center of A-endothety of a halfcone (or a strong halfcone)

is, if 0 1, an inverse star center. 36. Star and inverse star centers. A set with no cone center and at least one star center, strong star center, or inverse star center will be called star, strong star, or inverse star, respectively. The following theorems are proved in the same way as Theorem 41. THEOREM 44.

The sets s,

fl S.

of star centers, strong star

centers and inverse star centers of a set S.c V are convex. We have S, = st n = St = S

Sc

S

c

S,

S

S

S S is norm-closed, or linearly closed, so are

=

and

Here linearly closed means closed on every straight line; norm-closed, closed with respect to a (convex) norm. THEOREM 47. The set of centers of symmetry of a linearly closed star or inverse star is a flat.

T. S. MOTZKIN

The proof makes use of the fact, that by Theorem 44 every center of symmetry of a star, strong star or inverse star is a star center, strong star center or inverse star center. —1; an inverse star for —1 < 2 <0; a star with a center of symmetry none for 2 > 1; an inverse star with a center of symmetry none for 1 > A > 0.

TuzonaM 48. A star has no center of 2-endothety for 2 <

none

We see that inverse stars have no overstar center, and stars no inverse overstar center. TaaoaaM 49. Every point between (i.e., on the open segment connecting) a star center and an overstar center is an overstar center. Every point between an inverse star center and an inverse overstar center is an inverse overstar

center. 37. Ambicoitvex seta. The theorems and propositions of this section will be merely stated. If S1 and S1 are disjoint convex sets such that S1 c , S, c St., we call the ordered pair (S1 , S1) an ambiconvex pair. Ambiconvex pairs are the same as intersections of ordered pairs of opposite convex v-halfcones with a flat not through v'. The totality of ambiconvex pairs is intersectional in the sense that with all also (fl Sit, fl S11) is an ambiconvex pair. If (S1, S1) is an ambiconvex pair then S = S1 U S is an ambiconvex set. If v1 S. v, e S. v1 * v1, then S contains either the segment v1v2, or the straight line 1 through v1 and v2, or 1 — v1v2. If the ambiconvex set S is a flat then S1 and S1 are complementary se,niflots. For every pair of disjoint convex sets, and, in particular, for every ambiconvex D S,. Every pair pair (S1 , S1) there exist complementary semiflats S1' S1 , of subsets of complementary semiflats has an ambiconvex hull. The direct product (S1•S11, S .S') of two ambiconvex pairs is ambiconvex. Its union is a pairproduct of the unions S and S" of the two pairs. Tnsoaau 50. If S is a finite-dimensional ambzconvex set and $, the set of cone centers of the closure of S, then all ambiconvex pairs with union S consist of a unique partition of S — S fl S., into two sets, combined with an arbitrary ambiconvex pair with union S fl S fl S.,, etc., until arriving at a Set S with

S c S., i.e., a set whose closure is a flat. TasosEM 51. If the closure S of a finite-dimensional ambiconvex set S is a flat, and if. S — S spans_the fiat S1. then all ambiconvex pairs with union 5 consist of a partition of S into two open halfspaces disjoint from of a Partition of the remainder of S into two relative halfspaces disjoint from etc., combined with an arbitrary ambiconvex pair with union S n Applying Theorem 50 to S fl S1, etc., we arrive after a finite number of steps (not exceeding the dimension of 5) at the empty set. This number is one if and only if S is nonempty and either a flat less a flat, or a convex

ENDOVECTORS

set, or the union of two nonempty finite-dimensional convex sets whose closures are disjoint, and which form an ambiconvex pair. Such a union is a proPer ambiconvex set. CEAPTEB

V. TUE STRENGTH AND INVRUR STRENGTH

38. DefinItion of 2(v). A 2-overstar center v of S is also a 2'.overstar center for every 2' <2,2' > 0; the sup of all 2 for which v is a 2-overstar. center of S (the max If S is linearly closed) is called the strength 2(v) of v for S. Defining 2(v) = 0 for all star centers that are not overstar centers, we

have 0

2(v)

1.

Mutatis mutandis, similar statements hold and the inverse strength 2(v), 00, is defined for inverse overstar centers and inverse star centers. 2(v) 1 TnWREI 52. The strength (inverse strength) of v for S u the inf (sup) of the strengths (inverse strengths) of v for all intersections of S with straight lines through v. For empty S both 2(v) are 1 for all v V; for a nonempty fiat S both 2(v) are 1 for v S and undefined elsewhere. is not between two points 53. If S is not a flat, and :7 v e of S, then 2(v) = 0(00). the only points where both 2(v) are defined. This includes v e 39. ContinuIty properties of 2(v). TUB0REN 54. The set of points v e S with 2(v)> 2.(2. 0), i.e., the set of 2-overstar centers of S, 2> 2., is linearly open oo,S2) in S.". The corresponding statement (replacing >, 0, Sd," by holds for inverse stars.

For if v, is a Aoverstar center, 2 > 2., and Vt Se', then v S implies that S contains the segments from v2 to v, — 2,(v — v1), 0 2, 2. Hence v, + p(v1 — = 2(1 — p)/(l + 2p). But o < p < 1, is easily seen to be a 22-overstar center,

for small p.

The proof of the second statement is similar, enlarging v S to a subset of S this time first by consideration of v, and then of v,. 55. The set with 2(v) 2.(2. > 0) are linearly closed in S,I'. holds for inverse stars.

S

of points v €S

The corresponding statement

For if v5 = v1 + p(Vz — v1) e and v, , V2 Sc,", then v S implies that S contains the segments from Vs — 2c,(v — v,) to V5 — 2,(v2 — v3), 0 If p <0 can be arbitrarily small, it is easily seen that V1 The second set in the theorem is fl for 0 < 2,
The strength (inverse strength) isa linearly continuous function

T: S. MOTZKIN

380

of V.

are convex (TheoProperties 2(v) need not have. Although S0' and for 0 <2 < 1, need not be convex, as shown by constructing the rem 44),

, $3

S1

> <

-

This construction also shows that 2(v) need not vanish at all boundary points of S.

For

and its direct product with a segment, 2(v) is nonzero even on part of the boundary of S. <1, 0 < 22 < 1, is + 22), 0 < The open segment v0v3 of length p(l + where the lengths of vov2 and v1ve are c the smallest set with v1 e p(1 + and p(l + 28). If v is not on the straight line v0v8 and if v4 =

is obviously parallel to

and

for the convex hull of (v, v., v2, v4, v2 e to vv0; hence v1 e follows that also the smallest set S including v with v1 e

It

— 21(v — v1), V8 =

— 22(v —

v2), then v0v4

S has the form

and can be closed by inclusion of v. and v,. The closed segment v1v2 is for + s:; if = 12 then = (v1, v2}, and 2(v) has its minimum Denoting by SA' the set of inverse Aat the midpoint of v8v8. small overstar centers (1
outside S' or S

v

of all closed or open

ENDOVECTORS

381

segments in S including, or ending at, v; it is the largest set in S with v as the extended strength, as 2(v) for S.. if S.,. is star center. We define not a flat. If we associate, with each maximal open segment v1v, in S (not a flat), the closed segment from (v1 , 0) to (vs, 1) in the direct product V-R., and with each maximal open ray (haifline) r in S the closed ray (r, 0), and denote by S the union of these segments and rays in V.R,, then A*(v) = p(v)/(1 — p(v)) for

The v-inverse star S.... in S is the union of {v), if v e S, and of all closed or open rays in S on whose continuation v lies; it is the largest set in S with v as inyerse star center. We define A*(v), the extended inverse strength, as 2(v) for if S..,, is not a flat. If we associate, with each minimal closed segment v1v. whose continuations are contained in S, the closed segment from (v1 , 0) to (v1. 1) in VR0, and

with each minimal closed ray r whose continuation is contained in S, the closed ray (r, 0), and denote by the union of these segments and rays in p(v))/p(v) for p(v) = inf p, (v, p) S6. of § 40, the extended strength A*(v) is a continuous function of v vanishing on the boundary of S and riaing to a maximum of

V-R0, then At(v) = (1 For S = S1, or

—

1/3, 1/2,1, respectively,

attained at a single point in S1 and S., on two segments

in S2. However,

with a maximum of 1 within the segment So!, at its midpoint, but of 1/2 within S — SO", on four segments, shows: THEOREM 58.

The extended st,'engfh 2*(v) need not be a continuous function

of v.

Trivially, the set of points with A*(v) (0 < < 1) need not be connected, whether the set of points with A0(v) > 0 is connected or not. 42.

The strength for convex sets. In {vJ.R0, above or below a point (v, p) + p). Support, n.flat, bounded

+ p) or shall mean in have the usual meaning.

59. If S is a bounded convex set in with more than one point and ifS, consists of all points of above (S, 0) and below every n-flat through an (n — 1)-support of (S, 0) in (RoW, 0) and the opposite parallel support of (S, 1), and Sq of all points above these n-flats and below (S, 1), then S. consists of all points above S, and below Sq.

For the points below S. are those below all the defining segments. Now

'1'. S. MOTZKIN

382

the defining segments through a point (v, 1) of the relative boundary of (S, 1) form the conical part of the relative boundary of the convex set above (S, 0) and below every n-fiat through (v, 1) and a support 8 of (S, 0). Thus the points below Sa are those below all these n-flats, letting v and .3 vary. How-

ever, the points below all n-fiats through a fixed .3 and all (v, 1) are those below the n-flat through .3 and 8', the opposite parallel support of (S, 1). From here the theorem ensues easily.

It follows that the "level sets" on the slope of the relative boundary of The slope of S, S,, or of Sq, determine the relative boundaries of the can be extended and thereby 2(v) defined outside S, with —1 <2(v) <0. 60. If S is a bounded convex set spanning J4*, n > 0, then 2(v) =0 on the boundary of S and only there, all are convex, 2(v) is maximal on a set S+ of at most max (0, n — 2) dimensions, and = max 2(v) 1/n. Of the four statements, the first two are readily verified. Concerning the others observe that if S.f. had n dimensions, then the relative interior of the correspondIng part (S+ , p+) of the boundary of S, could not be on any 88'flat. If S+ has ,z — 1 dimensions then the relative interior of (Si. , is on just two öo'flats, both parallel to (S+ hence = 1/2, = 1. But by

Theorem 34 the bounded set S has only one center of symmetry, whence n = 1. If S+ has n — k dimensions, 2 k n, then (S+ , p+) is determined by k + 1 öô'-flats, with p+ 1/(k + 1), 1/k. The sequence 5, S+, S++, -.. leads after at most (n + 1)12 steps to a single point, the affine center of S. 43. The strength for special convex sets. Tuaoitait 61. If S is the direct product of two bounded convex sets S' and S". n'n" > 0, then = mm , 2!,.'), but

+ n" if A'÷
These statements follow easily from A(v', v") = mm (A(v'), 2(v11)). To verify the latter equality consider any maximal segment through (v', v"). It lies in the direct product of a maximal segment through v' and a maximal segment

through v" and can be replaced by one of these for the purpose of determining 2(v', v"). THEOREM 62. If the bounded finite-dimensional convex set S with more than one point has a center of symmetry s, then all Se', —1 <2 <1, are s-hoinothetic, and S for 1—2k 1—2k 1—2

1-f-A' in particular, + + = S'. The homothety follows from the conical shape of S, and S,, the relation between 2,, and A by considering maximal segments through s. Similarly we see: i.e., A

THEoREM 63.

If for a bounded finite-dimensional convex set S and a paint s,

383

ENDO VECTORS

some S, 0 < A <2+, is s-homothetic to S, then all Se', —1 <2 <2+, are s for = homothetic, S+ = {s}, and 1 — 222+

1—

1+2, + + 2122(1 — Theorem 61 implies

i.e., A =

A+))/(1

—

1 — 22+

1+2± — 1+2 +

THEOREM 64. If S is the direct product of two finite-dimensional bounded convex sets S' and S" with points s' and s" such that the S' are s'-kotnothetic A < A,. = 2',., are s"-homothetjc and if and the

(s', s")-homothetic.

then S+ has more than one point and thus,' by TheoHowever, if rem 63, no Sr', 0 < 2 <2+, is homothetic to S. Nor is any = as Theorem 63 applied to S' and S" gives two different values of A. A

44. The strength for convex polyhedra. THEOREM 65. If S is an n-dimensional convex polyhedron and has more than one point, then all are convex

polyhedra.

Here S, is determined already by those öö'-flats for which J contains an (n — 1)-face of (S, 0).

The number of (n — 1)-faces of SA* is constant for —1 <2 0 and decreases nonstrictly for increasing 2 < 2+. At 2+ the number of (n÷ — 1)-faces (defined as 0 if n.,. = 0) is at least by n — n+ + 1 smaller than for 2 < THEOREM 66. If S is an n-dimensional convex polyhedron and s a point in the interior of S such that, for every (n — 2)-face ço of S, there exists an affine

transformation T, other than the identity, with T(S) = S, T(ços) = ços, where —1 < 2 <4, are shomothetic, ços is the (n — 1)-flat through ço and s, then all and S.,. = {s}.

Indeed, for small A > 0 the faces of

corresponding to the two (n — 1)S, and

faces of S at ço must intersect on ços. Thus this we can apply Theorem 63.

Every regular polyhedron, as well as every semiregular polyhedron that is the convex hull of the union of two regular polyhedra with the same symmetries (and thus the same affine center), fulfills the assumptions of Theorem 66. All 4, 1/n 4 1, occur. By Theorem 61, if S is the direct product of regular polygons of k1, ••, sides, k1 < - -.
2)

and

For a convex quadrilateral S, the existence of an affine symmetry with

1, respect to a single diagonal ensures homothety of all All 4, 1/2 < occur. overlaps that of The problem of finding all pentagons with homothetic determining, in a plane over an arbitrary division ring, the pentagons perspec-

384

T. S. MOTZKIN

tive to the parallel pentagon through the opposite vertices. 45. The inverse strength for ambiconvex sets. THEoREM 67. If S is a above consists of all points of proper ambiconvex set in R, and if (S., 0) and below every n-flat through an (n — 1)-support of (S, 0) in (R05, 0) and

the opposite parallel support of (S. 1), and Sq of all points above these n-flats is the set of all points above (S2, 0) and below Sb, and below (S., 1), then 1) and above Sb. and Sq the set of all points below For the points below S6 are those below all defining segments. Now the

points below the defining segments through a point (v, 0) of the relative boundary of (S, 0) are those below every n-flat through (v, 0) and a support ö' of (S, 1). However, the points below all n-flats through a fixed ô' and all (v, 0) are those below the n-flat through o' and o, defined as before. From here the theorem ensues easily. The remark in § 42 on level sets holds also in this context. The slope of S, can be extended and thereby 2(v) defined outside s., with —co <2(v) < —1. then 2(v) = oo on THEOREM 68. If S is a proper ambiconvex set spanning the boundary of and 2(v) is minimal on S a set S- of at most n — 1, but for bounded at most max (0, n — 2) dimensions.

Proof as for Theorem 60. The affine center of S is also the affine center of S. 46. The inverse strength for special ambiconvex sets. THEOREM 69. If S , 2'.'), is a pairproduct of two proper ambiconvex sets S' and 5", then 2_ = max for A_ < 2 co, and the dimension nof S-. is 2n'... if 2'.. = =

but n'. + n" if 2'..> These statements follow easily from 2(v', v") max (A(v'), 2(v")). To verify the latter equality consider any minimal segment through (v', v"). It lies in the plane containing the direct product of a minimal segment through v' and a minimal segment through v" and can be replaced by one of these for the purpose of determining 2(v', v"). As in § 43 we prove: THEOREM 70. If the proper ambiconvex set has a center of symmetry s, then = if (1)? § 43, holds; in partiall S, 1
If for a proper ambiconvex set S and a point $ some S, <2 oo and —oo <2<—i, are s-homothetic, S.- is a flat through s, and if (2), § 43, holds = THEOREM 71.

L <2< co, is s-homothetic to S, then all

(with 2_ instead of At).

Theorem 69 implies THEOREM 72. If S is a /.'airproduct of two proper ambiconvex sets S' and S" with points s' and s" such that the S.b are s'-homothetic and are 2'.', then all S, A'.. =
ENDOVECTORS

385

Remarks as to Theorem 64 hold. 4?. The Inverse strength for ambiconvex polyhedral sets. THEOREM 73. If S is an n-dimensional proper ambiconvex polyhedral set then all S,' are convex polyhedral sets.

Proof as for Theorem 65. The number of (n — 1)-faces of S is constant for —00 A < —1 (A = —Co means A = oo) and decreases nonstrictly for decreasing A > A_. At A_ the number of (n_ — 1)-faces is at least by n — n- + 1 smaller than for A > A-. By Theorem 69, if S is the iterated pairproduct of plane proper ambiconvex sets with distinct L, 1 < L. < (all these values occur already for 5-sided sets), and possibly of a straight line less a segment, S_ will be a direct product

of plane convex sets and possibly of a segment, without representation of the plane ambiconvex set with largest A_; thus the bound max (0, n — 2) in Theorem 68 is exact. CHAPTER VI.

48.

PROPER END0THETY

Nearldesla. For every subsemiring R' c R with 0£ R', 1

R" cR' with leR" such that

R', a subset

if all A,eR', all p,ER",

is an R'-nearideal. THEOREM 74. The set R" — 1 of all p with p + 1 eR" has the property that p eR" — 1, A ER', 1 — A ER', implies Ape R" — 1. For by definition every A with A, R', A, E R", is an endovector of R". In particular, R" is endo-(A, 1—A), hence A(,p + 1) ÷ (1— A).1 = A/i + 1€R". THEOREM 75.

If R' is a ring then a set R" is an R'-nearideal :f and only

if R" is the residue class of 1 modulo an ideal in R'. If R' is a division ring, = (1) or R" = R'. If R' is a ring then a nearideal R" is endo-(1, 1, —1), hence by Theorem 14 a grid, and + 1€ R", Pt + 1 eR" implies Pt + + 1 R". This, with Theorem 73, proves the necessity of the conditions; the sufficiency is immediate. THEOREM 76.

If R' = R'(1) is infinite, then a set R" is an R'-nearideal if

and only if R" — 1 is an additive semigroup containing 0. The set R" — 1 — {0} contains a unique finite set p R" — 1 if and only if p = A1p1, A1 E R', while no p is thus representable by the others. There exist for every k = 0, 1,2,

For if 1 + p ER", 1 + ye R" and R" is a nearideal then R" is = (1, ..., 1), hence (1 + v) + 1 + ... + 1 = 1 + p + v R". This proves the first contention. If R" = (1) then k = 0. Otherwise there exists a v R', e.g., every E R" — 1 — {0}, such that ER" — 1 for all sufficiently large A e R'. Let be the smallest of all such Then every ER" — 1 is of the form since an element not of this form, together with leads to a v < If —1 for A A0, then every ER" —1 can be represented, in the sense of the theorem, by all A <2A0. Deleting all where

e R" —

1

representable by smaller ones we obtain the unique basis. In case

T. S. MOTZKIN

386

we have k = 49. Proper endothetic convex sets and flats. If R" is an R'-nearideal then the set A of A with all A, R' and 2€ R" is complete; u(4) is the family of Proper O-R"-endothetic R'-convex sets. In particular, R" o(4). R = R0, the For R" {1) we obtain the R'-convex sets, for R" = R' A

convex halfcones. In these cases all O-R"-endothetic R'-convex sets are proper.

In general, the set A need not coincide with H(R" U (4(U1) fl R')); e.g., if R' is the semiring of positive integers then 4(U1) fl R' = U1 while 4 H(R") for R" {1), {O, 1). However, we have: THEOREM 77.

If R' is a ring then all O-R"-endothetic R'-flats are Proper

O-R"-endothetic R'-convex sets.

For the set 4' of endovectors of the O-R"-endothetic sets includes (1, 1, —1). Now if A€4 and A' ...), u1 =(1,O,O, ...) then A'€A',u1E4',A + u, — 2'EA', hence A =(A + —A')+ 2' —u1e4'. Q.e.d. 50. Proper endothetic grids and sets. THEOREM 78. If R' = RU), then I?" consists of all 1 + 2i, A R', for some fixed nonnegative integer p. The corresponding A is

H(I,1,—l),

u=0; p=1;

H(1,—l,—l), p=2; H(—1M_I) , p 3 This follows from: THEOREM 79. For infinite R = R(1) and v minus ones), = 0, the set H =

(1) (2) (3) (4)

all 2eUU(O)forp=0,,.=O; all 2€ UU—U for p =O,v all AER with A * 0,2,

all AeR wth

p+

= (1, ..., consists of

1,

..., —1) (p

1;

1(modp —

1(modp — 1)for p 1)

1,v=0;

in all other cases (i.e.,for

1,

2).

Of course, (mod 1) means "arbitrary," means (mod p).

(mod 0) means =1,

(mod —p)

Cases 1 and 2 are trivial. Contention 3 follows by noting that U c H and that any vector stays in H(lM.o) when increased by p — 1 units anywhere. Now let

then AeH implies (2,0, ...)+(0, minus H. Taking A = we have H, hence 1°: c H, and 2° a vector stays in H when increased by + 1 units. But also 3°: A H implies —A + (v — 1) minus ones H; from 1°, 2°, 3° we obtain the result stated for p = 0. For p = 1, = 1, we have (I, — I) — (1, 0) = (0, —1) e H, hence (1, 1) e H; ones

but H(1,1)cHand (1,—1)eHimply H=R. For p=2,i.'=l, see Theorem For p we have 1°: A€H implies A — A —A = —A€H; hence 2°: H(1,1, —1)cH; 3°: 2€!! implies A + two minus ones €H; from 1°,2°,3° we obtain the statement in this case. For arbitrary p 1,i. 1 note that, by 15.

ENDOVECTORS

387

this gives (4) for p + 1. But for p H and includes elements A all of whose coordinates are large, 2' = ••• —2€ H has negative large coordinates; to these, p — L' — 1 units may be added repeatedly, and the proof thus completed. Whereas (4) gives all the complete endovector sets of Theorem 78 (also for compression + 2, since

finite R(1)), (3) does not exhaust the complete sets belonging, for infinite R'(l), to families of proper O-endothetic sets. These familes correspond to the nearideals of Theorem 76; of the latter, only the cases k = 0 and k = 1 are associated with (3). UNIVERSITY OF CALIFORNIA

REPRESENTATION OF POINTS OF A SET AS LINEAR COMBINATIONS OF BOUNDARY POINTSt By

T. S. MOTZKIN AND E. G. STRAUS 1. Introduction. Let L be a linear space of dimension at least 2. For a set S in L, denote by L2 any two-plane such that S fl L2 is bounded and by The outer boundary B' of all S fl B the union of the relative is the union of the relative boundaries of the unbounded components of all We call S 2-bounded if it is the union of all S fl L2. — S. , of real numbers is called admissible for S if for An n-tuple (a1, every point P S there exist points x1, - - •, (not necessarily distinct) in B' such that

=

a1x1 +

-

-+

-

In this note we prove the following two theorems. THEOREM 1.

The n-tuple (a1, - - -, a,) is admissible for all 2-bounded sets in

Lii and only if

(i) a1+•--+a,,1,

j = 1, (ii) A more detailed description can be obtained as follows. Define S' as the union of the relative complements of the relative interiors of Thus, — S. if S is 2bounded then S c S'. For any p €S' define s as the maximal subset of 5' which is starlike with respect to p and the strength (see [11) of n s as the maximal number A such that p in L2n[(1+A)p_ASp*JcL2nS,? -

The 2-acentricity 2(P) of p in We say that 2(P) is attained if

is the greatest lower bound of all

of S is the least upper bound of 2(t) for all P S. We say that A is attained if for every P e S there is an L, through p so that A

2L2(P) A.

It is clear that all the numbers 2(p), A lie between 0 and 1 (inclusive). THEOREM 2. Let S be a 2-bounded set in L and let p be a Point of S. Then for each n-tuple of non-negative real numbers (a1, - -, a,) which satisfy (i) and -

2(p)a,, where equality in (iii) (iii)

--

j

= 1, - - -, n,

is permitted if 2(p)

is attained,

•,x,,eB so that

1 Sponsored (in part) by the Office of Naval Research, U.S.A. 389

there exist points

T. S. MOTZKIN AND E. G. STRAUS

P = a,x1 +

-

+ a,,x,,

Conversely, for convex 2-bounded S all non-negative (a1,

•,

which permit

a representation (2) must satisfy (iii). Let (at, - - -, COROLLARY. Let S be a 2-bounded convex set in L. non-negative satisfying (i). Then it is admissible for S if and only if

j

(iv)

= 1,

be real

•.-,n,

where 2 is the acentricity of S. Here equality is permitted if and only if A is attained. 2. Proof of Theorem 1. The necessity is easily seen. Condition (i) is obviously necessary if S consists of a single point p 0. Condition (ii) is necessary in case S is the unit ball of a normed space and p = 0 its center. In that case (1) implies

The sufficiency proof (in which we may assume dim L

2 and n

2) is

effected via five lemmas. LEMMA 1. Let S be a bounded set in the plane. Let p S and x, y B' so that the open segment xy is contained in S', and p = Ax + (1 — 2), with 0

1/2.

A

Then for every p with A

p 1—

A

there exist points u, v e B' such

that p = pu + (1 — p)v. In particular, every point in S is a midpoint of two boundary points. This establishes the sufficiency of (1), (ii) for n = 2.

PROOF. We may assume A

Conditions (1), (ii) are sufficient for n =

3

maximal segment in / fl S' that contains p, and assume p = Aa o <

A

+ (1 —

A)b,

1/2.

If A a, then by Lemma 1 there exist points x, y B' such that ft = a,x + (a, ÷ a,)y and the conclusion of the theorem holds wth x, = x,

= x. If a,
Conditions (i), (ii) are sufficient for n

PROOF. We may assume a, <0 < Since

a,

3.

a,. Define a, b, A as before.

a,
a,b

÷ (a, + a,)c.

REPRESENTATION OF POINTS OF A SET AS LINEAR COMBINATIONS

391

Thus by Lemma Hencec=(Aa+(1—A—ajb)/(a,+a.) where 1 there are points x,, x, B' such that (a, + a,)c = a,x, + a.x, and 1 =

+ a,x, + a,x, where x, = b. LErnIA 4. Conditions (i), (ii) are sufficient for a,, ••, We use induction on n. If 0 a, a, PROOF.

a,x,

a1+"•+a,,=1 where

then

Hence

0. a,,

1/2

and

((a,+a,),a,,••,a,,)

admissible for S. Thus for each p ES, (1) holds with x, = x,.

is

LEI1MA

5. Conditions (i), (jj) are sufficient for a1

a,.< 0

at,.

Again we use induction on n. Since ((a, + •, a,,) + ai), (1) and (ii) we may assume i = 1. We then have a, <0 a, a,, + a,,-, 1/2. The given n-tuple can be reduced to the admissible and a, + ,a,) if n >4 and also if n = 4 and a4 1/2. (n — 1)-tuple (a,,(a, + a,),a4, For n = 4, a4 < 1/2 we have a, I
satisfies

for Proof of Theorem 2. We now select L, so that Z,,e, . ,n with a1 0, and then prove sufficiency for Sfl L2. establishes the sufficiency of (i) and Lemma 1 with A = AL,(P)/(1 + (iii) for n = 2. In Lemma 2 the condition on the a, becomes 0 a, a, If we choose A as before, then the case A a, remains the a, ,_, (1 + A(p))'. same. In case a, < A we get ncw p = a,a + (a, + a,)c where c = ((A — as)a + 3.

all j = 1,

(1 — A)b)/(a, + 2

a, <

a,) so that 2(p)

a,

—

(a, +

a,

—

a, — a,

a,

Lemma 4 remains valid since (1 + A(p)Y' 1/2. To prove the converse, let S be closed, bounded and convex in the plane and let p = 0. The convexity of S implies that a,x, + -- + a,,-,x,-, = (a, + ... + where ye S. Thus a,x, +-" + a,,..,x,,_, + = 0 implies II y

It Ill x,, II

A(O)' or A(O)a,,

a,

+-" + a,,—,.

,a,,) for 4. An exnmple. It might be conjectured that the set of (a,, which a representation (1) exists for a given p in a given connected bounded set S is connected. This is obviously the case when S is starlike with re

spect to p. However, it need not be the case when S is the interior of a Jordan curve which is not starlike with respect to p. Consider the curve C

T. S. MOTZKIN AND E. 0. STRAUS

392

which consists of the unit circle with center (0, 0) modified by deletion of < 6 < it (3, the two arcs < 6 < 5it/6 and joining the endpoints 8 ir/3 to the point (lIe, 0) by straight line segments; similarly joining the endpoints 6 =

2ir/3, 5ir16 to (—e, 0).

Now for all representations a(xi, x2) -Iwe

have e

,

= (0,0) with (x1 , x2),

1/s with the exception (x1,x2)

C

= (1/e,0)

= l/e. 5. Remark. A slight modification of the proofs of Lemmas 4 and 5 shows that Theorems I and 2 remain valid for oo-tuples (a1 , a2, In Theorem 1 (i), the sum a, may converge conditionally; in this case (ii) is regarded as fulfilled. In the proof, the co-tuple is reduced to the k-tuple (a1 , a2, a') for sufficiently large k. ak_I, where

REFERENCE

1.

T. S. Motzkin, Endovectors, these Proceedings, pp. 361-387.

UNIVERSITY OF CALW0RrnA, Los

SUPPORT CONES AND THEIR GENERALIZATIONS BY

R. R. PHELPS C is a closed convex subset of a real Hausdorff Introduction. Suppose topological vector space EY A point x in C is called a support point of C if

there exists a nontrivial linear functional / in Et such that supf(C) = f(x). The functional I is called a support functional .of C. Note that a support point is necessarily in the boundary of C, and that any positive multiple of a support functional is again a support functional of C. This paper is, in many respects, a sequel to (1] in which it was shown (among other things) that if E is a Banach space, then the support points of C are dense in the

boundary of C, and the support functionals of C are dense among those functionals which are bounded above on C. These results were proved by first showing the existence of certain "support cones" of C. This method, which is so useful in Banach spaces, does not yield the existence of support points or functionals in more general spaces, since it rests heavily on a characteristic property of normed spaces—the existence of bounded convex sets with nonempty interio,r. Lemma 1 of [11 (which asserts the existence of support cones) does not depend on this property, however, and in § 1 of the present paper we extend this lemma to more general spaces and more general "cones." The extended result has several applications, are contained and 3. in In § 2 we consider support questions for C in a complete locally convex space E. It is unknown whether every closed convex set C must have any support points, but we can prove density theorems for two classes of sets which are known to have support points: those C with nonempty interior and those which are weakly compact. In the former case, every boundary point

of C is a support point; we show (as a corollary to a more general result) that the support functionals are dense (in an obvious and appropriate topology in Es) among those which are bounded above on C. In the latter case, every functional in E5 supports C; we show that the support points of C are dense in the boundary of C. In fact, we obtain this conclusion assuming only that C is locally weakly compact, a result previously obtained by V. L. Klee [3, p. 438] by a somewhat different method. Generally speaking, the results in this section are straightforward generalizations of the analogous results in [1].

In § 3 we use our main lemma to obtain a new and simple proof of the existence of relative extreme points for a bounded, complete convex subset of a topological vector space. (The notion of relative extreme point, introduced by Klee in [4], holds promise of application to questions of weak compactness

16) as well as to the question of the validity of the Krein.Milman theorem without the assumption of local convexity.) A tool which we develop for this 393

R.

394

R. PHELPS

purpose is interesting in itself; it is a concave functional which is associated with each closed convex set S which does not contain the origin. (The definition of by means of S is similar to the manner in which the Minkowski (or gauge) functional is defined for a convex set which does contain the origin.) In what follows, the topological vector spaces E under consideration will be assumed to satisfy the Hausdor(f separation axiom, and all subsets of E which we discuss will be proper and nonempty. Whenever we introduce an element

/ of E*, we assume that it is not identically zero on E. 1. The main lemma. Since we will not assume the existence of a norm in our space E, we make use of the obvious substitute—the Minkowski functional.

We start by reviewing its definition and basic properties. Let T be a subset of E which is storshaped (with respect to the origin that is, if xeT then 2xeT if Define, for each x in E, Pr(X)= inf (A > 0: xe A T}. Note that p(x) = oo if (and only if) x AT for each A > 0. (In order to make use of the extended real number system, we adopt the following conventions: If a > 0, then (±oo)a = ±00 = (±oo) ± a, (±oo)(—a) = moo, (±oo)O = 0 and 0° — 00 = 0.) The properties of listed in the following proposition are well known and not difficult to prove. PaoposrrloN 1. If T is a starshaped subset of the topological vector space E, then p = Pr has the following properties: (i) For alix in E, and p(ax)=ap(x) !fx€T,then p(x)

1.

(ii) If T is closed, then p is lower semi-continuous, and p(x) 1 implies that xET. (iii) p(x)=0 if and only if AxeTfor all A >0; hence p(x)=Oimpliesx=çts if T is bounded (or merely linearly bounded). (iv) If T is convex, then so is p; that is, if x, y e E and A e [0, 1], then p[Ax + (1 — A)y] Ap(x) + (1 — A)p(y). (v) If T is closed and convex and E is locally convex, then p is lower semi-

continuous in the weak topology. (vi) If T is closed and f€ with supf(T) 1, then 1(x) p(x) for all x in E. We now recall the definition of a support cone: A set K is a convex cone

with vertex

(or simply a convex cone) if K is convex and if Axe K whenever

xeKand A If K is a convex cone and ifxeE, thenK+xis convex cone with vertex at x. We say that the cone K + x0 supports the set X at x0 if (K + x0) n X = {x0). In El] we considered cones of the form K = (x: I x II kf(x)},wherefeE* and k >0. The set K+ ythentooktheform(x: Ux —yfl k[f(x) — f(y)]}. This suggests the following generalization: Let be an extended real valued function on E, let T be a starshaped subset of E, and suppose k > 0. Partially order E by setting x y if kp1.(x — y) co(x) — p(y). If ye E, let k, T, y) fr: this is the set of x such — y) — that x y. If is linear and T is convex, then j(ço, k, T, y) is a convex cone, which we denote by K(ço, k, T) + y. (In case T is the unit ball of the normed space E, this is the cone K(ço, k') + y of [1].) When there is no I

SUPPORT CONES AND THEIR GENERALIZATIONS

ambiguity as to k and T, we will simply write 1(y) for J4s, k, T,y) and K + y for K(ço, k, T) + y. Note that if T is bounded, then x y and y x imply that — y) =0 and hence x = y. LEMMA 1. Suppose that X is a closed subset of the topological vector space E, that is an upper semi-continuous extended real valued function on E which is bounded above on X, that k > 0, z X, and that T is a closed, starshaped

subset of E. If X is complete and T is bounded, then there exists x0 in X such that x0 J(z) and X fl J(x0) = If we assume that X is compact (but not necessarily that T is bounded) then there exists x0 in X such that x0 1(z) and 'p is constant on X fl J(xo). PROOF.

It follows from the semi-continuity of z and ço that J(z) is closed.

now, that X is complete and that T is bounded. In this case, a point x0 in X will satisfy the conclusion oi the lemma if x0 is a maximal element of J(z) n X. We will apply Zorn's lemma to obtain this maximal element; to do this we need only show that any totally ordered subset U of

J(z)nXhas

an

upper bound in X. I' W={z}, we can take x0=z, so we

If xe W,x*z, then 0 z

can assume > 'p(z)

—co.

>

Similar reasoning shows that {'p(x): XE W— (z}} is a mono-

tonic net of real numbers. Since 'p is bounded above on X, this net converges and hence W itself is a p-Cauchy net, that is, given £ > 0 there exists w in

Wsuch that if then p(x—y)<e, i.e., x—yeeT. Now, since T is bounded, given any neighborhood U of the origin in E there exists c > 0 such that cT c U, which shows that W is a Cauchy net in E, and hence converges to an element Yo in the complete set X. To see that Yo

is an upper bound for W, let xe W and suppose ye h,y

x.

Then 0

for each such y; since W converges to Yo, we have (taking all limits over those elements y in W such that y x and using the kp(y — x) ço(y) —

ça(x)

semi-continuity of p and .p) kp(yo—x) urn sup ço(y) —

ço(x)

S°(Yo) — ço(x).

liminfkp(y—x)

Thus, y.

limsup[co(y)—so(x)J =

x, which completes this part of

the proof.

Suppose, now, that X is compact. Then J(z) n X is compact and hence there exists a point x0 6 f(s) n X such that 'p(x0) = sup'p( f(s) fl X). Suppose that xe J(x0) n X. Then xc f(s), so that ço(x) 'p(x.). On the other hand, x e J(x.) implies that ço(x) çc(x0), which hows that 'p is constant on J(x0) n X. 2. Support theorems. In this section we prove our density theorems for support points and support functionals in a locally convex space E. It is possible to define a number of topologies for E but the one which is an

immediate generalization of the norm topology in E5 (when E itself is formed) is the topology of uniform convergence on bounded subsets of E. As is well known (see, e.g., L2, p. 17]) a neighborhood system of the origin for this

topology is given by the polars B° of the members B of the family of all bounded closed convex sets in E which are symmetric about the origin. (For such a set, B° = {f:f€ E5 and f(x) 1 for all x in B}.) Thus, a subset F of E5 is dense in E5 if for each f in E5 and each B in there exists

R. R. PHELFS

g in F such that! — g e B. We now prove two lemmas (analogous to Lemmas 2 and 3 of Lii) which give conditions ubder which two functionals are "close" in this topology. The lemmas themselves require no topological assumptions, so we state and prove them for arbitrary real vector spaces. LEMMA 2.

Suppose that B is a convex symmetric subset of the vector space E,

and suppose that / and g are linear functionals on E with 0 < a = sup! (B) = supg(B) < co. If I g(y)l a whenever 1(y) =0 and y e4aB, then either /— g€(1f2)B° orf+ge(1/2)B°. PROOF. Let h' be the restriction of g to the subspace !_I(O). By hypothesis, if y B n f_*(O), then I h'(y) I 1/4, so, if p is the Minkowski functional for B, then I h' I (i/4)p on I _1(O). By the Hahn-Banach theorem there exists a linear functional h defined on E such that I h I (l/4)p on E and h = h' = g Since g — h vanishes on f1(O), there exists a real number ft such on I E (1/4)B°. We (1/4)p or ±(g — that g — h = ,9f and therefore Ig — must now examine the cases corresponding to the possible values of ft. Note first that if P = 0 then g = h and hence (since sup h(B) 1/4) a 1/4. fore 1,9 aB° C (1/4)B° so that f — 9€ (1/2)B°. We can assume, then, that ft*O. Suppose that ft>O; then either O 4+a'. I! g is a linear functional on EsuchthatO < r =supg(B)< 00

I

and g(x)

0 whenever kpB(x) f(x), then there exists A > 0 such that! —

Ag

e B°.

Since k(4a + 1) < a, we can choose x in B such that f(x) > k(4a + 1).

Suppose

that ye 4aB and f(y) =0. Then kp(x ± y) k(1 + 4a) <1(x) =

f(x ± y) so that g(x ± y) 0. Thus, 19(Y) I g(x)

and

(letting A = air)

we have sup Ag(B) = a Ag(y) I. It follows from Lemma 2 that! + Age (1/2)B° or f — Ag (1/2)B°. Now, choose z in B such that 1(z)> max (1/2, k(4a + 1)); then kp(z) <1(z) so g(z) 0. Hence Cf + Ag)(z) f(z) > 1/2, and therefore ! + Ag (i/2)B; we conclude that J — Age (i/2)B° c B°.

We now prove a theorem (analogous to Theorem 2 of [1)) which yields, as a corollary, a density theorem for support functionals. TasossE 1. Suppose that C :s a closed convex subset of the complete locally convex space E, and assume that C has nonempty interior. Suppose further that X is a bounded subset of E and that! in E5 is such that supf(C) < inf 1(X). Then if B is any bounded symmetric convex set in E, there exists g in Es and

x0 inC such that supg(C)=g(xo)< infg(X) andf—geB°. PROOF. Let r = sup! (C), o = inff(X) and choose ft such that r

<

ft < o. We

SUPPORT CONES AND THEIR GENERALIZATIONS

397

can assume, without loss of generality, that 4. mt C, so that r > 0. Let V = and Choose z inC such that r —f(z) — cony (X u (4.1) ri (x:f(x) let B' be the closure of the bounded symmetric convex set B + cony (V U — V) + [—z, zJ. Let p be the Minkowski functional for B'. If v e V then v e B' and —z B' so that p(v — z) p(v) + p(z) 2. Since X c V c B' we have

o <ö sup! (B') = a, and we can choose M such that M> 2 and M> Let k1 = 2M(ft — rY'; then k1 > 4 + a' > 0 and by — r)(4 + a1). Lemma 1 there exists x0 in the complete set C such that x0 — z e K(f, k, B') and(K+x0)n C= {x0}. Notethat VcK+x0; indeed, if ye p(v — z) + p(x0 — z)

2 + /C1f (xo — z)

<M +

—1(z)] <M+ k121(j9 — r) =

k1f(v — x0). Since K + x. misses the interior of C the such that 0 < sup h(C) = separation theorem shows the existence of h in suph(B') < oo (since B' is bounded). Since h(x0) = inf h(K + x0) inf h(V) ZM =

k'(fl — r)

k' > 4 + a', Lemma 3 applies to show that there exists A > 0 such that f — ge (B')° c B°, where g = Ah. Finally, to see that inf g(X) > g(x0), note so that v=ô'j9x€V and hence that if x€X, then But then h(x) the proof. h(x0).

ö191h(x0) > h(x0),

so inf h(X) >

h(x0),

which completes

The corâllaries below follow from Theorem 1 in the same way that the analogous corollaries in [1] follow from Theorem 2 of that paper. C is a closed convex set with none,npty interior in the complete locally convex space E, then the support functionals of C are dense (in the topology of uniform convergence on bounded sets) among these functionals in

which are bounded above on C.

CORoLLARY 2. If C is a closed convex set having nonempty interior in the complete locally convex space E, and zf K c E C is a compact convex set, then there exists a functional F in E * which strictly separates K from C and

supports C.

To obtain our theorem on the density of support points, we first state a condition under which our support cone will have nonempty interior. Suppose

that 1€ E and that U is a closed symmetric convex neighborhood of the origin in E such that a supf(U) < oo, Let p be the Minkowski functional for U and suppose k> 0. If there exists x in £ such that kp(x) <1(x), then there exists A > 0 such that Ax + U c K(f, k, U), and hence K has nonempty interior: We choose A > 0 such that k[Ap(x) + 1] = 21(x) — a. It follows that if ye U then kp(Ax + y) k(Ap(x) + 1) = 21(x) — a f(Ax + y), sincef(y) —a

for all y in U. THEOREM 2 (KLEE).

Suppose

that C is a closed, convex and locally weakly

compact subset of the locally convex space E. dense in the boundary of C. PRooF'.

Then the support points of C are

If z is in the boundary of C, choose a neighborhood V of the origin

such that C' = (z + V) n C is weakly compact. Choose a closed, symmetric convex neighborhood U of the origin such that U + U c V. We first show that z + U contains a support point of C'. To this end, choose y cE C

R. IL PHELPS

398

such that y e z + (112)U and choose (by the separation theorem and local con vexity of E) f in E* such that supf(C)
and let p be the Minkowski functional for U'. Looking at E in its weak topology, we see that p is lower semi-continuous, C' is compact,f is continuous

and 0 < a = supf(U') < oo. Letting k = a/2, the second part of Lemma I shows that there exists x0 in C' such that x0 — z e K(f, k, U') and! is constant on (K + x0) n C'. Since k < a, there exists x in U such that 1(x) > k, so that kp(x) <1(x) and therefore K has nonempty interior. Applying the separation

theorem to K + x0 and C' shows that there exists a nontrivial element g in E* such that supg(C') = g(x0). Furthermore, p(x0 — z) 2a'f(xo — z) < 2a'f(y — z) 2p(y — z) 1, since sup a1f(U') = 1. Now, p(x0 — z) 1 implies that x0 — ze U' c U so that x0€z + U. To see that x0 is actually a support point of C, suppose that w C. Then for sufficiently small 2 > 0,

xo+2(w—x,) is in (z+ U)+ Ucz+ V and is in C; therefore it is in C'. Hence g(xo + A(w — g(xo).

x0)1

g(x0) so that g(w) g(x.), which shows that sup g(C) =

Since we can find such an element x, for each such U, the proof is

complete.

The problem of the existence of support points for a closed convex subset of a complete locally convex space remains open. The following remarks shed a little light on the role which completeness might play in this problem. There exist locally convex spaces E with the property that for every closed convex set C in E, the support points of C are dense in the boundary of C, and the support functionals of C are dense among those bounded above on C, but E is not complete. Such examples are obtained by noting that some Banach spaces are not complete in their weak topology (e.g., C (0, 1] or c0) and recall-

ing that for convex sets, norm closure and weak closure are the same, and that weakly bounded sets are norm bounded, so that the appropriate topology in E* is the same as the norm topology. The results of (ii then apply. 3. RelatIve extreme pointa. We now introduce a functional on E which is of considerable use in applying our main lemma and which is of independent interest. If S is a subset of E which does not contain the origin in its closure, define, for each x in E, = sup (A 0: x AS). (If there is no question as to the set S involved, we write 'p for 'ps.) For such a set 5, let S' = [1, oo[S. Parts

(iv) and (vii) of the following proposition are not needed in the sequel, but they illuminate the connection between and more well-known functionals. PIWPOSrFiON 2. With S and defined as above, we have ço(x) < oo for each x in E. (i) (ii) If and co(ax)=aso(x) for all x in E; ço(x)=0 :f and only if x = 4'. (iii) 'p3 = 'p3.; if S is closed, so is S' and ço(x) = max (2 0: x€ AS) if ço(x) 0. (iv) If p is the Minkowski functional for the starshaped set E p(x) = ço(x) for each x such that ç(x)> —0°. If x * 4', then p(x)

only if

S', then =0 if and

SUPPORT CONES AND THEIR GENERALIZATIONS

(v) (1 S is closed, then 'p is 'p(x)

semi-continuous at each point x such that If, in addition, S is convex and E is locally convex, then 'p is weakly 0.

upper semi-continuous at such Points.

in E and hence

(vi) If Sis convex,

(using (ii)), 'p is concave. (vii) If feE* and if S is the closed half-space (x:f(x) a > O}, then 'p on {x:f(x) > 0) U (viii) If S is closed, convex and has nonempty interior, then 'p is continuous and xeintS} defined by and the interior of S. in the cone (Ax:2 (ix) The function 'p is bounded above on any bounded set.

for all A 0. The Paoop. Observe that ço(x) = if and only if rest of part (i), as well as parts (ii), (iv), and (vii), follows easily from the definitions. (iii) Since S c [1, On the other hand, if 0 < A <'p,.(x), there exists A' such that A A' ço3.(x) and x C A'S'. Hence x = A'ry for some r 1

and y in S so that 'p8(x) A', which shows that To see that S' is closed, suppose that y. is a net in S', with y, —by. Then y. = A,z,, for A 1 and z. S. Choose an open neighborhood U of such that rU c U if and such that UnS is empty. There exists A0>O such that yeA0U andhencethereexistsanindexa0suchthaty.€2,Uifa a0. IflimsupA, = 00, there

would exist a subnet yp of y, such that

This would mean, then, that assumption that A,*y, = z, eS. Thus, urn sup A. < 00

> A0 and hence A0 U

U, contradicting the since each A0 1, there exists A 1 and a subnet 2, of 2. converging to A. Hence z = A1y, S and therefore yeS', so that S' is closed. r'y; since S is closed, Finally, if ç(x) >0, choose A, > 0 such that 2,, — ç(x) and xc A,,S. Then 4tx çc(xy1x S. so sup (2 0: XC AS) is actually attained. (v) We must show that if ço(x) 0 and x, —' x, then urn sup 'p(x,) ço(x). = we see that x is not in the (closed) Suppose, then, that A > co(x); since a0 and thereAS' if a set AS'. Thus, there exists an index a0 such that fore 'p(x) A if a a0, so that urn sup ço(x) A. Since this is true for each such A, 'p is upper semi-continuous at x. Suppose that —' x in the weak topology. With A as before, we have x€ AS'. Since S is closed and convex, so is AS' and, if E is locally convex, there exists / in E0 such that f(x) < inff(AS') = c. Hence there exists a0 such that 1(x) < c if a a0, so that AS', and the same reasoning as above shows that 'p is weakly upper semicontinuous at x. (vi) Clearly 'p(x + y) 40(x) + 'p(y) if 'p(x) or ço(y) equals —oo or 0. II both are positive and e > 0 choose positive numbers A and 2' such that 2> ApU.

40(x) —

i/2, 2' > ço(y) —

£12

and,

and xc AS, ye A'S. Then (A +

(AS+A'S)cS so that

+ y) €(2 + A'Y' and therefore

(viii) In light of (v), we need only show that 'p is lower semi-continuous at each point x such that x = A0y for some y in the interior of S and A0 > 0

(so 'p(x)> 0). Suppose, then, that x0

x; we must show that 'p(x)

urn mi ç(x).

R. R. PHELPS

400

To do this, it suffices to show that for each r such that 0 < r < 1 there exists Since S is closed, we know that x= if a0 such that the interior of the Since y for some z in S, so that z = 4 and the half-open segment Jz, y) convex set S, we have and Ar 4. Then mt s. Choose A such that 4 < A = is in the interior of S. Since so that

a, such that

if a

e S for a a, and therefore ço(x1)

yJ C < < 1, x there exists rço(x)

— Jco(x)1Aoy,

a,, which completes the proof.

(ix) Suppose that B is a bounded set, and choose a neighborhood Uof the

origin such that rUc U if Id

1 and such that Un S is empty. Choose

A, > 0 such that Bc 20U. If x B, then U if A A. so that x AS if A A,, which shows that ço(x) A, if xeB. If B is a convex subset of a vector space E and if C is a convex subset of

B, we say that a point x in B is an extreme point of B relative to C if x is not interior to any 'nontrivial line segment in B which has one çndpoint in C, that is, if x = Az + (1 — A), (for distinct points z in E,y in C and 0 <4 <1) then z B. Note that x is an extreme point of B if and only if it is an extreme point of B relative to B. This notion was introduced by KIee in [4J, who proved, among other results, the following existence theorem. Suppose that B is a bounded closed convex subset of the THEoREM 3 complete topological vector space E, and that C is a proper closed convex subset of B. Then there is at least one Point in B C which is extre,ne relative to C. PROOF. We can assume, without loss of generality, that B C. Let p be the Minkowski functional for —B and let = Since B is bounded,

is bounded above on B and by Lemma I there exists x, in B such that 1, — B, and J(x,) n B = {x,}. To see that x, is a relative = extreme point of B, suppose there exist distinct points y in C, z in B and A x, e

in 10, 1[ such that x, = If ço(x,) = 0, then x, = — A)

z

*

qS.

and ço(z)

+ (1 — A)y. Since x, ço(x.). we have 0 and hence 2(1 — A)1z = —ye —Cc —B. Thus, p(z)

Az

— A),

If ço(x,) > 0, then x,

so that

e —C. Since —ye —C and

or

—C is convex, we have (letting a = + 1 —A) Aa1z = (1 — —C c —B. Thus A'a and Since —y p(—y) 1. (1 — A)(A1a

Now, z — x. = + 1). But co(z)

(1 — AXz —

+ —B,

y) so p(z — x,) = (1 — A)p(z — y) AJ implies that — ço(x,) Thus, p(z — x,) ço(z) — 40(x,), so

+1—

— 2)[ço(xo) + 11 = (1 — 2)[21a + 1]. z e J(x,) n B, a contradiction which completes the proof.

4. Further consequences of Lemma 1. It is possible to show the existence of support cones in spaces E which do not necessarily admit continuous linear functionals. Indeed, suppose that D is a bounded clOsed convex subset of E which does not contain the origin, and let K,, = {x: x = Ay,2 0, ye 1)). Let T be the convex hull of D U (this set is closed and bounded), let p = Pr and let so = SOD. Then K,, = {x: p(x) If andy in D), then while, on

_______

SUPPORT CONES AND THEIR GENERALIZATIONS

the other hand, if p(x)

then x e o(x)D c lCD. Since p is lower semi•

continuous and ip is upper semi-continuous, we see that K0 is a closed convex cone with vertex origin. (In terms of our earlier notation, K1, = 1, T, LEMMA 4. Suppose That Xis a closed subset of the complete space E and that I). If ço.,, is bounded above D is a bounded closed convex subset of E, with

on X, then for any z in x there exists x0 in X such that XO—Z€KD (Kb + x0) A

X=

and

{x0).

PROOF. (The condition that be bounded above on X is easily seen to be equivalent to the condition that there exist > 0 such that X and [A, oo[D be disjoint.) To prove the lemma, we simply note that X' = X n + z) satisfies the hypotheses of Lemma 1, so there exists x0 in X' such that x0 — z e and X' n J(x0)= {x0}. Suppose thaty€(KD+xO)n X; theny—z=(y—x0)+ (xo — z) is in K,, + Kb = Kb, so y€ X'. Furthermore, y — x0 K.,, implies that

p(y — x0) — x0). Since D is convex, ço(x0) + ç'(y — x0) ço(y), so that p(y — x0) ço(y) — ço(x0), i.e.,yeJ(x0). Thus, + x0) fl Xc X' n J(x0) = which completes the proof.

In a complete space E, each cone of the type K1, has a property which is sometimes called "regularity": If x. is a monotone net (i.e., x,, if a> fi) which is bounded above, then the net x,, converges (and its limit is the least upper bound for the set (x.}). This fact can be proved using the method of proof of Lemma 1. Lemma 4 can be used to show that if E is a lattice under

the partial ordering induced by Kb, then every set which is bounded above has a least upper bound. If E is a normed linear space, a possible choice for the function cc in Lemma 1 is the distance function to a subset Z of E: = inf {lI x — z II: z Z}. This function is Continuous and Lemma 1 can be applied using p = II .. II. This combination for cc and p can be used (in an obvious way) to replace the transfinite induction in Klee's theorem on Chebyshev sets in [5, p. 302]. (In this instance, take k = r.) REFERENCES

1. Errett Bishop and R. R. Phelps, The support fuiwtionals of a convex set, these Proceedings, pp. 27-35. 2. M. M. Normed linear spaces, Springer, Berlin, 1958. 3. V. L. Klee, Convex sets in linear spaces, Duke Math. 3. 18 (1951), 443-466. 4. Relative extreme points, Proceedings of the International Symposium on Linear Spaces, Jerusalem, 1960, Jerusalem Academic Press, 1961, pp. 282-289. 5. , Convexity of Chebyshev sets, Math. Ann. 142 (1961), 292-304. 6. , A con'ecture on weak compactness, Trans. Amer. Math. Soc. 104 (1962), 398-402.

UNIVERSITY OF CALIFORNIA, BERKELEY

CONVEX SPACES ASSOCIATED WITH A FAMILY OF LINEAR INEQUALITIES BY

H. PORITSKY 1.

IntroductIon. We consider a family of linear inequalities in the n

variables x1,

•, x,,,

of the form

+ xj,,(e) g(0),

x1fi(8) + x2f2(8) +

• •, g(8) are (n + 1) given functions of 0, a proper interval which is either finite,

where f,(6),f2(8),

00

and 0 varies over

0

or infinite. Of special interest is the case in which of the same period 1!: + 17) =f,(0),

g(0 + 11)

g are all periodic in 0,

= g(0).

in n-space to the "adThe inequality (1.1) restricts the point (x1, - •, missible side" of the locus xjf1(0) + = + which represents for each 0 an (n — 1)-flat. We consider the region R whose points satisfy (1.1) for each 8. The boundary of this region we denote by B. If n = 2, equation (1.4) represents a family of straight lines in the (xi , plane, while R consists of the points which lie to the admissible side of each of these lines. If n = 3, equation (1.4) represents a family of planes in 3space, and R consists of the points lying to the admissible side of each of these planes.

It will now be shown that the region R is always convex. To prove this, consider two points

,xi(1) , j, both in R. Every point P on the straight-line segment P1!'2 has coordinates given by ,-

(1)

p

P: [px1t" + (1—

p

the point P1 lies in R, its

0

coordinates satisfy (1.1):

+ ... + for every value of 0 in (1.2). Likewise, since P2 is in R, its coordinates, too, (1.7)

403

H. PORITSKY

404

satisfy (1.1) for every value of 8 in the same interval:

+ ...

+

g(8).

Multiply both sides of (1.7) for a particular 8 by the positive number p.0 < p < 1, and both sides of (1.8) for the same 8 by the positive number (1 — and add. It follows that the point P given by (1.6), for that p. satisfies (1.1) for the value of 0 under consideration. By allowing 0 to vary over its complete

interval, one proves that the particular point P for that p satisfies (1.1) for all 0. By then varying p over its interval, the proof of the proposition is completed.

It follows from the above that any straight line cuts B, the boundary of R, in at most two points. If it does Cut B in two points P1 , F2, then the interval P1P2 lies entirely within R, while the portions of the line outside P1)'2 lie outside R. If the straight line cuts B in just one point P1, then either the

half of the straight line to one side of P, lies inside R and the other half outside R, or else the line just touches the boundary of R at P1 and has only this point in R. Finally, if the straight line does not intersect B, then the whole line either lies outside R or entirety inside R. Similarly, if 0 is a point in the interior to R, any half-line OP through 0

either cuts B in one point, or else does not cut B at all, in which case the whole half-line lies in R. g(8) Throughout the following, it will be assumed that the functions are (real and) analytic in the (real) variable 0. Actually, much less stringent assumptions can be used for each theorem, for instance, that g be a class C

where

k is the highest order derivatives occurring in the statement

and proof of the theorem, and the reader will have no difficulty in supplying such—or even less stringent—conditions.

Thus in the above proof of convexity

of the region R, no derivatives occur, and the theorem is certainly valid if g are of class C"1; in fact the proof may be extended eYen foç 4iscontinuous ft, g; however, the subsequent statements regarding the intersections of straight lines with R would have to be examined very critically before their validity for discontinuous g could be assured. On the other hand, in the definition of the "envelope curve" C which appears below, derivatives of , g of order (n — 1) occur, and the assumption that f, g are of class C will suffice. Moreover, since, in the sections follow-

ing, various differential invariants of C are introduced—such as the curvature of C for n = 2, the curvature and torsion of C for n = 3—the existence of even higher derivatives must be assumed. However, as stated, we shall suppose that g are analytic in 0, and leave the extensions to broader classes of functions to the reader.

Actually, we are concerned with the intrinsic properties of the region R whose points satisfy the one-parameter family of inequalities (1.1). Such properties do not depend on any particular parameter 0, and 0 could be replaced by any other parameter = J(8), provided , g be correspondingly transformed into functions of &

While Euclidean differential invariants (such as arc length s) will be used

CONVEX SPACES ASSOCIATED WITH A FAMILY OF LINEAR INEQUALITIES 405

in the following, there is no intrinsic connection between the family of inequalities (1.1) and Euclidean geometry. No doubt many of the statements of the theorems can be given a broader foundation by introducing invariants under linear afilne transformations (which transform flats into flats and convex

regions into convex regions), but no attempt in this direction will be made in the -following. It is of interest to point out, however, that in the example considered in § 9, involving the conditions that a finite Fourier series be positive, Euclidean space rotations and Euclidean differential invariants do come in in a natural way. Of interest in describing B is the "envelope curve" C, obtained by solving simultaneously the n equations derived from (1.4) by differentiating both sides k times with respect to 0; k = 0, 1, -••, n — 1. The following is devoted

primarily to an attempt to describe the boundary B in terms of the curve C and its tangent and osculating flats of various dimensionalities. For ,z = 2 and for the periodic case (that is where (1.3) holds), it will be shown in § 2 that under certain auxiliary conditions on C this curve represents the boundary B, and R reduces to the interior of C, which is also the least convex area containing C. It was conjectured that for n > 2 and for the periodic case, R likewise reduces to the least convex solid containing C, under certain auxiliary conditions on C. However, this conjecture turned out to be wrong. There is a basic difference between even and odd dimensions. For odd n, the points of C, in general, lie outside R. For n = 4, for the particular example considered in § 9, the boundary B consists of intersections of pairs of planes (2-flats)

osculating C at two arbitrary points. The author's interest in a (periodic) family of linear inequalities arose in connection with a technical problem relating to the field at a great distance due to varying currents in n antennas. Such a family of inequalities may arise, however, in analysis, as illustrated by the following problem. Consider a polynomial in a = x + iy:

and let us inquire what the conditions are on the (complex) coefficients D1, ..., A, so that the real part of P be positive for a lying on and within the unit circle. Since the harmonic function Re [P(z)J

attains its optima over any region on the boundary of the region, it is sufficient that this condition be applied for j a I = 1. Putting

= Ak — iBk, there results (1.12)

which reduces to

a=

H.

406

(1.13)

A1 cos 0 + B1 sin 8 + ... +

cos nO +

sin nO

—

4,

which is of the form (1.1) in the 2n variables A1, B1, •-•, B. with f1(8) = cos 0, = sin nO, g(0) = —1/2. Thus (1.1) reduces to the condition f2(0) = sin 0, - .

that the Fourier series (1.12) represent a positive harmonic function. Strictly speaking (1.13) differs from (1.1) in that the inequality sign has been replaced by it could, of course, be restored to by changing the signs of both sides of (1.13), thus changing the definitions of , g to their negatives. However, this will not be done in the following, since most of

the descriptions and properties of the admissible region R could be used equally well if the inequality in (1.1) had been reversed ab initio. Similarly, the requirement that + D1c'j(z) + ... + 0 hold over any simply connected region A in the z-plane, where %Oo,SOj, . . ., (1.14)

are

Re [çog(z)

given analytic functions in A, leads to a one-parameter family of in-

equalities of the form (1.1), in the 2n constants A1, B1,. •, A,, B,,, where 0 is a parameter along the boundary of A. 2. Two-dlmenalonal cue. We consider the inequality (1.1) for n = 2: (2.1) x1fj(0) + g(0). Now we may plot (x1 , x2) in a plane. As shown in Figure the inequality

FIGURE 2.1

(2.1) restricts the points to one side of a family of straight lines, one such line resulting for each value of 0. It is evident from Figure 2.1 that the envelope C of the family of straight lines is of interest in describing the admissible region R for the point (x,, x,). To find the equations of the envelope C, we start with equations (1.4), which reduce to (2.2)

Putting

x1f1(O) + x2f2(8)

g(O).

CONVEX SPACES ASSOCIATED WITH A FAMILY OF LINEAR INEQUALITIES 407

0=0=00+40,

(2.3)

there results x1fj(6o) + x2f(Oo) = x1f1(00) + x2f(61) =

(2.4)

+ x,f(02) =

If one obtains the intersection point Pk of the lines 6 = 0,

0=

(k

=

1, 2, .•-) then, letting

limdO—'O, limkdO=6', will, in the limit, approach the envelope C. By subtracting

(2.5)

the points from each equation (2.4) the preceding equation, dividing by 46, and allowing 46 to approach 0, it is evident that the resulting point of intersection P,, will, under the assumption that 1.' g are analytic in 0 (or even that they are of class C'), approach the point P which, at 6 = + 0', satisfies simultaneously the two equations x1f1(0) + x0f0(0) =

(2.6)

= g'(6)

x1f11(8) +

These are linear equations in x1, x,, and if the determinant .0 of the coefficients does not vanish, they admit the unique solution: = F1(6) N1(0)/D(0), x2 = F2(8) = N,(0)/D(o), where N1(0) =

(2.7)

—

= — D(0) =f1(6)f(0) As 0 varies, this solution describes the envelope C. Summarizing, the envelope C is given by N1(8)

C: x1 = FL(0),

(2.8)

x2 = F2(6),

where F1, F2 are as in equation (2.7), and is finite provided D(6)

(2.9)

—f*(0)f(6) * 0.

Of interest in describing the relation of R and C is the concavity of C and its curvature, defined by (2.10) k = dço/ds, where ço is the angle that the tangent to C in the direction of increasing 0 makes with the positive xt-axis: ço

(1=

d/dO),

(2.12)

=

= tan1F2'/F11,

and ds is the length element oil C, defined by

ds=vd6,

H. PORITSKY

408

The angle ço may also be defined as = (2.13) with the x1-axis. since the normal to the line (2.2) makes an angle ± It will be seen from Figure 2.1 that if at 8 = 8' the envelope C is concave to the admissible side of (2.1), the portion of C near 8' forms part of the boundary B of R, unless as 0 varies sufficiently to either side, this portion of C is "cut off" by lying on the inadmissible side of member tines of (2.1). We shall now indicate some of the possible relations between R and its boundary B, and the curve C, but no exhaustive enumeration of all the possible cases will be made. Suppose first that 0 is confined to a finite interval (1.2), and suppose that the envelope C for that interval is finite, always concave toward the admissible side of the straight lines, and such that the total rotation of C, 4co =

(2.14)

ds =

= tan_hftlfl]0

is numerically less than it. Then, as shown on Figure 2.1, the point (x1, x1)

is confined to the region R bounded by the envelope C, and halves of the terminal lines (i.e., corresponding to the end points 0 = 0., 0 = of the

interval), proceeding away from C. If the same conditions regarding the envelope C hold, but the total rotation of the envelope lies between ir and 2ir, then R is bounded by C and finite segments on the terminal lines, as shown in Figure 2.2.

FIGURE 2.2

g(8) in (2.1) are periodic, of the same period (see Then the envelope C, if finite, is a closed curve. If C is equations (1.3)). always concave toward the admissible side and is a simple ctirve (i.e., if it does not intersect itself), then the region R consists of the interior of C, as

Suppose next that

shown in Figure 2.3. In Figures. 2.1, 2.2, 2.3, the condition regarding concavity

can be replaced by the statements: the curve C is always concave in the same direction, and one point P1 of C lies on the admissible side of the tangent to

Cat some other point P2. Stating the result illustrated in Figure 2.3 more formally, we have the following: THEoREM 1.

Let It, f. g in (2.1) be of class C";

suppose that they are

CONVEX SPACES ASSOCIATED WITH A FAMILY OF LINEAR INEQUALITIES 409

FIGURE 2.3

periodic in 6, with the same (smallest) period ii, and are such that

(F1')' + (F,') * 0,

(2.15)

(2.16)

D=

* 0,

—

(2.17)

=±2r,

(2.18) (2.19)

[F1(61)f1(6,) + F,(61)f,(O,) — 9(02)1 <

0,

are any two different fixed values of 6, not corresponding to the same point of C. Then C is a finite, closed, simple curve, always concave in the same direction, while the region R of points satisfying (2.1) consists of the points of C and its interior. where

By way of remarks, it will be noted that from (2.13) follows dç'

D

Hence, in view of (2.16), (2.17), D, ço', and k are always of one (and the same) sign. Hence, C is always concave the same way and, in view of (2.18), C is

a simple curve, i.e., it does not intersect itself. Equation (2.19) shows that equation (2.1) with 6 = 6, holds for the point on C corresponding to 81; this shows that the point 6 = lies on the admissible side of the line (2.2) at 6,. Hence, C curves toward the admissible side. On the other hand, if C is a finite, closed, simple curve, but only partly concave toward the admissible side, as over A1A,A, in Figure 2.4, but convex toward it elsewhere, as over then the admissible region R, jilt exists, is bounded by a portion of C and by the two straight lines tangent to C at the points of inflection, at which the curvature of C changes sign. On Figure 2.4, the admissible region is the region a,a,A,a,a1, free from shading

caused by drawing the lines tangent to C.

H. PORITSKY

410

Fiouas 2.4

Fiouna 2.5

Ficuaa 2.6

If k changes sign, but C, while finite and closed, intersects itself once as in Figure 2.5, then no admissible region may exist. Suppose that a finite, closed C is concave toward the admissible side, and that increases by 4ir as 0 describes its period. Then C must intersect itself an odd number of times (counting tangency as a multiple intersection). Figure 2.6 indicates a case with one point of intersection; Figure 2.7 with three points. In either case, R consists of the points which are surrounded twice as C is described once. = 6,r, an envelope C, with two intersections is possible, as shown For in Figure 2.8. However, for this example, no admissible region R is left, even

when C is concave toward the admissible side. If C satisfies the cohditions of Theorem 1 except for (2.19) which is reversed

FIGuRE 2.7

FIGURE 2.8

CONVEX SPACES ASSOCIATED WITH A FAMILY OF LINEAR INEQUALITIES 411

in sign, then C is convex toward the admissible side, then no admissible region R exists, that is no solution of (1.1) exists for periodic ,f2, g. This case is

shown in Figure 2.9 where only one tangent line to C, at P., is shown, and the inadmissible side of the tangent to P. is shaded. The tangent line at Po leaves the upper half-plane for the admissible region, but as P moves along C, from P0 to P1 corresponding to 4ço = the admissible region is completely cut off.

FIGURE 2.9

Suppose that 6=6 is a simple root of D, but that in (2.7) N1(6.) *0, N.(8.) *0.

Then as 6 the envelope C recedes to infinity in a fixed direction. The envelope, as shown schematically in Figure 2.10, consists of two parts, one to

R

FIGuRE 2.10

each side of. the asymptote corresponding to If the admissible side is below the lines, then as shown in Figure 2.10, R may be bounded by the lower portion of C and by parts of either or both of the Further possible singularities of C, not considered above, are cusps. The curve C will be said to possess a "simple cusp" at 0 = 0,, with edge parallel

H. PORITSKY

412

to the x1axis, if the Taylor series of F1, F, at 8 = 0. are given by = A,(8 — 0.)' + F1(8) — (2.15) + F,(0) — F,(8.) = B,(8 — where A, * 0, B, * 0. Such a simple cusp is shown in Figure 2.11, with the

R

FIGURE 2.11

at P., corresponding to 0 = 0. In this case, for the finite interval case, the boundary B of R may consist of a Part of C and parts of either or both but it will be noted that the neighborhood of the terminal lines 8 = 0,,0 = of the cusp P. is always excluded. If the admissible side on Figure 2.11 is below the lines, then a possible boundary for R is shown on Figure 2.11, as the boundary abCcd, consisting of a part of the envelope below the cusp and As stated above, the neighborhood of the parts of the lines 8 = 00,8 = cusp is completely excluded from R. cusp

3. Three-dImensional cue. We now proceed to the inequalities (1.1) for the three-dimensional case ,z = 3: x1f1(0)

(3.1)

+ x,f,(8) + x,f,(0) g(8),

the region R of the 3-dimensional (x1, x, , x,)-space satisfying this oneparameter family of inequalities. Equations (1.4) now become and

+ x,f,(0) + x,f,(8) =

(3.2)

yielding a one-parameter family of planes. The region R consists of points

(x1, x,, x,) confined to the "admissible side" of each plane of this family. Consider the planes (3.2), for the finite interval (1.2), corresponding to the sequence of values (2.3) of 8. Each successive pair of planes 8 = 8= will intersect in a line 4. Each line 4 will be cut by the succeeding plane

$=8,+, in a point F,,. On Figure 3.1 are shown schematically six planes of the sequence (2.3), the lines of intersectionl,,l,,l,,l,,l4of adjacent plane pairs; and the points P., F1, P,, P, of the intersection of 1, and 0,, and 8,, These points are also points of intersection of lo, I, and 84, and 1, and 11;

1,;

1,; 1, ,

1.

As the limits (2.5) are taken, the lines 4 approach a ruled surface S, and the points P,, approach a curve C, the "curve of regression" of S. that is the

CONVEX SPACES ASSOCIATED WITH A FAMILY OF LINEAR INEQUALITIES 413

FIGURE 3.1

curve C to which all the lines 1 of S are tangent. Thus the ruled surface S is a developable surface, with its lines 1 all tangent to C, while each plane of the family is an osculating plane of C. We shall also refer to C as the "osculating curve" of the family of planes (3.2), because the planes (3.2) are the osculating planes of the curve C. The proof of these statements will be briefly indicated. Start with equation (3.2) for the planes 0 = of the sequence (2.3): (3.3)

xIfi(Oh) + x$fl(Ok) + xsfs(Oi)

9(Oh)

= 0.

The line ik is the intersection of the planes Hence, it satisfies (3.3), and the linear combination of the equation (3.3) for °h and for 34 (.)

L(Ok+i) — L(Ok)

-, —

where L(Ok) is the left-hand member of (3.3).

o

As the limits (2.5) are taken,

equations (3.3), (3.4) approach the first two equations x1f1(0) + (3.5)

+ x1fa(0) =

x1f11(8) + x1f11(0) + xsfs'(O) =

= + evaluated at 0, + 0', provided the matrix of the coefficients of x11x,, x, in these two equations is of rank 2. At the same time, approach the family of lines 1 represented by these two equations. The point Ph is the intersection of the planes +

(3.6)

= 0,

= 0,

= 0.

From the first two equations follows that Ph lies on ik and that, as the limits (2.5) are taken, its limiting position (if any) will lie on 1. The second difference of equations (3.5), 3 (.7)

L(Ok+,) — 2L(Ok+I) +

—

0

H. PORITSKY

414

is also satisfied by Pk. Taking the limit (2.5) of (3.7), one obtains the third equation (3.5). If the determinant D(0) = fj'"(O)f of the coefficients of equations (3.5) does I

not vanish, then their unique solution for each 0 is given by = F1(0) = N1(0)ID(8) (3.8) where N1(0) is obtained by replacing the jth column of D by

Equations

(3.8) represent the coordinates of P, the limiting position of Pk, and are the equations of the curve C. Summarizing, the lines 1 of the ruled surface S are obtained by solving simultaneously the first two equations (3.5). The curve C is obtained by solving simultaneously all three equations (3.5), yielding, for x1, x2, x3, the ratios of Wronskian determinants of triplets of the functions 11,12 ,f3, 9' rep. resented by equations (3.8). To prove that the ruled surface S formed by the lines 1 (described by the first two equations (3.5)) is a developable surface,

it is sufficient to prove that (3.9)

urn d(O,0+40)

i.-.o

48

=0,

where d is the minimum distance of the lines 1(0), 1(0 + 40). In this connection,

it will be recalled that for lines 4,

intersect each other so that for these

lines d(8k, Ok+1) vanishes. Hence, (3.9) is valid for the limiting process (2.5).

To show that the planes (3.2) form the osculating planes of the curve C, it is sufficient to show that (3.10)

where

tim n(0, 0 + 40)1(40)2

(i8)-.O

0

n is the normal distance of the point P(8 + 40) on C from the plane

of the family (3.2) corresponding to 0. Equation (3.10) is proved in a similar manner to (3.9) by recalling that for the point Ph and the plane Oh the normal distance n vanishes. We turn to Figure 3.1. Adjacent lines 4, form angles 40j with each other; the ratio where is the distance P1P1+1, approaches the curvature k1 of C under the limiting process (2.5); adjacent planes 0, 0,41 likewise form dihedral angles with each other, and the ratio dçoJ4s1 approaches its torsion k2. The curvature k1 will be defined as positive near 0 = in absence of cusps on C it will remain positive. The torsion k2 is positive when

it represents a right-hand screw motion as P advances in the direction of increasing 0 and arc length s along C. Suppose that the admissible side of the planes on Figure 3.1 is toward the reader. It will be seen that the planes (4, 01, cutting each other in the line la., cut space into four "dihedrals" or "quadrants," only one of which is "admissible," since only it lies on the admissible sides of and 0,. Thus at most only half of the plane to one side of 4, toward the reader, forms part of the polyhedral approximation to the boundary B of R. The planes 00, 81 , 82 cut space into eight "trihedrals" or "octants," each pass-

ing through P., but only one of these—a rather "open" one—lies to the

CONVEX SPACES ASSOCIATED WITH A FAMILY OF LINEAR INEQUALITIES 415

in the schematic Figure , 0,; it is shown as admissible side of of intersection of 00, 0,. 3.2, where P,E is along the line

FmuRE 3.2

The introduction of 0,-plane may cut off that part of the admissible trihedral of Figure which lies on the inadmissible side of 0,, as well as parts of its boundary. Similarly, the planes 04, 0,, may cut off portions of the previous admissible polyhedral region and its boundary, and add new facets to the latter. It is evident that at most only one-half of each line I, forms part of the l*undary. For 0 restricted to a finite interval, if neither k1 nor k2 vanishes, it would appear from Figure 3.1 that the boundary B of R consists of portions of the end-planes, the curve C, and the portion of S to one side of C. However, neither Figure 3.1 nor Figure 3.2 indicates the complete intersections of the various planes. As shown in Figure 3.2, the plane 0, may cut off part of along P,E, and further planes 0,, cut off even more. 4. DifferentIal space curve geometry. In connection with curve C, we now proceed to recall certain differential geometric concepts and relations of space curves.

It will be noted that the (unit) normal (vector) to the plane (3.2) has components proportional to

fi:f,:f,;

I,.

Since 1 satisfies the first two equations (4.1), the (unit) vector along the line I has components proportional to (4. 2)

,,

11

J2

JO

• •

11

IS

JO

.

Jiel

'

the planes (3.2) are the osculating planes of C, the (unit) vector normal to (3.2), whose components are proportional to (4.1), is also the binormal I, to Since

C. The unit vector

tangent to C lies along 1. and its components are

proportional to the second order determinants in (4.2). The direction of h is chosen to correspond to increasing 0; that of I, will be specified presently. The principal normal I is given by

416

H. PORITSKY

(4.3)

i,=l,x11.

Consider the curve C whose equations, in parametric form, are given by (3.8) which we repeat, (4.4)

C: x1 = F1(8), x, = F,(0), x, = F,(0),

or in vector form by (4.5)

r = (.r1 , x,, x1) = [F1(0), F,(e), F,(0)J

where r is the position vector OP of any point P on C, 0 the origin. Let s be the arc-length of C. Then the following relations hold for C: (4.6)

as in (4.2), is the unit tangent vector of C, pointing in the direction where of increasing $ and 0. Hence, —

—r'/v

+ (F2')' + (F,')1 —

—

ds

= V(F)' + (F,')' + (F,')' dO = vd(I. the following Fresnet equations

Furthermore, between the vectors h, hold: d11

(4.8)

4k =

+k,I,,

—k111

= a where, as above, k1 is the curvature, k, the torsion. If I, is chosen in the direction of dl jdO, it follows from the first equation (4.8) that k1 will be positive; this is always possible, provided dIJde does not ever reverse in direction. The direction of I, is chosen as in (4.3) so that 1,, 1,, 1, form a right-hand system, and equations (4.8) then determine the sign of k,. Its sign agrees with the sign convention stated above.

In terms of the vector r and its 0-derivatives, we obtain from (4.4)-(4.8) ('= dide)

v=jr'I=ds/dO, (49)

r" =

+ i,k1v',

r" = k,kjv'Is + where, in the last equation, - - - refers to h- and 1,-components. Hence, follows x r" = v'k11, [r'r"r"] =

CONVEX SPACES ASSOCIATED WITH A FAMILY OF LINEAR INEQUALITES

417

where label indicates the triple scalar product a b x c. To obtain k1 , k, in scalar form, we introduce the matrix

F (4.11)

F,' F,"

F,"

F1"

Frn"

of

components of r', r", r". It will be seen that I, lies along the vector

given by the first row; the vectors corresponding to the first two rows of this matrix are both parallel to the osculating plane; I, lies along the vector whose

components are the second order determinants formed from the first two rows. The above equations yield = (F,')' + (F,'f + (F)', = (r' x r")' (4.12)

F1'

= F,"

FF

= F," F," I

F:

F,'

F:,

F:'

2

2

I,, 1

FFI'

+

F," F:' r

+

F,"

F:'

I'

l,i

From (4.5)-(4.9) follows

d'r_

dr

d.c

(4.13) ds

=

''

+ k,(—k111 + k21,).

Hence, the following Taylor expansion in s (4.14)

r = r0 +

sh + -'I: +

holds

at a point P, of C:

+

+ k,k,k] +

where s is the distance on C from Po, and indicates d/ds. The functions constitute the point coordinates of C; the functions f./g may be considered to constitute its plane coordinates. The curvatures k1 , and vectors 1,, 1,, 1, formulated above in terms of may, in turn, be expressed in terms of g by substituting from (3.8). However, they may also be expressed simpler in terms of by means of (4.1)-(4.3) and application of the Fresnet equations (4.8). 5. The boundary B for the finite Interval cue. Example. Suppose that, for n = 3, 0 is restricted to a finite interval, equation (1.2), and suppose that the curvature k, of the osculating curve C is positive and that k, * 0. Then it was conjectured in § 3 from inspection of Figure 3.1 that the boundary B of the region R to which the inequalities (3.1) confine the point (x1, x,, x,) consists of part of the ruled surface S formed by halves of the tangent lines to C, proceeding from C in the direction of increasing s (and 0), and the two terminal half-planes of C, at pointing away from S. This may be expected to be true provided that the interval — 0, is not too large, so as

H. PORITSKY

418

to cause S to intersect itself. We shall now examine the region R and its boundary B for the case when C reduces to a segment of a circular helix. It turns out that for this case, B does consist of parts of the terminal plane halves and Parts of the lines I forming the ruled surface S, but these portions differ from the conjectured ones, inferred in §3 from Figures 3.1, 3.2. Consider the one-parameter family of inequalities (5.1)

Solving equations (3.5), one obtains for C the segment

O
(5.2)

of the circular helix (5.3)

x1=acosO,

x3=caO.

This helix lies on the cylinder (5.4)

4+

4 = at

and has the lead angle tan' c. The ruled surface S, formed by the tangent lines to C, is now an involute

FIGURE 5.1

CONVEX SPACES ASSOCIATED WITH A FAMILY OF LINEAR INEQUALITIES 419

helicoid.

To find the nature of B, we consider its sections B by the planes x8 = const = E

(5.5)

normal to the helical axis. As is well known (see, for instance, reference [41), such a section is an involute of the circle (5.4), (5.5) with cusp at 6' = 0' = Elac.

(5.6)

Figure 5.1 shows the quarter-circle projection AB (A is also labelled P1, D1)

The tangent lines to C which constitute S of C on a plane of constant project into lines tangent to AB; only one such line, J0J1J2J, is shown on Figure 5.1. Also shown there are three involutes, corresponding to the inter, sections of S with the planes (5.5) for three values of 0' in (5.6), satisfying spectively.

<

<

= 0,0<

<2r/2, with cusps at

P1, P2, re-

The angle 0' at the cusp is proportional to the height of x3 and can be arbitrary, but the pertinent lines tangent to the involutes, that is, the intersections of the osculating planes with the planes (5.5), make contact with the involutes -only between A and E,, corresponding to the initial and terminal tangent lines to C. The section R5 of the region R by each plane (5.5) lies to the right and above the tangent lines to the respective arcs DE4 of the involutes. The boundary B5 of R5 in the plane (5.5), (5.6) thus consists of: (a) the half-line DiG or PiGi along the section of the initial plane (5.1) (0 = 0) with the plane under consideration; (b) the part of the involute arc D1E, to the right of the corresponding to intersection of the and (c) the half-line line terminal plane (5.1) (0 = with the plane in question. Thus for 0 (=0) by for by GZFEIH,, where F is the intersection of the tangent at A with the involute, other than at A. In particular,- the point P, corresponding to lies on the inadmissible side of the line D1G1. Hence, P1 cannot form part of R,,, nor of R. This is also evident from the fact that P2 is the cusp of the tangent lines to the involute; it was shown in § 2, Figure 2.11, that the neighborhood of a cusp of C always lies on the inadmissible side of some of the tangent lines of C. It is concluded from the above that, except for the end point 0 = 0, no

point of C lies on the boundary of R. Thus, the conjecture regarding B for n = 3 stated in § 3 does not quite hold for this example; nevertheless B does consist of parts of the tangent lines to C and parts of the two terminal osculating planes.

Conjecture regarding the boundary B for the periodic case, for n = 3. g periodic no terminal planes exist, and C, if finite, is a closed curve. One may, therefore, conjecture, that at least for this case, by analogy to n = 2, the inequalities (3.1) might confine , x2, x,) to a finite volume R which is the least convex solid containing C, provided proper restrictions apply to k1,k1. 6.

For

H. PORtrSKY

420

The following attempts at formulating such theorems turned out to be Unsuccessful. Yet the attempts made and the negative results obtained are of some interest in themselves and will now be described. The failures to connect the region R for a periodic family of inequalities (3.1) with the least convex region containing C stem from two causes. One has shown up in the example given in § 5, and is due to the fact that each point P of C (for k1 , k * 0) always lies on the inadmissible side of the Osculating planes of C to one side of P. The other cause of failure is that the particular restriction (positive curvature and torsion for C) made by the author to create auspicious conditions for the conjectured theorem turned out to be so severe that no periodic curve C satisfying them exists.

From this brief summary and the examples in § 7 some features of the boundary B for the periodic case (3.1) begin to emerge, and are hinted at at the beginning of § 7. In the following attempt to prove the conjecture stated at the beginning of this section for the periodic case (3.1) and a finite periodic envelope curve C, the following restrictions on C will be assumed: (a) The curvature k1 and torsion k2 do not vanish and are bounded. (b) The end point of the rotation vector (6.1) = k111 + k211 deseribés a simple closed curve dM in the (ii, There is no loss in generality in putting assumptions (a) in the form 0 < const < k1 < 00, 0 < coust < k <00.

(6.2) (6.3)

If the torsion k0 were negative throughout, a reflection in a plane can always be used to convert C into its mirror-image curve, with positive torsion. If k changes sign at a point P0 on C, the osculating plane comes up to a limiting position at P0 and starts turning back; the osculating plane at P0 could then form part of B. We now proceed to show that under assumptions (a), R cannot consist of the least convex solid containing C. Choose axes as implied in equatjons (4.13)7 with a particular point P0 on C as origin, with the x1-, x1-, x0-axes, respectively, along i1, h, h and let s, measured from P0, be used as the paranleter 0. Then, at Po in (3.1), g = 0, (6.4)

0,

so that at P0 (for s =

0)

0,11

(3.1) becomes

f0(0)xs <

0.

From (4.14) follows for the x5-component of r: = k1k2s0 (6.5) +

Hence, if f.(0) > 0, then (6.4) will hold only for negative s of sufficiently small numerical value, —€ < s < 0, but will be violated for positive s, 0 < s < *, for some s > 0. If < 0, then (6.4) will hold for positive s, 0 <s < s, and will

CONVEX SPACES ASSOCIATED WITI-I A FAMILY OF LINEAR INEQUALITIES 421

be violated for negative s, —e < s < 0. Thus for si sufficiently small, the points of C either to one side or the other side of P0 lie outside the admissible region R. Around each point of C, there is thus a small neighborhood which also lies outside R. Since P0 is essentially any point of C, it follows that under the above assumption no point of C can lie on the boundary of R.

To continue the search for the nature of the boundary B of the region R for the periodic case (periodic (3.1), and finite periodic curve C), we now add the above restriction (b).

To explain this restriction, we consider on a unit sphere Z (see Figure 6.1) the motion T of the representative spherical triangle whose vertices are

FIGuRE 6.1

the end points of , h. The Fresnet equations (4.8) imply that if s is considered as time, the motion T of the triangle relative to fixed space F can, at any instant s, be considered as a rotation represented by the rotation vector given by (6.1), which evidently lies in the plane determined by For periodic k1, k8, the vector describes a closed curve d1 (see Figure 6.1) in the (6.6)

,18)-plane, whose equations are given by d1: = k2(s) ,

are axes fixed relative to T and directed along the vectors It is of interest to note that the arc lengths of the curves c1, c2, c3 described by (the end points of) , i8 are given, respectively, by1 where

(6.7)

do=/e1ds=wsinçods, dc/' = dço = k8ds =

da'=Iwids,

This follows readily from equations (4.8). A more explicit statement of the restriction (b) is—the periodic plane curve

is a simple curve (that is non-self-intersecting), and that it is described The notation dii,

here is used in the same sense as in the text following Figure 3.1.

H. PORITSKY

422

once as 0 completes its period ii. Part of assumption (b) was suggested by the first example in § 7. It turns out, however, that THEOREM 2.

There exist no closed curves Cfor which assumptions(a), (b) hold.

The following proof of this theorem is kinematic. We start by recalling a theorem that spherical motion, that is the motion of a rigid body M relative to a fixed space F, with a fixed point 0, may be fixed in the produced by proper rolling without slipping of one cone moving body M, on another cone C, fixed in space F; the vertices of both cones are at 0. The cone CM is known as the "poihode"; the cone C, as the "herpoihode". The motion at any instance is, of course, also represented by The generators of the herpoihode C, lie along the vectors a rotation vector w. Likewise, the generators of the poihode CM lie along the vectors w, that is, along the vectors which at the time s come to occupy the positions coincident with w. (If the inverse motion were considered, that is, the motion of

F relative to M, with the latter assumed fixed, then

is replaced by —w, and —CM would become the herpoihode, —C, the poihode.) These cones cut I in two spherical curves CM, and the motion may be described by means of rolling without slipping of CM OH C,-.

it will be seen that in the present case the poihode cone CM reduces to the (i,, i3)-plane, and the curve CM to the arc AB of angle < ir/2, see Figure 6.2.

FIGUI1E 6.2

Under the assumptions regarding d% (see equation (6.6) and Figures 6.1, 6.2), is described twice by u, the unit vector along w: (6.8)

u

.41 a'

I=

cos go

+

sin go,

first from A to B, then from B to A. The herpoihode curve c, is indicated schematically on Figure 6.2; it consists of two curve segments, each of length joined at two cusps Li, These cusps correspond to the positions of- u

CONVEX SPACES ASSOCIATED WITH A FAMILY OF LINEAR INEQUALITIES 423

come to coincide with the extreme points A, B, where p attains its maximum (at B) and its minimum (at A). In the time element ds the angle of rotation (of c,) IS which

I

w I ds,

and is also equal to (6.10)

is the spherical curvature of c,, and w the velocity with which where c1 , c, are being described: w = du/ds I = I

Since, under the assumptions (6.1), (6.2), never vanishes, it follows that the curve c, is always convex as shown in Figure 6.2. After these preliminaries, we start on the proof proper by recalling equation (4.6). Upon integration this yields iL (6.12)

rI

Jo

dL

= I.10 I1ds

If C is closed and of length L, then r(L) =

r(O),

and there results

(6.13)

Hence, (6.14)

rz .10

i1ds=

çL

(j.11)ds=0,

Jo

where j is any fixed (unit) vector. A contradiction will• be established by showing that there exists a particular vector j for which the equation (6.15)

j•ij>O

always holds. Hence, (6.13) is violated, and periodic k1, requirements cannot lead to a closed curve C.

satisfying the above

We choose for j the vector Ja on Figure 6.2, which lies at a cusp furthest away from A in the position indicated. In the actual motion, C, is stationary (in the space F), and CM, along with the rest of the representative triangle, rolls without slipping on c,. However, for purposes of examining the inequality (6.15), it will do equally well if we consider the inverse motion, in which c1 and the spherical triangle are stationary, and c, rolls on CM. Starting with the position indicated in Figure 6.2, as the point of contact moves toward B, the vector will move away from A. Its maximum distance will be attained

when j,, coincides with the point B, and at that time the angular distance between j and 11 will be less than 42. As the rolling proceeds, contact is made between the second loop of c,, and JB will approach i1, its minimum distance being attained when iA coincides with A. Its distance will then increase until the period of the motion will have been completed at the initial position shown in Figure 6.2. Since the angle between h and Jo is always

H. PORITSKY

424

we have established the equation (6.15) for j = j8. This comless than pletes the proof.

It is of interest to note that in the direct motion T of

relative to c,,

are described by points of a great circle the curves c1, c8 described by (containing CM), which rolls without slipping on CF. Hence, C1 , are spherical involutes of c,.. On the other hand, c2 is the polar curve of c,; in other words, it consists of the poles of positions of the above great circle, that is, poles of the great circles tangent to c,. It is readily seen that for c, having the appearance shown in Figure 6.2, the curve c1 has the appearance as shown in Figure 6.3, and intersects itself at least once. I am grateful to the referee for pointing out that space curves whose torsion does not change sign, as well as curves fulfilling other differential geometric inequalities, have been studied before (cf., e.g. [5; 6; 7)).

FIGURE 6.3

7. Examples. From the last result established in § 6, it follows that for a closed-space curve C to exist, the curve dM, given by equation (6.6), must be either described several times or be self-intersecting, or if k1* 0, then must be both positive and negative. It is conjectured that if k2 changes sign, then parts of the osculating planes at the points at which k2 = 0 form part of B, the rest of B consisting of portions of the tangent lines to C. An example2 will now be given of a closed curve C for which conditions (a) in § 6 hold, but for which dM is described back and forth several times before C is described once. For this example the region R is vacuous, that is no points lying to the admissible side of the osculating planes of C exist. Let r, z be cylindrical coordinates and consider the toroidal surface

described by

r=a+bcost9, x3=bsinO, 0
CONVEX SPACES ASSOCIATED WITH A FAMILY OF LINEAR INEQUALITIES 425

(a + b

sin 0.

We choose for C the closed curve lying on the toroid and obtained by putting = 0/n, (7.3) where n is a positive integer, n > 2. As q' increases by 2w, 8 will increase by 2nw, and the curve C will close, having gone once around the x3-axis and having wound n times around the toroid. It is evident that the portions of C corresponding to the increments 40 = 2w, = 2w/n, (7.4) in 0 and can be made to coincide with each other by carrying out rotations and the torsion k2 of amount 2w/n about the x3-axis. Hence, the curvature of C are periodic in 0 of period 2w: (7.5)

k,(0 + 2w) = k1(0), k3(0 + 2w) =

Moreover, both k1 and

are even about 0 =

0

and 0 = w, the values of 0

which correspond to points at which C crosses the plane X3 = 0. Consequently,

the curve d, collapses in the present case into a curved segment which is described back and forth over the interval 40 = 2w, and this is repeated n times before the curve C is completed. For properly restricted b/a (b must be less than a in order that the toroid should not intersect itself) and n, both k1 and k2 can be shown to be positive. Hence, C is a closed curve with positive 1i1 for which the curve dM, equations (6.6), is described back and forth n times. As regards the region R lying to an admissible side of the osculating planes

of the above curve C, it can be shown that no such region exists, since all of space is swept out by the osculating planes as C is completely described. Therefore, every point- of space will lie on the inadmissible side of some of the osculating planes. At this point, one becomes curious if non-vacuous regions R can ever exist

for any one-parameter periodic family of inequalities (3.1). That such is actually the case can be seen from the following example. In (3.1) put = sinacosO, f2(6) sjnasjn0, 76 fs(0)=cOsa, g(0)=a>O, O
(.)

(7.7)

OP:x1=af1,

x3=afs,

describes a right circular cone C1 of semi-vertical angle a and altitude a cos a (see Figure 7.1). The planes (3.2) now reduce to the family of planes which are perpendicular at P to the generators OP of this cone. The region R on the admissible side of these various planes lies below the cone C2 obtained by

H. PORITSKY

C2

FIGURE 7.1

revolving the line DP, normal to OP (see Figure 7.1) about the x8-axis. The solution of the first two equations (3.5) leads to the lines generating the cone C1 just described. However, upon solving all three equations (3.5), it is found that the curve C is now reduced to the single Point D. In a sense, the example is, therefore, degenerate. However, by replacing the function g by (7.8)

g(0) — a

+f(0),

where f is a periodic function of sufficiently small numerical value, a family of planes will be obtained with a non-singular osculating curve C and with an admissible region R whose boundary B is not far removed from the conical surface C2. 8. Linear inequalities for general n. We now proceed to consider the inequalities (1.1) for a general n, and will make certain conjectures relative to the solution of these inequalities. The constants x1, x1, - ., x,, in (1.1) may be viewed as the coordinates of a point in n-dimensional space, while the inequality (1.1) for each value of 0 restricts the point to one side of a proper (n — 1)-flat, that is, a linear space of (n — 1) dimensions. Thus the region R whose points satisfy the inequality (1.1) for each 0 consists of points lying to one side of a family of (n — 1)-flats. A geometrical and algebraic analysis somewhat similar to that of § 3 can be formulated. By considering the (n — 1)-flats corresponding to values of 0 given by (2.3), one obtains a sequence of (n — 2)-flats which form the inter-

sections of adjacent (n —

1)-Rats

of the sequence (2.3). Similarly, intersections

of successive pairs of these latter will form a sequence of (n — 3)-flats, etc., till one finally arrives at point intersections. In the limit, as 40 approaches

zero, the points lead to a skew n-dimensional curve C, while the flats

in

question consist of its osculating lines, planes, -., (n — 1)-flats. In algebraic form the equations of these flats are obtained by starting with .

x1f1(0) + x2f2(0) + .-. +

—

CONVEX SPACES ASSOCIATED WITH A FAMILY OF LINEAR INEQUALITIES 427 and

differentiating it a total of (n — 1)-times, obtaining the n equations: + + .•• +

x,f1(O) + (8.2)

-•

= + •-• + Assuming that the matrix of the coefficients is of rank n, the solution of the first k equations leads to a flat of n — k dimensions, while the solution of all n yields a point. By varying 0, the last point describes a curve C, (8.3)

of which the flats (8.1) are the osculating (n — 1)-flats. The functions are obtained as ratios of two determinants formed from the coefficients in the equations (8.2).

by r(O), we obtain for the unit tan-

Denoting the position vector gent

=

(8.4)

L.,

ds = vd8,

An n-dimensional curve C possesses not one but (n — •

. -,

These

1)

curvatures k1, k2,

a'elated to the curve C, its arc length s, and its unit

tangent vector i1 by

of (8.4) and as follows: di1

a

=

k1i2,

d12 —

— — —k1i ds (8.5)

d13

=

ds

— k2i2

——

+ k311

1. — "n_un_I

ds

is the unit tangent vector to C; lu , 1,, b, -• form a system of n orthogonal unit vectors, of which the first k are parallel to the limiting position of the k-flat passing through a point of the curve C and k — 1 "adjacent points," as the latter approach point P. Equations (8.5) correspond to the Fresnet equations for a curve in 3-dimensional space. They were obtained by the author independently a number of years ago, but they appear to be well known. By analogy to (4.14), the vectors i1, -. -, can also be determined by the requirement that the Taylor series for C near any point P1 have the form Here, as stated above,

r=r0+i1(a11u+

•]

+ 121a2!u2 + (8.6)

+ 13[assu3+

+. +

- -]

-. -)

-

+•

- -]

H. PORITSKY

428

where u = 8 — 0,, where the dots •-- following Uk indicate terms of higher if non-zero, may be chosen to be positive. powers, and where a,, , By measuring s from P, and transferring, parameter equations (8.6) may be given a form similar to (4.14) stk1

(8.7)

= (k,k2---

+

m n, are expressed in terms of k,, . •, and their derivatives, where the origin is at P,, and the xk-axis is along lk at P1.

where the coefficients of

In terms of the curve C, assumed finite, if 0 is confined to a sufficiently small

finite interval, one may conjecture that for positive k,, . . ., the region R, whose points satisfy the inequalities (1.1) for every value of 8 in its interval, is bounded by portions of the osculating (n — 1)-flats at the terminal points of C, and parts of the osculating k-flats; k = 1, 2, n — 2. A similar conjecture, but without the terminal osculating (n — 1)-flats, was considered for periodic g. A further conjecture considered, based on n 2, was that, for n > 2, under proper restrictions on k1 and C, the region R coincides with the least conve,C region containing C.

However, the results of 6, 7 show that the last conjecture is not true for n = 3. It may be shown, by means of the last equation (8.7) and using a proof similar to the one used in § 6 and based on equation (6.5), that for odd n the curve C crosses its osculating (n — 1)-flats. Hence the points of C itself are excluded from the boundary of R. References [5; 6) suggest that k,,-, must change sign for a closed curve C in for odd n; it is conjectured that parts of the osculating (n — 1)-flats at the points of vanishing will form part of the boundary B of R. That for even ii the last conjecture does not hold follows from the example of § 9 (positive finite Fourier series), where it is shown that R extends beyond It is shown in § 9 for the same example, for n 4, that the section of the boundary B of R, by each osculating plane to C, is bounded by an ellipse whose points correspond to the intersection of that plane with every other osculating plane to C. The general relation between B and the osculating flats of C for the example of § 9, for even n > 4, is also briefly outlined. 9. Example for periodic case for even dimensions. We shall now consider in detail the example indicated at the end of § 1, namely, the conditions on the coefficients for a finite Fourier series to be positive: (9J)

4- +

A1 cos

The coefficients A,,

8 + B1 B,,

sin

-•,

0 + -..

B,., take

+

A,., cos mO + B., sin mO

0.

the place of the variables x1, - - -, x,, and

_______________ CONVEX SPACES ASSOCIATED WITH A FAMILY OF LINEAR INEQUALITIES 429

the

problem reduces to a one-parameter family of linear inequalities in a of n — 2m dimensions.

(Euclidean) space

Equations (8.2) for the curve C now become

A1cosO+B1sinO+ ••• +A.,cosm0+Bmsinm0 —

A1

sin 0 + B1 cos 0 — 2A2 sin 20 +

—A1 cos0 — B1 sinO — 22A2 cos 20 — A1 sin 0 —

(9.2)

B1

—

cos 20 + sin 20 —

22B1

= 0, •.. = 0,

= 0,

cos 0 + 2'A2 sin 20 —

2°'"LA1sin2O —

B1cosO]

+

+

B2

sin mO —

cos 20]

cos mO)) =

0.

They can be simplified by introducing the variables = A4cosk8 + B4 sin kO,

(9.3)

—A4sink0+B4coskd,

whereupon equations (9.2) are transformed into linear equations with constant coefficients in Uk, V4. The solution of these equations yields —

(94)

c—

— (m — k)! (m + k)!

= 0. (Actually C4 depends also on m.) Substituting in (9.3) and solving (9.3) for

A4,B4, there results for the curve C C: A4 =

C4

cos ke,

B4 =

C4

sin kO.

Hence, the space curve C is closed and is given by equations (9.5), (9.4). Alternatively, equations (9.2) for any constant value of 0, 8 = a, may be solved by introducing the trigonometric polynomial

f(0)=A1cosO+B1sinO+ "- + B,,1sinm0,

(9.6)

corresponding to the left-hand member of the first equation (9.2), and which satisfies the condition of zero mean (9.7)

f(O)dO Jo

and

= 0,

noting that equations (9.2), for 0 = a, can be put in the form

(9.8)

f(0) Ie=.

=

=

0,

...,

= 0.

These conditions are satisfied by the trigonometric polynomial of "degree" m: (9.9)

where

f(8) =

—

++

sin2hhl(°

—

= ço(0 — a)

H. PORITSKY

430

22m- i(

Fm

(9.10)

m1

)2

(2m)!

The Fourier expansion of çt(O) by the binomial theorem yields: 1

2"

—

)

2i (9.11)

=

{cosmo

—

(2m)cos(m

— 1)8

2m(2m

+

+

+

1)

(m — 1)!(m

cos(m — 2)5

+ l),COSO].

The coefficient of cos kO agrees with Ak, Bk in (9.5) for 0 = a = 0. Similarly, the expansion of ço(O — a) as a trigonometric series in 0 yields coefficients that agree with (9.5), (9.4) for 8 = a. Corresponding to an arbitrary point P(A1, B1, ..., B,,) of we introduce, similarly, the trigonometric polynomial f(0) given by (9.6). Since f(S) satisfies the zero mean condition (9.7), its minimum fmin is negative. One of the following three alternative cases, respectively designated as cases (a), (b), (c), must hold: case (a) (9.12)

fmln

=

case (b)

—4,

case (C)

In case (a) (i.e., when the upper inequality in (9.12) holds), the left-hand member

is positive for all 0, the curve

of

y=f(0)

(9.13)

in the (0,y)-plane lies above the line (9.14) and

y

= —-4,

the point P lies in the interior of R. In case (b), the curve (9.13) lies

above the line (9.14) except for k values of 0, 0 =

.

. •, 0,,,

where 1

k

at which it contacts the line from above; the left-hand member of (9.1) is positive for all 0 other than 0,, •, 0,,, at which it vanishes; the point P now lies on the boundary B of R. In case (c), the curve (9.13) crosses the line (9.14), the inequality (9.1) fails to hold for some 0, and the point P lies outside R.

It is evident that the curve (9.15)

y=f(0—a),

obtained by translating the curve (9.13) in the (0, y)-plane a distance a in the direction of the 0-axis, has the same minimum as f(S), and belongs to

the same case (a), (b), (c) as f(0). Since

+ -.• + B,,, sin m(O — = A,(cos0cosa — sin Osin a) +

f(0 — a) = A, cos (0 — a) (9.16)

+

a)

ml) cos ma + cos ml) cos ma),

CONVEX SPACES ASSOCIATE!) WITH A FAMILY OF LINEAR INEQUALITIES 431

f(0

— a)

corresponds to equation (9.6) with Ak, = Ak cos ka I- Bk sin ka', Ta(P).

The

Bk

replaced by

= — Ak sin ka + BkcOSka,

k—1

•

—

transformation T represented by (9.17) constitutes an orthogonal

which consists of rotations of amounts a, 2a, •••, ma in transformation of the m mutually orthogonal planes containing the A,-, B,-axes; the A2-, 82-axes;

•.•; the

Bm-axes.

If a is varied, T forms a continuous periodic group

under this of transformations. Thus, any point P generates a curve group, every point of which belongs to the same case (a), (b), (c) as P. then P corresponds to the point 0 0 on C, namely to If f(0) reduces to

P0:A,=C,,B,=0, "•,Am=Cm,B,,,=0. From the form (9.9) for a = 0, it is evident that ço(O) belongs to case (b), so that P0 lies on B, and that the curve (9.13) now makes contact, from above, of order 2m with the line (9.14) at 0 = 0. From (9.9) follows that the curve C corresponds to T,,,(P0). Hence, the curve C lies in the boundary B. Recalling

the least convex the theorem of §1, it now follows that every point of solid containing C, lies in R. It is readily proved that each of the curvatures k,, . . •, k2,,,-, of C is constant, and that none vanishes. By proper choice of the direction of the vectors I,, •, 12,,,1, one may make k, .••, k2,,,_2 positive.

The author's first conjecture was that R consists of true for m = 1, when C reduces to the circle

This conjecture is

(9.19)

and R to its interior; it was further suggested by the limiting case, n —> considered by Carathéodory and described at the end of this section.

However,

it turns out that for m > 1 this conjecture is wrong and that, except for points on C, the boundary B of R extends beyond Indeed, the points P in the interior of may be represented as follows: —, —.. (9.20) OP = = 1, p 0P1, 0 < p, < 1,

where 0 is the origin (i.e., the point A, =

B, = ... = B,. = 0), and P. are (2in + 1) points on C, corresponding to arbitrary values of 0; 0 = 0,, . .., 020111. the boundary of If P lies on then a similar representation but with a reduced number k of points P1, k < 2m + 1 (and a correspondingly reduced set of p's and 0's) holds. The special choice (9.21)

0, = i40,

40 = 2r/(2rn + 1)

in (9.20) leads to OP = 0. Hence, 0 lies in the interior of It is readily shown that in every direction OP through 0 there are points in the interior of

a point

From the convexity of follows that in any direction OP, there exists on the boundary of provided R, does not extend to infinity—

H. PORITSKY

432

an alternative that is prevented by the (reduced) representation (9.20). Now recall the trigonometric polynomial f(8) corresponding to P in with 2m + 1 or less terms. In the (8, y)-plane, this equation is reflected into the relation

yf(O)

(9.22)

2

valid for each 0. From equation (9.22) follows that each point of y = f(0) is the centroid of the points for the same 0, on the curves y = = — 0), which correspond to the points in (9.20), with positive "weights" Now

a centroid of point masses on a line always lies between the extreme points. Since each curve y = — touches the line y —1/2 from above at one single value 8 = 0, of 0, every point of the curve (9.22) will lie above the line y = — 1/2, except when P reduces to a point on C proper, and all p's vanish except = 1. Hence, for f(8) in (9.22), fm1n > —1/2, and every point of other than points of C, lies in the interior of R. Turning to the boundary B of R and its relation to the curve C, we consider first the case of four dimensions n = 4 (m = 2). For m 2, the family of 3-flats is given by (9.23)

A, cos 8 + B, sinO + A2 cos 20 + B2 sin 20 =

—

There results now

c1=_f, C=+, and

C is given by = —f cos0,

C:

Ao=+cos20.

B,

0, = —f sin

B2=+sin20.

To obtain a description of R and its boundary B, we consider R0, B0, the section of these respective manifolds by the 3-flat (926)

Ai+Ao_—_+,

which is the osculating 3-flat to C at the point (9.27)

B,=0, Az=+, B,=0

on C, corresponding to 0 = 0. The rest of B can be obtained from B0 by

applying to it the transformation group 7',, which transforms (9.26) into (9.23) for 0 = a.

Indeed, as indicated in (9.12) (case (b)), for every point P on B, the trigonometric polynomial

CONVEX SPACES ASSOCIATED WITH A FAMILY OF LINEAR INEQUALITIES 433

f(0) =

(9.28)

A1

cos 8 + B1 sinO + A, cos 20 + B, sin 20

takes on its minimum,fmin =

(or at several 0's). Hence,

at some 0, 0 =

—112,

every point P on B lies on some one of the 3-fiats (9.23). In particular, for points on B,, equations (9.26), (9.28) show that — 1/2 is taken on (at least) at 0 = 0. Hence, the relations f(0)

(9.29)

= A1

+ A,

=

= —4,

f'(O) I.-,, = —(B1 + 2B,) = 0,

as well as the inequality (for 0 (9.30)

0)

f(0)=A,cosO + B1sinO + A,cos20+

—÷ must hold for every point of B0. Equations (9.29) show that B0 lies in the plane (2-flat) IT, which osculates C at 0 = 0. Each 3-fiat (9.23) cuts the plane H, in a line 1(0). The area to the admissible side of this periodic family of lines 1(8) is B0; we denote its boundary by C0. If the envelope of the lines 1(0) satisfies the conditions described in connection with Figure 2.3 and the accompanying text, then C, is that envelope. The roots of the equation (9.31)

4+f(0)=-f +A1cosO+B1sinO+A,cos2O+B,sin20=O,

for real A1, B0, A1, B,., are either real or occur in conjugate complex pairs. Multiplication of equation (9.31) by et' converts the middle member into a polynomial of degree 4 in e'9; hence, (9.31) possesses 4 roots per period 2ir in the 0-plane. If (A,, --•, B,) corresponds to a point on B,, then as indicated above, 0 = 0 is a double root of (9.31), or it may be a 4th order root. In the former case, there are two more roots. then If the other two roots are complex, at 0 = a ± (9.32)

[P+sin2 0—a±]

f(0)=

>0,

where

=

(9.33)

2

and where K is chosen so that (9.7) holds. This form applies to points P in the inferior of B,, since for slight variations of p and a, the minimum —1/2 is retained.

If the other two roots were real and distinct, then f(0) would take on a —1/2 between them, and f(0) would correspond to a point

minimum 1mm

outside R. Hence, if the roots are real, they are coincident, say, at 0 = a; and (9.34)

.20 .,0—a 1 f(0) = — — + Ksin — sin 2

2

2

,

K=

4

2+cosa

where K is determined so that (9.7) holds. The proximity of points with

H. PORITSKY

434

<

—

1/2

shows that P now lies on the boundary C0 of B0. As a approaches and (9.31) acquires a quadruple root at 1/ = 0.

zero, f(8) in (9.34) approaches

By expanding f(O) in (9.34) in a Fourier series, one obtains the following

parametric repi-esentation of C0: A1 = (9.35)

C,:

1+cosa —2

—

+ cos a ' cosa

——2 —

2-

Sina

+ cos a sifla

2(2 + cosa) = 2(2 + cosa)' The coefficients (9.35) can also be obtained by solving the four simultaneous linear equations consisting of (9.29) and 2

f(8)

2

(9.36)

=0. Equations (9.36) are the equations of the osculating plane to C at 0 = a. and for variable a. Thus the points of C0 form the intersections of The last two equations (9.35) show that the projection of C0 on the (A2, Ba)plane is an ellipse, with A2 = 0, 0 (i.e., the origin 0) at one of its foci, and a vertex at the point (9.27) on C. Its projection on each coordinate plane is either an ellipse or a straight line segment. Hence, C0 is an ellipse lying in passing through P0 on C and with 0 as focus. Similarly, one can obtain B1, the section of B by the 3-flat (9.23) for 8 = lies in and consists of the boundary and by applying 7'1 to B0. This the interior of an ellipse whose points form the intersection of lI-i with 11,, for arbitrary a. This ellipse may also be obtained by applying the transformation T1 to the ellipse C0 given by (9.35). This completes the description of the manifold B. Summarizing, we have thus proved the following THEOREM 3. Let R be the convex region whose points (A1, B1, A2, 82) satisfy

the one-parameter family of inequalities (930) for every 0, and let B be the boundarij of R; let R0 and

be the respective intersections of R and B with the 0 in (9.23); let C, given by (9.25), be the curve given by (9.27), be the of which (9.23) form the osculating 3-fiats, and let point 0= 0 on C; denote by the osculating plane (2-fiat) of C at 0= a; introduce the group of transformation 7', given by (9.17). Then, the manifolds R, B (as well as the curve C) are invariant under the transformation group and R, B -(and C) can be genera/ed by applying this transformation group to R0, B0 (and is a flat area lying in the P0), respectivetv. The two-dimensional manifold 3-flat (9.26) corresponding to 0

plane ho and consists of the points of the plane curve C0, whose pOints form and IL, and of the interior of C0. This curve C0 is an the intersection of ellipse, with P0 as a vertex, and the origin 0 as one focus of i/s projection on the (A, 82)-plane.

We now proceed to consider briefly (9.1) for m > 2 (n > 4), and to outline

CONVEX SPACES ASSOCIATED WITH A FAMILY OF LINEAR JNEQUALITI ES 435

a similar description of R, B, and R0, B0 their sections by the (2rn

— 1)-flat

(9.37)

corresponding to the first equation (9.2) for 0 = 0. For points of B0, the corresponding trigonometric polynomial f(0) takes on = —1/2 (at least) at 0 = 0, and hence, its points satisfy its minimum + mB,, 0, + as well as (9.37). For points in the interior of B0, the trigonometric polynomial f(0) can be expressed as follows: B1 + 2B2

(9.38)

(9.39)

f(0)

Ksin2 = —÷ +

+

0_at]

0

+

Pi > 0,

where a are arbitrary, are arbitrary and positive, and K = K(a1, - •, is adjusted so that (9.7) holds. For points on C0, the (2m 3)-dimensional boundary of B0, f(0) is obtained from (9.39) by allowing one or more P, to approach zero. If p1 —' 0, the points of C0 satisfy the equation .

=0,

(9.40)

=0,

as well as (9.37), (9.38), and thus the points of C0 lie on two (2m which osculate C at 8 = 0 and 0 = a1.

— 2)-flats

Similar descriptions apply to the sections of R, B by any other (2m — 1)-flats (9.1), 0 = a, and this section of B can also be obtained by applying to B0.

More details of the relation of B to C will be presented elsewhere. A somewhat related problem studied by Carathéodory and others (references [1; 2; 3]) pertains to infinite Fourier series. Carathéodory et al. consider the conditions on the constants B, so that (9.41)

u

=

÷

? (AR cos nIl + B,, sin nO) > 0,

for r <

1,

where the infinite series is assumed convergent for r < 1. The function u is harmonic in the (x, y)-plane, where x = r cos 0, y = r sin 0. Hence, if (9.41) holds for r = r,, it will hold for r < r0. However, one cannot put r = 1 in (9.41) and regard it as a limiting case of (9.4) for m — since for r = 1 the series (9.41) need not converge. Carathéodory obtained as a necessary and sufficient condition that, each m, the coefficients Ak, Bk; 1 S k m, lie'in the region K,, of the 2m-dimensional space (A1, B1; B2, ..., B,,) which is the least convex solid enclosing the curve (9A2)

Dm. A,cosO,B1 = sinO, A2 = cos20, •--,B,,, = sinm0.

It is natural to inquire whether the curve C, given by equations (9.5), (9.4), in some sense, approaches the curve (9.42). The alternate signs occurring in

H. PORITSKY

436

(9.4) can be eliminated by replacing 0 by U + x; making this change in 0 and introducing m, we put equations (9.5) in the form =

(9.43)

where (9.44)

—

(m — k + 1)

t(m)!lt

—

(,n

—

k)!

•

m

(m + k)! = (m + k) '-• (m + 1)

It will be noted, indeed, that (9.45)

lim

= 1,

k flnfle

that is, that the terms involving cosj0, sin jO,j =

1,

.

. ., k, in (9.43) approach

on the 2k-flat the corresponding term in (9.42). Thus, the projection of containing the A1, E1, •--, Ba-axes approaches the corresponding projection of D,1, for any fixed k, as m —a These considerations and Carathéodory's theorem led the author to the conjecture, disproved above, that the region R whose points satisfy the inequality (9.1) is identical with REFERENCES

1. C. Carathéodory, Ober den Variabiltätsbereich der Fourier'schen konstanten 'von positiven harmoniachen Funktionen, Rend. Circ. Mat. Palermo 32 (1911), 193; Supplemento

6 (1911).

2. 0. Toeplitz, (Iber die Fourier'sche Entwickelung positiven Fnnktionen, Rend. Circ. Mat. Palermo 32 (1911), 191; Supplemento 6 (1911). 3. E. Fisher, Uber das Carathéodory'sche Problem, Potenzreihen inst positiven reelen Tell betreffend, Rend. Circ. Mat. Palermo 32 (1911), 240; Supplemento 6 (1911). 4. H. Poritsky and D. W. Dudley, Conjugate action of involute helical gears with pa#allel or inclined axes, Quart. Appi. Math. 6 (1948), 193. 5. T. S. Motzkin, Sur lee arcs plans dent lee courbet ne se cou pent pas, C. R. Acad. Sd. Paris 206 (1936), 1700-1701. 6. I. J. Schoenberg, An isoperi,netric inequality for closed curves convex in evendimensional Euclidean spaces, Acta Math. 91 (1954), 143-164. 7. T. S. Motzkin, Coinonotone curves and polyhedra, Bull. Amer. Math. Soc. 63 (1957), 35. GENERAL ELECTRIC COMPANY

A COMBINATORIAL LEMMA ON THE EXISTENCE OF CONVEX MEANS AND ITS APPLICATION TO WEAK COMPACFNESS BY

VLASTIMIL PTAK

In the present paper we present a theory of weak compactness based on a simple combinatorial lemma which gives conditions for the existence of certain convex means.

To explain the advantage of such a treatment let us consider the following example. Take a compact Hausdorif space T and a bounded sequence of continuous functions such that converges to zero at each point 1€ T. It is a well-known fact that, in these conditions, the function zero may be arbitrarily well uniformly approximated by convex means of the x1. The easiest way of proving this is to show first that converges to zero weakly in C(T) and then use the Mazur theorem according to which weak convergence of a sequence implies strong convergence of convex means. To show that x converges to zero weakly it is possible to use the Riesz representation theorem for linear functionals on C (T) and the Lebesgue dominated convergence theorem to show that equiboundedness and pointwise convergence imply convergence of integrals. Although simple, this proof is by no means an elementary one. A direct proof of this result has been given by Gillespie and Hurwitz [61 and Zalcwasser [131. Their proof is based on the notion of the oscillation of the sequence at a given point and a transfinite induction. The combinatorial lemma on convex means furnishes a straightforward proof of this theorem which avoids the use of integration. A thorough analysis of weak compactness shows that a similar result usually forms an essential part of the proof. Secondly, let us recall the double limit condition. of Banach. In his book, he considers weak convergence in the space B(Q) of all bounded functions on an abstract set Q. A bounded sequence B(Q) converges weakly to zero if and only if I

for each sequence q

Q such that the limits urn, exist for each i. It turns out that the combinatorial condition for the existence of convex means has a similar "double limit" character. This enables us to obtain from "double limit" assumptions conclusions about the existence of convex means which makes it possible to avoid the use of integration in proofs of theorems on weak compactness. Especially, it is possible to give a simple proof of a result (see Theorem (3.3)) which includes both the Eberlein theo437

VLASTJMIL PTAK

438

rem and the Krein theorem on convex extensions. The combinatorial lemma appeared first in 1959 in the Czechoslovak Mathematical Journal [11]. The author's original proof of Krein's theorem (based on showing that if a set A satisfies the double limit condition then so does cony A) is reproduced in the book of G. Köthe [8] on topological linear spaces. We intend to collect applications of this method to the study of weak compactness of operators in another communication. 1. The combinatorial reBult. Definitions and notation. Let S be a nonvoid set. We shall denote by M(S) the set of all realvalued functions 2 defined on S and satisfying the following conditions: 10.

A(s)

0 for each seS,

the set N(2) of those s eS for which A(s) > 0 is finite, = 1. A(s). If A c S, we define 2(A) to be the sum be a family of subsets of S. If K c S. we denote by Now let WnK*O. If e >Oand HcS are given, thesetof those we shall denote by M(H, e) the set of those A eM(S) such that N(A) c H and 2(W) < s for each We The main purpose of this section is to give necessary and sufficient condie) to be nonvoid. tions on S and for M(S, The results are based on the following essential lemma. be a family of subsets of S. Let (1.1) Let S be a nonvoid set and let H c S be nonvoid and suPpose that M(H, e) = 0. Let R c H be nonvoid 0 < e' < e. Then there exists a nonvoid finite K c R such that and let 2°. 3°.

M(R,7/'(K),c')—O. PRooF. Suppose that M(R, e') is nonvoid for each nonvoid finite M(R, K c R. Take a nonvoid finite A1 c R. Take c') and put A1 U N(21) c R. Take 2,e M(R, e') and put A, = A, U N(A,) c R. Putting A,,+1 = A1 U N(21) and choosing M(R, e'), we obtain M(R) and a sequence of finite sets 0 A1 c A, c -.- R. Now a sequence A2

let n be a natural number large enough so that (1 + (n — < e. If we M(R, e), we obtain a contradiction and +- -+ the proof will be complete. To prove this let us take an arbitrary and show that the sequence A1(W), ),(W), - -. contains at most one term Indeed, suppose that 2,(W) c' for some p. It follows that W A N(A,) 0. If q > p, we have N(2,) c A,4-1 c A, so that W fl A, * 0 whence We %I"(A,) This implies 2,(W) < s' since A, M(R, ,'). It follows that 2(W) = (21(W) + - -. + < (1 + (n — I)e')/n < e. Since W was arbitrary in we have A M(R, 7/', c M(H, e). The proof is complete. The following lemma is technical and almost obvious.

-show that A =

-

(1.2) Let K1 , K,, . -. be a sequence of nonvoid finite subsets of a set S. For each n, let 1',, be a nonvoid subset of K1 x --- x Suppose that the sets P,. fulfill the following condition: jfk < n and Es1, - -, e then [Si, •, Pk. Then there exists a sequence such that for each n we have [Si, .. -, P1. -

- -

A COMBINATORIAL LEMMA ON THE EXISTENCE OF CONVEX MEANS PROOF.

439

For each s K1 let Ps(s) be the set of all p e P,, whose first element

are nonvoid and is s. We have I',, = U P,,(s) for s 6 K1 and since the sets K1 is finite, there must be an eK1 for which P,(s2) is nonvoid infinitely many times. With view to the "restriction" property of the sets P., we nonvoid for each n. We can repeat the same procedure with the have

since it fulfills the same conditions. This yields s, sequence P2(s1), P1(s1), and so on. We are now able to prove the theorem on the existence of convex means. (1.3)

Let S be an infinite set and

a family of subsets of S.

Then the

following two conditions are equivalent to each other: (1) there exists an infinite Hc S and e > 0 with M(H,

e) = 0; S and a sequence of

(2) there exists a sequence of distinct elements such

that (SI,

.

•, s,,}

c W%.

Suppose that (2) is fulfilled. Let H be the set consisting Suppose that A M(H, e) for some Since N(A) is finite and N(A) c H, we have N(A) c {s1, . . ., for some

PROOF.

of all elements of the sequence e

< 1.

n so that N(A) c

It follows that 1 =

A(N(2))

A(

<

< 1 which is a

contradiction.

On the other hand, suppose that M(H,

£) =

0

for some infinite H and

some e>0. By (1.1), there exists a finite nonvoid K1 such that = 0. Since K1 is finite, we have H — K1 * 0 and it follows from (1.1) that £14) = 0 for some nonvoid finite K2 c H — K1. M(H — K1, ii Take now H — K1 — K2 * 0 and repeat the process. In this manner we obtain a sequence of pairwise disjoint nonvoid finite sets K1, K2, ... such thats * 0 for each n. Let P,, be the subset of those fl x [s,, x K,, for which there exists a We with '(S., W. •

It follows that each P,, is nonvoid and that the

I',,

fulfill the "restriction"

property. By Lemma (1.2) we obtain a sequence K1 such that Es1, . . ., s,,] P,. for each n. The sets being mutually disjoint, the sequence is distinct.

In the sequel we shall be frequently dealing with the following situation: We are given two sets S and A and a subset W of S x A. For each aeA, let W(a) be the set of those s €S for which Es, a] W; similarly, if s€ S, we denote by W(s) the set of those a A for which Es, a] £ W. The family will consist of all sets W(a) with a e A. In this situation, the condition for the existence of convex means may be reformulated as follows: (1.4)

COROLLARY.

The following conditions are equivalent:

(1) there exists an infinite HcSandan e>0 with (2) there exists a sequence of distinct elements S,, S and a sequence a,, such that '(Si, ... s,,} c W(a,.) for each n; (3) there exists a sequence of distinct elements s,, S and a sequence a,,

A A

such that W contains the "triangle" consisting of points [se, a,,1 for i it; (4) there exists a sequence of distinct elements s,, e S and a sequence a,, A such that s,,E W(a,,) fl W(a,+1) fl •.. for each n;

VLASTIMIL P'rAK

440

(5) there exists a sequence of distinct elements

S such that W(s,) fl

A

is nonvoid for each n. PROOF.

Immediate.

2. An illustration. In this section we intend to show how the theorem on convex means can be applied. We shall apply it to the proof of a classical theorem; although the result we are going to prove is by far not the best possible, the proof represents a typical application of the lemma on convex means and puts into evidence the idea underlying all further applications.

Let T be compact Hausdorff, Xn C( T) and 1. Suppose that — 0 for each t T. Let e > 0 be given. Then there exist non-negative numbers A,,-• -, with = 1 such that < (2.1)

I

lim

PROOF. Let S be the set of all natural numbers and define a set W c S x T in the following manner: [s, tI W if and only if x,(t) e. Let be the family of all W(t) for t T. Let us show first that our problem will be solved if we find a A eM(S, e). Indeed, we intend to show that To see that, take an arbitrary t T. We A(s)x,1 <24 if A M(S, I

have A(s)x,(t) I

I

*€W U)

I

A(s) I x,(t) + E

.€3—W(t)

A(s) < 2e.

A(s) + £ IEW(t)

A(s) I x,(t) I

IES

It remains to show that M(S, e) is empty, e) is not empty. If M(S, it follows from (1.3) that there exists an increasing sequence sn €S and a e W(s1) A ...

e

W(s) being

A

closed subsets of 7', there exists a 10 T which belongs to all This, however, means that £ for each n, which is impossible. The proof I

is complete. 3. Krein's theorem on the convex extension of a weakly compact set. The lemma on the existence of convex means states that M(S, e) empty implies the existence of a sequence eS and a sequence e A such that s,, e A A . .. It gives, however, no information about the relation between and W(a,), - . -, W(a1_,). A slight refinement of the argument yields a more precise result. The following theorem shows that may be chosen "far" from W,, -j-, (3.1) Let S and A be two sets and let W and M be two disjoint subsets of S x A. We supPose that, for each finite a,, -, the intersection M(a,) fl -.- fl is nonvoid. Let be the collection of all W(a) for a A. Suppose that, for some e > 0, the set M(S, s) is empty. Then there exist two sequences S and A suck that

M(a,) fl ... (spice s,, ..

s,,

and

A

M(a,,_,) A E

A W(an+j) A

the sequence

is distinct).

A COMBINATORIAL LEMMA ON THE EXISTENCE OF' CONVEX MEANS

Since M(S,

PROOF.

that M(S,

e12)

€)

= 0.

0,

441

by (1.1) there exists a nonvoid finite K1 such

Let P1 be the set of those s

K1

for which

W(s) * 0. Clearly P1 0. For each p e P1 , p = s, choose an a(p) e W(s) so that s€ W(a(p)). Put M1 = fl M(a(p)) for p E P1 so that M1 * 0 by our assumption. By (1.1), there exists a nonvoid finite K2c M1 with M(MI, e14) = 0. Let P2 be the set of those pairs [s1 , s2J e K1 x K2 for which W(s1) fl W(s2) 0. Clearly P2 * 0. For each p e P2 , p = Es2 , s,], choose an a(p) W(s1) fl W(s2) so that s1 , s2 W(a(p)). Put M2 = fl M(a(p)) for p £ P1 U P2 so that M± * 0. Suppose we have already defined the sets K1, •, K,, and P1, •, P,,-1 and a mapping a(p) on Pi U U P,,-1 to A such that M(M,,_1, (1f2")e) = 0 where M,,_1 — fl W(a(p)) for fl ••- fl pEPI U U P,,-1. Let P,, be the set of all [SI, x .•. x K,, for which W(s1) fl ... Ii W(s,,) *0. Clearly I',, * 0. For each P€P,,,p = [s1,.. - -

chooseana(p)eW(s1)fl ••-fl W(s,,). PutM,,=flM(a(p)) forpEP1U-.-LJP,,, so that M,, * 0. By (1.1) there exists a nonvoid finite K,,+1 c M,, with M(M,,, Ii fl = 0. We have thus obtained a sequence of nonvoid finite sets K,, c M,,_1; the sets P,, satisfy the "restriction" property of Lemma (1.2). Hence there exists a sequence s, such that = [s1, -, s,,] e P,, for each n. Put a,, a(p,,). It follows that {s,, •, s,,} W(a,,). Further, s,, K,, c M,,..1 c M(a,) fl - - - fl M(a,,1). The proof is complete. -

-

We are now able to prove a general result which includes both the Eberlein theorem and the Krein theorem. We begin with a definition.

(3.2) DEFINITION. Let E be a convex space. A set A c E is said to fulfill the double limit condition if it is impossible to find a neighborhood of zero U in E and two sequences a1 e A and 4 U° such that lim, and

urn1 lim. both exist and are different from each other. (33)

Let E be a complete convex space, let A be bounded and

THEOREM.

fulfill the double limit condition.

Then A°° is weakly compact.

PROOF. The space E being complete, it is sufficient to prove that A° is a t(E', E) neighborhood of zero on each U° or in other words: If Sc U° and 0 belongs to the a(E',E) closure of S then, for each £ >0, there exists a y€ convS such that II Let > 0 be fixed and let W be the subset of S x A consisting of those Es, Cl for which I e. Let M be the set of those [s, a] for which I <ef2. Since 0 belongs to the c(E', E) closure of S, the set M fulfills the condition of (3.1). Let us show now that the theorem will be proved if we show that M(S, e) is nonvoid. Indeed, if 2 M(S, take y = I

If ceA, we have

•EW(a)

A(s)II+ E

M being a number such that II M for 2eA and x'€ U°. Suppose now that M(S,

=0. By our theorem, we obtain two sequences

VLASTIMIL PTAK

442

e for i n and

a1, s, such that
I I < o/2 for i < n.

4. On double sequences. In this section we present the combinatorial treatment of double sequences. Let us begin with two definitions: DErINrrIoN. Let a,0 be a double sequence and suppose that urn, a,, = exists for each p. We shall say that the convergence is almost uniform with respect to p if, for each s > 0 and each infinite set R of indices q, there exists a finite K c R such that for each P. — a,, I < s mm

(4.1)

kEk

(4.2)

DEFINITIoN.

I

Let a,, be a double sequence and suppose that lim, a,, =

a,, exists for each P.

We shall say that the convergence is uniform in the mean with respect to p if, for each infinite set R of indices q, there exists a finite Kc R and nonnegative At such that e

kek

for each peP.

Let us describe almost uniform convergence in combinatorial terms. Let If is a positive number, denote by W1 the set of all (p, qj P x Q for which I a,, a,,

P and Q be two copies of the set of natural numbers.

The definition of almost uniform convergence means that for each infinite R c Q there exists a finite K c R such that Pc Uk€K (P — W(k)) or in other words fltex W(k) = 0. (4.3)

THEOREM.

Let a,, be a bounded double sequence such that limq a,, = a,, Then the following are

exists for each p and lim, a,, = a,, exists for each q. equivalent: 1°. 20.

the convergence urn, a,, = a,, is almost uniform with respect to p; the convergence urn, a,, a,, is almost uniform with respect to q;

3°.

both limits lim,a,, and lim, a,, exist and are equal to each other; the convergence urn, a,,, a,, is uniform in the mean with respect to p; the convergence lim, a,, = a,, is uniform in the mean with respect to q.

4° 50

PROOF. Let P and Q be two copies of the set of natural numbers. Let W,cPx Q be the set of those [p,q] for which o. IfqeQ then

W(q) will be the set of those indices p e P for which I a,1 — a,, (1° 4°) Suppose now that 10 is fulfilled. Let R c Q be infinite. Denote )7' by the family of all W(p) with We intend to show that * 0. Suppose, on the contrary, that there is an increasing seM(R, P such that p,, W(r1) fl ... fl for each n. quence r,1ER and According to 1°, there exists a finite K c R such that W(k) = 0. This is a contradiction since K c {r,, .. ., for n large enough. There exists, accordingly, a A M(R, )7', s). Let B be a constant such that I a,, I B for each p and q. If P is given, we have I

A COMBINATORIAL LEMMA ON THE EXISTENCE OF CONVEX MEANS

aoo)]

—

k€W(P)

OEN(A)

41 a,,,

—

a,, I ±

A,,+e

k5NA—w(p

A,, I a,,, —

a,,

443

I

A,,<(2M+ kEN(A)

that 4° is fulfilled. 3°) Suppose now that 40 is fulfilled. Take a limit point a of a,0 and suppose there exists a a with a I = 1 such that a(a0,. — a) e for each r from an infinite R c Q. By assumption, there exists a finite K c R and non-negative A,, = 1 such that A,, with so

(4°

I

4(a,,, — a,,)

<

or eac

C

P

It follows that Ak(a0k — a)

£

2

Since K c R, we have a)

—

e

a contradiction.

Hence a,, converges to each limit point of a,, so

that 3° is satisfied. (3° —÷ 1°) Let 3° be satisfied and suppose there exists an infinite R c Q such 0 for each finite Kc R. that Take a fixed r R. I contend that the set W(r) is infinite. Suppose W(r)

is finite. Since W(r) fl W(r') is nonvoid for each r' R, there must be a p e W(r') for infinitely many r' R. This, however, is impossible since this means r' W(p) for infinitely many r'; the set W(p), however, is finite. e for infinitely many The set W(r) being infinite, it follows that I a,,. — a,, e; this, however, being true for each r R, we obtain a p so that a,, — a I

contradiction with 3°. We have thus proved the equivalence of conditions 1°, The proof is concluded by observing that conditions 2° and 5° are obtained from 1° and 4° by interchanging p and q and that condition 3° is self-dual. 5. Weak convergence. In this section we intend to apply the lemma on

the existence of convex means to describe weak convergence in spaces of continuous functions. (5.1)

DEFINITION.

Let H be a set and

a sequence of functions on H

converges to zero for each h H. We say that the convergence such that is almost uniform on H if, for each infinite set R of indices and each e > 0, there exists a finite K c R such that foreachh€H. minlx,,(h)I<e kEg (5.2)

Let H be a set and let

be a bounded sequence of functions on H

which converges to zero at each point of

F!.

Then the following conditions are

YLASTIMIL PTAK

444

equivalent:

(3) for each

there exists a convex mean > 0 and each subsequence < e for each h H; exists for each n we have (4) for each sequence h., H such that

such that

I

I

= 0;

urn urn a

(5)

converges to zero almost uniformly on H.

PROOP. Assume (5) and suppose that urn, I 0 for an infinite set R of indices n. By our assumption, there exists a finite Kc R such that I x,,(h,) I < €/2 for each j. Take q, such that I

Xk(hq) —

for each ke K whenever q

q0.

lint xk(h,) <

q, and kEK we have

For q

limIxk(hf)I

which is a contradiction since I < €12 for some k K. This proves (4). Suppose now that (4) is satisfied. Take an £ > 0, denote by S the set of I

natural numbers and consider in S x H the set We consisting of aH Is, h] such that I x(h) €. Let be the family of all sets of the form with h H, and let us show that M(S, e) is nonempty. indeed, suppose it is empty; it follows from (1.4) that there exists an increasing sequence S and points h,1 H such that h,, W(s,) fl ... fl so that for j n. We can extract a subsequence such that urn, exists for I

each s e S. It follows that urn1 I Hence there is a 2 M(S, , e).

a fixed heH. We have I

y(h)

A(s) x.(h)

I

.CE

for each sN which contradicts (4). 2(s)x. and

Consider the function y = A(s)

I

a€W(A)

M€ + .e S

I

= (M+

x, I +

A(s) x1(h) I

1)c

M is a number such that x M for each n. This proves (3). Assume (3) and suppose that the convergence is not almost uniform. It fol-

where

I

I

lows that there exists an infinite set R and a positive c such that for each finite Kc R there exists an with c for each n€K. If R is the sequence r1 < r, < there exist h H such that s for i = 1,2, .. . ,j. Pick a subsequence such that lim, exists for each r eR, so that lim, x,(h') I 6 for each r R. There exists a a with I a = 1 and an infinite subset Pc R such that a urn1 e for each p P. By our I

I

I

assumption, there exists a finite F c P and a convex mean LEF that I < €/2 for each h H. It follows that a for each I

whence

lim1 x1(h7)

the proof. Which is a contradiction since each a Iimj

€/2.'

such < €12

This proves (5) and concludes s.

A COMBINATORIAL LEMMA ON THE EXISTENCE OF CONVEX MEANS 445 (5.3)

Let H be a dense subset of a compact Hausdorff space T

THEOREM.

and let x,, be a bounded sequence of continuous functions on T which converges to zero at every point of H. Then the following conditions are equivalent: converges to zero at each point t e T; (1) converges to zero in the weak topology of C(T); (2) there exists a convex mean (3) for each e > 0 and each subsequence < e; such that exists for each n we have (4) for each sequence h, H such that urn5 urn urn = 0; I

I

3

converges to zero almost uniformly on H; converges to zero almost uniformly on T.

(5) (6) PROOF.

Assume (1), take an infinite set R of natural numbers and a positive

< t}. These sets are open in T each r R, let U(r) = T; I and form a covering of Tso that there exists a finite Kc R with = €.

For

I

It follows that the convergence is almost uniform on T and we have (6) which, in its turn, implies (5). We have shown in the preceding (5) implies (4). If (4) is satisfied and an infinite R and a positive given, it T.

follows from the preceding lemma that there exists a finite Fc R and a

such that < e for each he H. Since H is dense in T, we have so that (3) is proved. Let (3) be fuifihled and suppose there exists a functional x' C( T)' of norm one and a positive £ with for each n in an infinite index set R. , x'> Take a convex mean y = with Fc R such that lyl < €/2. We have t x' < e/2 which is a contradiction. <x,, x'> = This proves (2) and concludes the proof, the conclusion from (2) to (1) being

convex mean

I

I

I

I

trivial. (5.4) THEOREM. Let 0 be a set and let B(Q) be the Banach space of all bounded functions defined on Q with the norm x = supq5Q Let be a bounded sequence in B(Q) such that xn(q) converges to zero at each point of Q. Then

the following conditions are equivalent: (1) x,, converges to zero in the weak topology of B(Q); (2)

for each e > 0 and each subsequence

(3)

such that < €; for each sequence q, Q such that I

there exists a convex mean

I

R

exists for each n we have

3

converges to zero almost uniformly on Q.

(4)

PRooF. This is an immediate consequence of the preceding theorem; it is sufficient to observe that B(Q) where is the tech-Stone cornpactification of the discrete set 0. The theorem follows if we put T = P0

and

H=Q.

(5.5)

DEFINIrION.

Let E be a normed linear space.

A set M c E' is said

VLASTIMIL PTAK

446

to be norm-generating if it is bounded and if there exists a constant a > 0 such that for each xeE there exists a yeM with A bounded set M c E' is (5.6) LEMMA. Let E be a normed linear space. E)-closed absolutely convex extension norm-generating if and only if its contains a multiple of the unit cell of E'. PRoOF. Let M be norm-generating and suppose that xeM°. We have

aix I sup (x, M> 1 so that x e (1/a)U where U is the unit cell of E. Hence c (1/a)U whence aU° c M°°. On the other hand, if aU° c M°°, we have sup(x,M> sup(x,M°°> sup<x,aU°> = alxt. (5.7)

THEOREM.

Let E be a normed linear space and

a sequcnce in E.

Then the following conditions are equivalent: converges weakly to zero; (1) (2) x,. is bounded and converges to zero on a norm-generating subset of E' which is a(E', E)-com pact;

is bounded and converges to zero almost uniformly on some norm(3) generating subset of E'; (4) x% is bounded and there exists a norm-generating subset A c E' such that urn urn <xx, a> = 0 n

j

for each sequence aj e A such that the urn, <xx, a'> exists for each n. converges to zero in the weak topology of E. Then PROOF. Suppose that is bounded and converges to zero almost uniformly on any a(E', E)-compact

subset of E'. Now any norm-generating subset of E' is bounded in the

norm and hence contained in a c(E', E)-compact set. Suppose now that (4) is satisfied. Since A is norm-generating, there exist positive numbers a and such that sup<x,A>

for each xe E. It follows that the mapping T which assigns to each xe E the function Tx ye B(A) defined by y(a) = (x, a> is an algebraic and topo= converge logical isomorphism. Since (4) is fulfilled, it follows that to zero in the weak topology of B(A) by (5.3). If fe E', there exists a g e B(A)' such that <TI,9> = for each x e E so that = converges to zero. (5.8)

Let M be the Banach space of all bounded measurable functions on

(0, 1> modulo functions zero almost everywhere with the norm I x Let

I

ess sup x(t) I.

A be the set of all functions of the form (1/p(B))cfi, where

characteristic function of a measurable set B. Let

is the

be a bounded sequence in

A COMBINATORJAL LEMMA ON THE EXiSTENCE OF CONVEX MEANS

M.

447

Then x,, converges weakly to zero if and only f =

Jim Jim

0

xN(t)aJ(t)dt exists for each n. for each sequence a A such that It is easy to see that the elements of A constitute a norm-generating

subset of M'. be a measure space with p such that be a sequence of sets in

(5.9) Let (1',

p)

0 and p(T) < co and let a > 0 for each n. Then

there exists a subsequence n1 < n2 < ••- such that AN1 (1 void for each k.

n

fl

is non-

PROOF. Let S be the set of natural numbers, and let Wc S x T be defined as follows: (s,t]e Wif and only if tEA,. Let /1 be the family of all W(t) 0, we can with t T. If we show that M(S, , e) is empty for some use (1.4) to show that there is an increasing sequence < s2 < ... such that

is nonempty for each k. Since obviously W(s) = A., the W(s1) fl fl theorem will be proved. To show that M(S, Y/', e) is empty, suppose there is a AG M(S, Consider the linear subspace E of B(T) spanned by the of A,,. characteristic functions It is easy to see that it is possible to define a bounded linear functional f on E by the formula

<Eaixs,f> = Since AeM(S,7/',€), we have 0

e

for each I T. It follows that = ep(T)

On the other hand A(s)p(A,) a

so that we obtain a contradiction if ep(T) < a. 6. Weak colnpactues8. A slight technical refinement of the preceding methods yields characterizations of weak compactness in linear spaces. (6.1) DEFINITIONS. If S is a completely regular topological space, let Ce(S) be the Banach space of all bounded continuous functions on S with the norm 'Cs

If E Ce(S), let be the unit ball of E' in the topology o(E', E) so that S may be considered as a topological subspace of S. if P and Q are two topological spaces, a function h(p, q) on P x Q is said to be separately continuous if it is a continuous function of each variable, the other variable is

VLASTIMIL PTAK

448

kept fixed. We shall say that h satisfies the double limit condition if it is impossible to find two sequences p P and q e Q such that the limits lim, other.

,

qi)

q,) both exist and are different from each

and lim,

All results of this section are based on the following proposition. Let T be a completely regular topological space and let A be a bounded T) such that the double limit condition is satisfied on A x T. subset of E Then A is weakly relatively compact in a(E, E'). (6.2)

PRooF. Consider A imbedded in a cartesian product P of real lines, one for each point t e T. Since A is bounded in E, the closure of A in P is compact. Take any r in this closure. Our theorem will be proved if we A show that, for each £ > 0, there exists a convex mean b = all 1 e T. I £ for Indeed, this shows first that r(t) is such that 1 r(t) — b(t) continuous on T and, further, that it also belongs to the a(E, E') closure of A in E.

To prove the possibility of approximation, take a fixed e > 0 and consider in

A x T the sets

M={[a,tj€A x T;II <€12). is nonvoid for each finite set of points •. ., T. Clearly M&1) fl ..• fl be the family of the sets W(t) with I T. It is easy to see that it Let

c) is nonvoid. Suppose that M(A, is sufficient to show that M(A, is empty. By (3.1) there exist two sequences and t, such that M(11) fl

A

fl W(t,,) fl

Pick a subsequence flr such that if at = tim

p,

exist. We have thus

and

lirn =

,

e)

fl

= 1,,,, the limits

lim =

and 3€14 for large j so that — p1 — €14 for each I and large j. it follows that the double limit condition is violated and the proof is complete. The following lemma is obvious.

all

p,

I

I

I

(6.3) Let E and E' be two convex spaces equipped with the topologies a(E, E') and or(E',E). Let A c E and BCE' be two sets such that their closures are countably compact; then the double limit condition is satisfied on A x B.

This lemma together with (6.2) yields the following result: (6.4) Let T be a countably compact completely regular topological space and let P c C,(T) be bounded and countably compact in the product topology (cartesian Then P is relatively weakly compact in

product of real lines, one for each point I T).

It is not difficult to see that it is sufficient to assume T pseudo-compact only.

A COMBINATORIAL LEMMA ON THE EXISTENCE OF CONVEX MEANS

Lemma

449

(6.3) remains true in this case as may be easily established using the

method of [9]. Most of the weak compactness results are, however, contained in the following theorem. (6.5) THEOREM. Let S and T be two completely regular topological spaces and <s, t> a separately continuous function on S x T. Suppose that <s, t> is bounded on S x T and satisfies the double limit condition on S x T. Then there exists a unique separately continuous linear extension of <s, t> to S x T. C8(T) in the following PROOF. For each s eS define an element h(s) of F

manner:

t>; this is possible since <s,l> is bounded. Clearly h(S) and T fulfill the double = <s,

limit condition so that by (6.2) the set h(S) is relatively compact in the topology o(F, F'). We may now extend <s, I> to S x T by putting = for each y€ If s€S is fixed, the function <s,y> = is Continuous on T by definition of the topology on T. Consider now, in F', the linear space L(T) algebraically generated by the evaluation functionals I e T. Let W be the closure of h(S) in F in the topology a(F, F'). Since W is compact in this topology and the topology ci(F, L(T)) is coarser, these two topologies coincide on W. Now if s0 —' s0 then h(s0) in o(F, L(T)) and consequently, in o(F, F') as well. It follows that <S0 ,Yo> =

=


h(s0)

<So , Yo>

for each Yo e 1' so that <s, y> is separately continuous on S x 1'. By Lemma (6.3) the duality extended to S x T again fulfills the double limit condition and is clearly bounded. We complete the proof by observing that the same construction may now be repeated with T and S instead of S and T.

It is easy to see that this general theorem contains as a particular case the Eberlein-Krein theorem (3.3). The proof is immediate and is left to the reader. The present Theorem (6.5) enables us to weaken the assumptions since now we need not assdme the double limit condition on the whole of A x U°. It is sufficient to have the double limit condition on A x T only, T being

some norm-generating subset of U°. A number of similar results may be obtained, the formulation and proofs of which are left to the reader. Let us conclude by listing some conditions for weak compactness in spaces of continuous functions. (6.6)

DEFINITION.

Let T be a compact Hausdorif space and A c C(T). The

family A is said to be quasi-equicontinuous if the following condition is satisfied: given a directed set t0 — t0 in T then for each a0 and e > 0 there such that exists a finite set K of indices mm

I a(t0) — a(t0) I <

€

VLASTIMIL P'rAK

450

for each aEA. (6.7)

THEOREM.

Let T be a compact Hausdorff space and A c C(T). Then

the following conditions are equivalent: (1) the set A°° is weakly compact; (2) the set A is relatively weakly compact; (3) the set A is relatively weakly countably compact; (4) the set A is relatively weakly sequentially compact; (5) the set A is bounded and relatively compact in the product topology;

(6) the set A is bounded and relatively countably compact in the product topology;

(7) the set A is bounded and quasi-equicontinuous on T; (8) the set A is bounded and for each sequence £ T which converges to to on a countable set B c A the convergence —b 4 is almost uniform; (9) the set A is bounded and the double limit condition is satisfied on A x H for some dense subset H of T. PROOF.

Most of the implications are immediate and the conclusion from

(9) to (1) follows from (6.2) or (6.5). REFERENCES

1. S. Banach, Th4orie des operations Unéairea, Monogr. Mat. 1 (1932). 2. R. G. Bartle, On compactness in functional analysis, Trans. Amer. Math. Soc. 79 (1955), 35-57. 3. M. M. Day, Normed linear spaces, Springer-Verlag, Berlin, 1958. 4. 3. Dieudonné, Sur un tkêorème de Sr,tulian, Arch. Math. 3 (1952), 436—439. 5. W. F. Eberlein, Weak compactness in Banach spaces, Proc. Nat. Acad. Sci. U.S.A.

33 (1947), 51-53.

6. D. C. Gillespie and W. A. Hurwitz. On sequences of continuous functions having continuous limits, Trans. Amer. Math. Soc. 32 (1930), 527-543. 7. A. Grothendieck, CritEres de corn paciti dana lea eapaces functionnels gJntraux, Amer. J. Math. 74 (1952), 168-186. 8. G. Köthe, Topotogische lineare Rclume, Springer.Verlag, Berlin, 1960. 9. V. Ptók, Weak compactness in convex topological vector spaces, Czech. Math. 3. 70 (1954), 175—186. 10. , Two remarks on weak compactness, Czech. Math. J. 80(1955), 532-545.

11. , A combinatorial lemma on systems of inequalities and its application to analysis, Czech. Math. J. 84 (1959), 629-630. 12. V. Srnulian, On the principle of inclusion in the spaces of type (B), Mat. Sb. (N.S.) 48 (1939), 77-89. 13. Z. Zalcwasser, Sur une propriétê du champ des fonctions continues, Studia Math.

2

(1930), 63-67.

CONVEX CONES AND SPECTRAL THEORY BY

HELMUT H. SCHAEFER Introduction. It has been known for more than fifty years that if A is a finite square matrix over the real field with non.negative elements, then the spectral radius r(A) is a characteristic value of A, with at least one characteristic vector that has its coordinates 0. If A has positive elements throughout, then r(A) is > 0, of algebraic and geometric multiplicity one, and has a characteristic vector with strictly positive coordinates. The theorem found its analytic analogue in the Theorem of Jentzsch on Fredholm integral equations with positive (compact) kernel. In an abstract ordertheoretic setting the result appeared first in the important memoir of Krein

and Rutman [5), who considered continuous endomorphisms of a Banach space

that leave invariant a convex cone of vertex 0. Since the advent of that paper by Krein and Rutman, their basic results have been extended to a wider and wider class of mappings (including non-linear maps [9]) and spaces; contributions are, among others, due to Bonsall [2; 3], Karlin [41, and the present writer [7; 8; 10]. Somewhat apart from these investigations, and apparently only superficially related to them, has been standing the well-known and long recognized fact

that the positive Hermitian operators on a Hilbert space H form a convex cone of vertex 0 which is normal (i.e., of a certain topological type to be described below) in for the topology of bounded convergence (the "uniform operator topology"). Is there any intrinsic relationship between the topological and algebraic properties of this cone and spectral properties of Hermitian and normal operators? The answer is quite affirmative. In fact, if A is any ordered locally convex algebra1 whose unit interval is weakly semi-complete (= sequentially complete), then the relation c1e a c2e im-

plies that the spectrum of a is contained in

[c1, c2] (c1 ,

c2

R), and that

there exists a (unique) resolution of the unit of A: 6—. p(8), defined on the dp(A). This entails the presence of a Borel sets of R and such that a = spectral theory for a as is familiar from Hermitian operators (with the difference, of course, that 8 —p takes its values in a Boolean c-algebra of idempotents in A rather than among the orthogonal projections on some Hilbert space H).

A unification of the two aspects just outlined that each relate spectral theory with ordered topological vector spaces, can be achieved through the study of certain convex cones or, equivalently, certain (partial) orderings of Formal definitions are given in the next section. By the interval of a locally convex algebra, we understand the set J = K fl (e — K) (K the positive cone, e the unit of A). 1

451

452

locally convex algebras.

H. H. SCHAEFER

More precisely, if K is the positive cone of an

ordered locally convex algebra, then all elements of K whose spectrum is bounded show a spectral behavior which implies, in finite dimensions, the results on positive matrices quoted above while the elements in the unit interval / of K (corresponding to positive diagonal matrices in the classical case) are precise analogues of positive Hermitian operators.

Accordingly, the

elements contained in the real (complex) linear hull of J show the spectral features of Hermitiar (normal) bounded operators in Hilbert space. Thus, in the case of operator algebras, the unifying step is in considering cones of operators rather than studying a single operator leaving invariant a cone in the space on which it is defined. The results in Part I are mostly generalizations of earlier results of the author or of theorems from sources indicated in the references; several proofs (in particular, that of Theorem 2) are methodically new. The material in this part is intended to give the reader an impression of what is known on the spectral properties of positive elements under very general assumptions. Part II, dealing with spectral elements and algebras, parallels the first three sections of [12] in spirit but the approach is different and more algebraic in the present paper. The concept of spectral measure employed here is in conformity with the general notion of vector measure as given in [1]; what is usually called a "spectral measure" appears in § 4 as an extension of a spectral measure. A connection to and the literature quoted there is provided by Theorem 8.

The author wishes to express his gratitude to the U. S. Army Research Office under whose sponsorship most of the results in this paper were obtained. Definitions and auxiliary theoremB. For the convenience of the reader, we

collect in this preliminary section a number of basic definitions that will be used throughout this paper. Also listed are a few theorems from the theory of ordered topological vector spaces that are needed below. Let E be a Hausdorif topological vector space over R or C;' E is ordered (= partially ordered) if a convex cone K of vertex 0 is specified in E which is closed and proper (i.e., such that K fl —K = l0}). The order relation x y in F is then defined to mean y — XE K, and K is referred to as the positive

cone of F. Clearly ""

is reflexive, transitive, and anti-symmetric. We shall use the word "cone" exclusively for convex cones of vertex 0. A cone K in F is normal if for every filter on E, urn = 0 implies urn = 0 where is the filter with base ([F]: here [El = (F + K) fl (F — K) is the "full" hull of F c E with respect to K. (If K is .

the positive cone for an ordering of E, then [F] is the union of all order-

intervals with "end-points" in F.) When E is locally convex (and Hausdorff), K is normal in E if and only if x, y E K imply p(X + y) for every element p of a family of real semi-norms generating the topology of E; thus, 2 By N, R, C we denote the non-negative integers, real numbers, and complex numbers, respectively. Topological vector spaces are assumed to be }lausdorff unless the contrary is stated.

453

CONVEX CONES AND SPECTRAL THEORY

a iiormable space, a cone K is normal if and only if x '+ y II II x II for all x, y K and some topology-generating real norm. Normality, especially for the weak topology, is without doubt the most important concept in the theory of ordered topological (in particular, locally convex) vector spaces. Sc a family of bounded subsets of E such that E = U Denote by as (If E is locally convex, it is no restriction of generality to assume saturated, i.e., such that contains all subsets and positive multiples of each of its elements, and with each finite collection of elements the closed convex circled hull of their union.) A cone K in E is an s-cone if the family is a fundamental system for C; an C-cone is said to (K fl S — K fl S: S be strict if the family (Kfl S — Kfl S: SeC) is already fundamental for C. Let <E, F> be a (nondegenerate) dual system over R or C. If K is a cone in E, the dual cone K' in F is the set of all y E F such that xe K implies Re <x, y> 0. (Here (x, y) —s <x, y> denotes, as usual, the canonical bilinear form on E x F.) When C is a saturated family of weakly bounded subsets of F whose union is F, it is well known that the polars of the members of C form a neighborhood basis of 0 for a locally convex topology on E, called is consistent with <E, F> if F, identified with a subthe C-topology space of the algebraical dual E* of E, is the topological dual of E for in

I

The duality between normal cones and C-cones is established by the following

theorem which, in somewhat more general form, was proved in [6, 1] (see also [7, § 6]). THEOREM A. Let <E, F> be a dual system, C a family of weakly bounded subsets of F such that the C-topology on E is consistent with <E. F>. A cone K in E is normal for the C-topology if and only if K' is a strict C-cone in F. In particular, K is weakly normal in E if and only if F = K' — K'.

Assume that E, F are locally convex spaces over the same scalar field, and denote by 2'(E, F) the vector space of all continuous linear maps on E into F. Equipped with the topology of uniform convergence on a family C (as above) of subsets of E,. we shall denote this space by 22€(E, F). If K and H are cones in E and F, respectively, the set of all Te .9'(E, F), for which F); conditions under which this cone is proper TK c H, forms a cone in F) were given in [7, § 8]. We shall need the following and closed in result whose simple proof may be found in 17, (8.3)]. THEOREM B.

If K is an C.cone in E and H is normal in F, then

.

=

F). fT 9'(E, F): TK c H) is a normal cone in A third result needed later in this paper is a theorem [7, (7.2)] on the convergence of section filters of a directed set In an ordered locally convex space. Let E be an ordered locally convex space with positive cone K, and let M be a (nonempty) subset of E directed for ". The family of sections M. = (y e M: y x} forms a filter base in E; the corresponding filter is called the filter of sections of M and denoted by THEOREM C.

Let E be an ordered locally convex space with normal Positive

H. H. SCHAEFER

454

cone K, and let M be a nonempty directed subset of E. The weak convergence implies its convergence for the given topology on E. of

The remainder of this section is concerned with the definition of several concepts related to the spectrum of an element in a locally convex algebra. A locally convex algebra is a locally convex space and an algebra such that multiplication is separately continuous; unless the contrary is explicity stated, we assume such an algebra A to be defined over the complex field C, and we shall always assume that A has a unit element e. A locally convex algebra is ordered if its underlying locally convex space is ordered with a weakly normal positive cone K such that K contains e and the product of any commuting pair of its elements. An example of an ordered locally convex algebra is furnished by the algebra of continuous endomorphisms of Hilbert space (with K the cone of positive Hermitian operators), under the topology of either bounded or pointwise convergence; other examples are the algebra of space and, under certain condicontinuous compjex functions on a tions (cf. Theorem B), the algebra of continuous endomorphisms of an ordered locally convex space (under a suitable s-topology and the induced order). If A is a locally convex algebra and a e A, the spectrum o(a) of a in the one-point compactification of the complex plane (the Riemann sphere) is the complement of the largest open set in which A —* (Ae — ay* is locally holomorphic; this function, unless its domain is empty, is called the resolvent of a. the algebra of continuous, Let X be a compact (Hausdorif) space, complex-valued functions on X under the uniform topology, A a locally convex algebra. A spectral measure (on X into A) is a continuous homomorphism of the algebra ct'(X), with values in A; the supPort of- a spectral measure p

is the complement in X of the• largest open set U such that p(f) = 0 for every fe whose support is contained in U. A spectral algebra is the range of a spectral measure, and a spectral element of a locally convex alge-

bra A is an element contained in a spectral subalgebra of A. When A is a the (real) Banach algebra of all locally ctnvex algebra over R; and continuous real functions on X, one defines in a completely analogous manner

the notions of real spectral measure, real spectral algebra, and real spectral element. PART I.

GENERAL PosiTivE ELEMENTS

1. The generalized Pringeheim theorem. The theorem that we shall prove first was established, under restriction to Banach spaces, in [11]. The proof in made use of the classical Pringsheim theorem3 which asserts that a

power series with non-negative coefficients and radius of convergence 1 defines

an analytic function I such that the element of 1, represented by the power series in question, is singular at z = 1: To this, the result can be added that = 1; both assertions if this singularity is a pole, its order is maximal on persist as long as the coefficients of the series remain in a sector, vertex I

Cf. Landau, Dar8tellung und Begrundung einiger neuerer Ergebniue der tionentheorie, Springer, Berlin, 1929; §17.

CONVEX CONES AND SPECTRAL THEORY

455

at 0, of central angle <2r (for a more detailed discussion, see [11]). Although the proof in [11] carries over to. the locally convex case without important modifications, we give a simpler proof here that has the added advantage of including the classical result of Pringsheim. THEOREM 1. Let E be a semi-complete locally convex space (over C), K a weakly normal cone in E. If a,. K (n N) and if a,.? has radius of convergence 1, then the analytic function represented by the Power series is singular at z = 1. In addition, if this singularity is a pole, it is of maximal order on

IzI=l.

Let f be a functional element given by 1(z) =

PROOF.

Z' a,,,? when z I < 1,

and let the radius of convergence of this series be 1. Denote by any continuous linear form on the underlying real space E,, of E; the radius r, of convergence of the series Z
{r,: cc e K'}

=

1,

the first of these equalities

K' by Theorem A. For otherwise, the series a,t' would converge in E for all t, < I< where 1, and hence f would

holding since

—

have a holomorphic extension to the open disk z < which contradicts the Let p. 0 < p < 1, be fixed and let cc be non-negative on K. I

assumption. Set

=

(keN)

Since for t> p, all terms in the three series t' are

+ p]' =

[(t — 0

0

(t —

p)'

(p <

< 1)

0

non-negative, it follows that the series (t —

so)'

0

has

radius of convergence r, —

p,

and hence that b,(t —

radius of convergence 1 — p. This implies that z = 1 is singular for 1. Assume now that the singularity of I at z = 1 is a pole of order k. If = has etG

is any complex number of modulus 1, and if z = limf(IzI)Iz— CI'=O

0 < I < 1, we have

t—.1

whenever

P> k. Because K is a weakly normal cone, this implies, for any

p>k,that

a,t' cos nO and

(1 — 0

converge to 0 for o(E, E') as t —

(1 —

t)'

aj' sin nO 0

1.

Thus if C is a pole of I of order m, it

H. H. SCHAEFER

456

follows that m

k and the theorem is proved.

COROLLARY 1. Let A be an ordered, semi-complete locally convex algebra and r(a) c(a). The spectral radius r(a) is in a(a), and let a be positive with is a pole of the resolvent of a, it is of maximum order on (2 I r(a). PROOF. Since c,5 is not in the spectrum c(a) of a, the resolvent R(A) of a exists in a neighborhood of that point, and its expansion at 00 is necessarily of the form

(00 = e)

R(A) =

and f(z) = R('A). If r(a) = 0, then f is an entire function and the Set z = first part of the corollary follows from Liouville's theorem while the second Theorem 1 becomes applicable after a simple becomes trivial. If r(a) > normalization, since by the definition of an ordered locally convex algebra, all coefficients of I are contained in a weakly' normal cone of A. The preceding corollary to

a

of

2'(E). This, in turn, occurs

T€2'(E)

invariant

a cone K in E that satisfies certain assumptions. When E is a Banach space,

it was first shown in [2] that TKc K implies r(T)€c(T) if K and K' are both normal for the respective norm topologies of E and E'. We show that TK c K implies the assertions of Corollary 1 when K and K' are normal and weakly normal, respectively; the spectrum of T is understood with respect to the topology of simple convergence for which topology we to be semi-complete. When E is a Banach space, this notion of spectrum coincides with the usual one. assume

COROLLARY 2. Let E be an ordered locally convex space whose Positive cone is normal and generating,' and let be semi-complete for the topology of simple convergence. The assertions of Corollary 1 apply to every continuous, Positive endoinor/ihism T of E such that co a( T). PROOF. On account of the preceding remarks, we have only to show that for its induced order and the topology of simple convergence, is an ordered locally convex algebra. But it is clear that the positive cone in is closed, and that it contains the identity map and the product of

any commuting pair (in fact, of any pair) of its elements; it is normal by Theorem B and hence weakly normal (Theorem A). Perhaps the best known result on spectral properties of positive operators

is the theorem that (under suitable, but very general assumptions) every positive compact endomo(phism of an ordered Banach space with positive spectral radius rCT.) as a characteristic number with (at least) one positive characteristic vector. This theorem, which has a comparatively long history, was first proved in the stated generality in [5]. A more general A cone K is generating in E if E K — K. By Theorem A, this property is equivalent with the weak normality of K'.

CONVEX CONES AM) SPECTRAL THEORY

457

theorem—but still in the framework of Banach spaces—with what might be called a geometric proof, appeared in [31 and was extended to locally convex spaces in [71. We are going to give a new proof of an extended version of the Krein-Rutman theorem. It is surprising that no normality condition has to be imposed in Theorem 2 (see below); but this is only apparent and the Krein-Rutman theorem is in fact a corollary of the generalized Pringsheim theorem, if not an obvious one. THEOREM 2.

Let E be an ordered, semi.complete locally convex space whose

positive cone is total in K. If T is a Positive, compact endomorphism of E with r(T) > 0, then r(T) is a characteristic number of T with (at least) one positive characteristic vector. Furthermore, r( T) is a pole of the resolveni of maximal order on

I

Al

= r(T).

The essential part of the proof consists in showing that r(T)Ea(T). is equipped with the topology of simple convergence.) Since the positive cone K of K is closed and proper, it follows from routine considerations (or from [6, (1.7))) that its dual K' is total in K' for a(E', K); hence if we let = K' — K', <E, is a nondegenerate dual system and K is normal for a(E, by Theorem A. TK c K implies that T is continuous for this latter topology. Denote by and respectively, the spaces of endomorphisms of E continuous for o(E, K') and o(E, E), each equipped with the topology of simple convergence on E and K, respectively. Since T"Kc K for all natural numbers n, it follows that PROOF.

(*) is

the expansion, valid for I Al> r(T), of R(A) both as a member of 2'(E)

This implies, in particular, that r0(T) r(T) if r0(T) is the spectral radius of T as a member of (the completion of) On the other hand, Theorem B implies that the cone C whose elements leave K invariant, is normal in thus by Theorem 1, r0(T) is in the spectrum of T€ .5f0(E). On the other hand, since T is compact (for the given topology of E),s there exists a number E, I El = r(T), which is a characteristic and

.

value of T; this implies r(T) = r0(T). Let us show that r(T) cannot be regular

for the resolvent of Te &'(E). It follows from (*) that R(A)K c K for all A> r; if r were regular we would have R(r)K c K since K is closed, hence R(r) e . WO and the expansion of R(A) e in a neighborhood of A = would be R(A) = R(r)[I + (r — A)R(r) + (r — A)2R(r)2

+ ...];

since the series converges to an element of it is clear that the r would in fact imply that R(A), as a member of is holomorphic at A r(T) = r0(T). Since this is impossible, we have shown that r(T) is in the spectrum of Te Since T iscompact, r is a pole of R(A) in .Y'(E) and, therefore, in it follows convergence of (**) at any point A

7' is compact if 7'U is relatively compact in E for a suitable neighborhood U of 0.

H. H. SCHAEFER

from Theorem I that, when A R(A) is considered as taking its values in the this pole is of maximal order on (A = r; clearly then completion of maximal among the poles, on (A = r, of the resolvent of T the order of r is I

I

in

to be shown that there exists at least one characteristic vector of T belonging to r(T). But if P is the leading coefficient in the x0 K principal part of R(A) at A = r, every nonzero element in the range of P is a characteristic vector of T with respect to r; on the other hand, since It

R(A)Kc K for all A > r and

P = lim (A — r)kR(A)

k is the order of the pole r), one concludes that PK c K. Thus, K being a total subset of E, there exists Yo K such that x0 = Py0 is a (nonzero) characteristic vector of T in K. (where

CoRoLLARY.

Under the assumptzons of Theorem 2, r(T) is also a characteristic

value of the adjoint T' of T in E', with (at least) one characteristic vector in the dual K' of the positive cone K of E. PRoOF. It is easily verified that T' is a compact endomorphism of E' for E) of uniform convergence on the compact convex subsets the topology Since this topology is consistent with <E, E'>, it follows that K' is a of E. closed, proper, total cone in E' with respect to ic(E', E). The observation that T'(K') c K' and that r(T) = r(T') concludes the proof. 2. Quasi-interior positive operators. While Theorems 1 and 2 of the previous section appear to reflect the essence of what could be expected of positive elements, or compact positive endomorphisms, with respect to their spectral behaviour at A = r, it is known from the finite dimensional case (ct. Introduction) that for certain types of positive endomorphisms (e.g., matrices with strictly positive entries), considerably stronger conclusions hold. Most of these additional results on the spectral radius r( T) could be established in [51 for "strongly positive" compact endomorphisms of an ordered Banach

space; here T is strongly positive if for each nonzero x€K, there exists n =n(x) such that T'x is interior to K. Thus this definition excludes, a

priori, many of the concrete naturally ordered Banach spaces (such as Hilbert from the discussion; moreover, in a non-formable space, no weakly space normal cone can have interior points [7, (7.6)]. In an unpublished paper by R. E. Fullerton,6 and independently in [!O], a weaker concept was introduced which covers most cases in which an intuitive notion of "strictly positive" elements is present: An element x of the positive cone K of an ordered topological vector space E is quasi-interior to K if K fl (x — K) is a total

subset of E. The relation of this concept to those of "support point" of K and (in the case of a vector lattice) of "weak order unit," has been discussed 6 Quasi-interior points of C0fl88 in a linear space, University of Maryland, 1957. In this paper, the basic properties of quasi-interior points are developed but no applications to linear mappings are made.

CONVEX CONES AND SPECTRAL THEORY

459

in some detail in [8]. With the aid of this notion, one defines [10] a positive endomorphism T of E to be a quasi-interior map if 00 c(T), and if there exists p > r(T) such that

y = R(p)Tx= is quasi-interior to K for every nonzero x E K. Since, when K has nonempty

interior, every quasi-interior point of K is an interior point and conversely, the present notion is weaker than that of strong positivity. A typical example of a quasi-interior map on 4 (in its natural order) is given by a bounded such that ti,k 0 and such that for each matrix T k) of subscripts, there exists n with > 0. As the principal available result on quasi-interior positive maps we quote the following theorem whose proof, which can be carried over verbally from [10, Theorem 2), will be omitted. Recall that a linear form f on (an ordered vector space) E is strictly positive if Re /(x) > 0 for every nonzero x eK. A locally convex space is called a locally convex vector lattice if it is lattice ordered such that its positive cone is normal and the lattice operations are continuous. THEOREM 3.

Let E be an ordered, semi-complete locally convex space, T a

quasi-interior Positive endomorphism. If the spectral radius r is a Pole of the

resolvent of T, then:

(1) r> 0 and r is a simple Pole of the resolvent. (2) Every characteristic vector Pertaining to r, of T in K (of the adjoint T' in K'), is quasi-interior to K (is a strictly positive linear form). (3) Each of the following assumptions implies that the dimension d(r) of the nullspace of (rI — T) is one:

(a) K has nonempty interior; (b) d(r) is finite; (c) E is a locally convex vector lattice. 3. A Tauberian theorem. Problem8. S. Karlin [4] has applied several theo-

rems on power series with non-negative coefficients to the resolvent of a positive operator in an ordered Banach space. Since the method Qf proof applies immediately to the case of an ordered locally convex algebra, we wish to formulate, as an example, the Tauberian theorem proved in [4] under the present more general conditions. THEOREM 4. Let a be a Positive element in an ordered, semi-complete locally convex algebra such that r(a) = 1 and r(a) is a Pole of the resolvent of order k. If e0 is the residue of the resolvent R(A; a) at A 1, then (weakly)

urn

=

(a

—

e denotes, as always, the unit element of the algebra A.) The proof of this theorem is, by Theorem A, immediate from the fact that the leading

coefficient of the principal part in the expansion of R(A; a) at A = 1 is (a —

H. H. SCHAEFER

460

and from the classical result' that a,

a

whenever a, 0 are the coefficients of a power series with radius of convergence 1 and representating a scalar function f which has a pole of order

k (k 0) at 1, with a the leading coefficient in the principal part of 1. Application of Theorem 4 to quasi-interior maps yields the following ergodic theorem. COROLLARY. Let E be an ordered, semi-complete locally convex space whose Positive cone is weakly normal and generating. Let T be a quasi-interior Positive map such that r(T) = 1 is a pole of R(A; T), and denote by P the residue of R(A; T) at 1. Then one has Jim

E =

for all xeE and The expansion at infinity of the resolvent of a positive element in an ordered locally convex algebra is distinguished from a general power series having its coefficients in a weakly normal cone, by a strong relationship between these coefficients and, in addition, by the fact that local elements of a resolvent have a single-valued extension (or monodromy) property. It is therefore natural to ask whether the spectral nature of the spectral radius r(a) has a stronger influence on the remainder of the spectrum on A I = r(a) than is expressed in Theorem 1. We formulate two questions that were raised in for positive operators on a Banach space, in a more general I

form. * Let A denote an ordered locally convex algebra, and let a be positive in

A with r(a) = 1. We assume A to be semi-complete and denote by RA the resolvent of a. (I) If 1 is an isolated singularity of RA, is every other singularity of RA on I A I = 1 necessarily isolated? (II) If 1 is a pole of RA, can RA have nonpolar singularities on the circle 121 =

1?

PART II. 1.

SPECTRAL ELEMENTS AND ALGEBRAS

Spectral measures. We recall that a spectral measure on a compact

space X into a locally convex algebra A is a continuous homomorphism p of with values in A. If p(1) * e, p can always be extended to a spectral measure fi on a larger space X into A such that = e; it is sufficient to take for the topological sum of X and a one-point space {w}, and to extend p to X by defining = e — e1 where e1 p(l) and denotes the characteristic function of (w} c X which is continuous on X. I Cf. E. * Added

C. Titchmarsh, The theory of functions, Clarendon Pres8, Oxford, 1939. in proof. In the stated generality, the answer to both questions is negative.

CONVEX CONES AND SPECTRAL THEORY

Let 58 denote the family of all bounded subsets of a locally convex algebra In accordance with the general definition of hypocontinuity for separately continuous bilinear mappings,8 we shall say that multiplication in A is left (right) 58-hypocontinuous if for each B €58 and every 0-neighborhood V in A, there exists a 0-neighborhood Uc A such that BUc V (UB c V). We denote by ..21' the positive cone {f:f(t) O} in THEOREM 1. If multiplication in A is left or right F8-hypocontinuous and if p is a spectral measure with values in A, then K = p(.2') is a normal cone. A.

PRoOF. Leaving aside the trivial case p = 0, by Lemma 1 below we can assume that p is one to one (hence an algebraic isomorphisrn); since the range

of p is commutative, we can suppose multiplication to be left $8-hypocontinuous.

If U is the neighborhood filter of 0 in A, we have to show that [U] converges to 0 (cf. the preliminary section). Let U€U, then [U] =(U+ K) fl (U— K)=

if ce[0,b—a], c=p(q) and U{la,bl:a,beU}. Now that is, 0 g p(f), then g [0,11, 1. On the other hand, the ele= ments of a dense subset of [O,f]_are of the form g = hI where 0 h 1, Hence [0, b — a] c J(b — a) where I p{[0, 1)) is bounded; thus h LU] c U+ J(U— U). if V is a given 0-neighborhood in A, it follows from b— a

the left 58-hypocontinuity of multiplication in A that there exists U€ U such that [U] c V and the proof is complete. If A is an ordered locally convex algebra with positive cone K, a spectral measure p on X into A is called Positive if fe implies p(f) K. COROLLARY. If multiplication in A is left or right 58-hypocontinuous, every spectral measure with values in A is positive for a suitable ordering of A This condition is, for instance, satisfied if A is the algebra of continuous endomorphisms of a locally convex space under the topology of bounded convergence; for the topology of simple convergence it is satisfied if the underlying space is tonnelé. To obtain simpler results we shall, in the remainder of this paper, restrict attention to locally convex algebras in which multiplication is either left or right Y&hypocontinuous. However, it should be noted that Theorems 3, 5, 6 and the sufficiency parts of Theorems 2 and 4 are independent of this assumption. LEMMA 1.

Let p * 0 be a spectral measure on X with support X0

No/p is a full ideal in

-

The kernel

and is identical with the kernel of

where

is the restriction of 1 to Xo. PROOF.

It is clear that N is a (closed) ideal in

to show that N is

full, it is sufficient to prove that 0 / g and p(g) = 0 imply p(f) = 0. Defining (n 1) by f = h,,(q + we obtain p(f) = since p(g) 0 by assumption. Because h, II 1 in it follows that {p(h,j} is I

bounded (in A), and therefore is bounded; hence, by a well-known theorem of Banach, it follows that p(f) = lime.... = 0. a

N. Bourbaki, Eapaces vectoriets topologiques, Hermann, Paris, 1955, Chapter III, §4.

H. H. SCHAEFER

462

To prove the second assertion, we observe first that if Jo = 0 for some f, whose support then f can be uniformly approximated by functions g in is contained in X X0 so that p(q) = 0; by the continuity of p, this implies we have to p(f) = 0. Assume, conversely, that p(f) = 0 for some 1€ show that = 0. Since N is the complex linear hull of its non-negative elements, we can assume that I 0. Now if f(to) > 0 for some e X0, there exists an open neighborhood U of t0 such that 1(1) e > 0 whenever t U. Also, since X0 is the support of p, there exists a function g 0 whose support is contained in U and for which p(q) = a * 0. But since N is a full ideal in p(f) 0 and fg eg imply p(€g) = ea = 0 which is contradictory. Thus p(f) = 0 implies fo 0 and the lemma is proved. LEMMA 2.

Let A be an ordered, semi-complete locally convex algebra. If p,

are commuting, Positive spectral measures on X, Y, respectively, into A such that

p(l) =

v(1)

=

e,

A with A(1i =

e

there exists a unique positive spectral measure A on X x Y into and such that A(f® g) = p(f)v(g)

for all!

and

Since the tensor product Y) is uniformly dense in ® x Y), the unicity of A is clear; we have to show that A exists under

PROOF.

the present assumptions.

Denote by A the range of v in A, equipped with the order and normed structures carried over from (1's the support of to which it is algebraically isomorphic by Lemma 1. Thus A is an ordered Banach algebra such that, if a = v(g) where g is real-valued, hail 1 is equivalent to —e a e. Let denote the space of continuous functions on X into A, equipped with its induced order and the uniform norm. We define a continuous mappng from into A as follows. Let Fe and fixed. Denote by be an arbitrary partition of unity, j i.e., a set 1, ..., n} such that 0 f,e and = 1 for all t X. Let Q = (t1, . .., be a set of points such that is an element of the support

of f, (1 =

1,

ofFinanyS1. We set fl(F;

n),

and denote by

the maximum oscillation

Q)

Q) and Q') be any two pairs with functions ft and g, (and points and s,), respectively (1 = 1, ..., n; j 1, . --, rn), such that €/2 and I €12. Put h,, = f,g,. Since II — F(s5) II whenever h,,. * 0, it follows from the properties of the norm in A (see above) that Let

I

I

I

—ee

[F(t1)

—

F(s5)]p(h15)

for the order relation in A (the latter being coarser than that of A because is positive); here the middle term is Q) — Q'). Hence if Q0) is a sequence with I

—o 0, it follows that ii(F;

Q,,) converges in

CONVEX CONES AND SPECTRAL THEORY

A to an element 1Ei(F) e K which is clearly independent of the particular sequence chosen. Also, F—+ fl(F) is positively homogeneous and additive, and

into A which we denote thus has a unique linear extension to all of fact that 4ü maps the unit interval (F: 0 F le} in The again by which contains interior points, into the unit interval of A which is bounded since K is weakly normal, implies that is continuous. x Y) and if, for each s X, h, We now define A as follows. If h e denotes the partial mapping I —÷ h(s, I), then clearly s v(h.) is an element of Set A = o where denotes the map h —* (s —'

v(h,))

Obviously A is a positive, continuous linear mapx Y) into gE ping satisfying 2(1) = e. We omit the easy proof that for fe from which it follows that A is multiplicative on A(fØ g) = 0 x Y), A is a homomorphism ',t'(Y); since the latter subspace is dense in of

x Y) into A and the proof is complete. of the algebra be an arbitrary family of compact spaces; X = H. X. is compact. Let is said to depend only on the variables of index a E H if 1. = An f€ (a H) implies f(t) = f(s) (t.—4 10 denotes the projection of X onto X0). The consisting of those functions that depend only of subalgebra on one variable of fixed index a, is clearly isomorphic with let onto When p is denote the obvious isomorphism that maps a spectral measure on X with values in a locally convex algebra A, a family of spectral measures (respectively on X0) into A, p. is said to be the and all a of this family if p(f) = for each fE product of the index set over which that family is defined. It is rather immediate that each spectral measure p on X is the product of a uniquely determined the converse question is settled by the following theorem. (A family family of spectral measures with values in one and the same algebra A is called abelian (or commutative) if the union of their ranges is contained in a commutative subalgebra of A.) In this theorem, A is assumed to be a semicomplete locally convex algebra. THEOREM 2.

Let {X0} be an arbitrary (non-empty) family of compact spaces,

a family of spectral measures, respectively on X. into A and such that = e for all a. In order that there exist a spectral measure p on fl X0 it is necessary and sufficient that be an into A such that p = 0. abelian family, and that there exist an ordering of A for which all are positive. PRooF. In view of the corollary of Theorem 1, the necessity of the condition is clear. To prove its sufficiency, we observe first that Lemma 2 extends immediately to an arbitrary finite number of compact spaces and Let H denote any finite subset 0 of the index set spectral measures the subalgebra of containing all functions and denote by depending only on the variables of index a e H. For fe define

H. H. SCHAEFER

464

p(f)

= (®

it is quickly verified that this definition is unambiguous. Thus p is a homointo A, positive for of morphism on the subalgebra V = UH This implies, are positive. any ordering of A with respect to which all by an argument completely analogous to one near the end of th& proof of by the StoneLemma 2, the continuity of p; since V is dense in with Weierstrass theorem, p has a unique continuous extension to Clearly this extension is a spectral measure with the required values in A. properties.

Let p be a spectral measure on X into A, and let 1€

be fixed. It

is quickly verified that the mapping g

p(g

of)

a spectral measure i.' on 1(X0) into A. v is called the associated real or complex spectral measure of the element a = p(f), accordingly as f is realor complex-valued. Since f(X0), which is the support of v, is identical with the spectrum of a (Theorem 3 below), every spectral element a of A can be represented as a = v(1) where 1 denotes the identity function on a(a); v is uniquely determined by a. In the case of a real spectral element, the unicity of o' is an easy consequence of the Weierstrass approximation theorem; in the complex case the proof (given in [12]) is much more complicated. If a A is a spectral element and v its associated complex spectral measure, the elements in the range of v are called "functions of a" and g —* g(a) = v(g) an operational calculus for a. 2. Spectral elements. Let p be a spectral measure on the compact space X into A and denote, as befose, by X0 the support of p. THEOREM 3. If a = p(f) is a spectral element of A, then u(a) = 1(X0) (hence is

a(a) is bounded)?

It is easy to see that a(a) c f(X0). For if A0 f(X0) and if denotes the restriction of I to Xo, the function A —, (A — with values in is defined and holomorphic in a neighborhood of A0. By Lemma 1 (N the kernel of p) can be identified; if po denotes the and spectral measure on Xo into A thus associated with p, then A —* — bY'] = p0(fg), clearly the resolvent of a. is holomorphic near A0 and, since Conversely, let A, be a point of the resolvent set of a. The assertion will X with = A,,, there exists an be proved when we show that for each = 0 whenever g open neighborhood U of to such that has its support in U. To this end, let s > 0 be a number such that A —* R(A) = (Ae — is holomorphic in IA— A01 < €; let U= {t€X: 11(t) — A01 <e/2} and let ge be a function whose support is contained in U. For each A satisfying by setting I A — A0 I > 3e/4, we define an element g(2) e 0

It is assumed that t'(l)

e.

If p(1)

e then c(a) = fiXo) U 101.

CONVEX CONES AND SPECTRAL THEORY

g(A;

t) =

if

A

1(1) * A

iff(t)=A.

0

It is readily verified that A — g(A) is holomorphic in P = (A: A — A0 I > 3e/4}, in the annulus and so is A 12[g(A)). We wish to show that ii[q(2)] 3e/4 < A — A0 < e; but obviously, the latter identity is valid in a neighborhood of infinity, hence (Ae — a)j4g(A)] = p(q) in that neighborhood. This implies, in by the identity theorem for analytic functions, that (Ae — a)u[g(A)] = D and, therefore, the assertion. Since 1 R(A) is holomorphic in A — A0 I < s, I

has a holomorphic extension to all of the complex plane; from

A —o

g(A)

= 0 throughout. proved.

p[g(A)J

= 0 and hence, by Liouville's theorem, = 0, the theorem is Since this clearly implies

= 0 we obtain limA...

It follows from Theorem 3 that a real spectral element of a locally convex is algebra is a spectral element with rea.l spectrum; conversely, if a = an element with real spectrum, then f must be real valued on the support of A spectral element is Positive if its spectrum is contained in the nonnegative real numbers; such an element is clearly positive for every ordering of A for which any presenting spectral measure (i.e., a spectral measure such that a = p(f)) is positive. It is evident that each spectral element a can be written as a = a1 + La2 where a1 , a± are real spectral elements such that the product (Theorem 2) of their associated real spectral measures exists; this condition, as may be concluded from Theorem 2, is also sufficient for a1 + ia1 to be spectral; furthermore, it makes the representation unique [121. Similarly, every real spectral element b = pCI) has a (minimal) decomposition b = b1 — b where b1 = and b2 = are positive. The following theorem

is concerned with the existence of positive spectral elements in a semicomplete locally convex algebra A. THEOREM 4. For a A to be a positive,spectral element, it is necessary and sufficient that there exist an ordering of A and a constant r > 0 such that

o

a

ye.

PROOF. If a = p(f) is a positive spectral element of A and if denotes the restriction of f to the support X0 of then it follows that 0 (Theorem 3). By the corollary of Theorem 1, there exists an ordering of A for which p is positive. Since it is-clear that 0 a iifo lie for any such order, the condition is necessary. To prove its sufficiency we assume, without loss of generality, that 0 a e for some order structure of A. We construct a real spectral measure presenting a as follows. Denote by P,, the polynomial on [0, 1]

It is well known that these polynomials (which are the Bernstein polynomials

to within a positive factor) permit the representation of any non•negative polynomial P as a linear combination a,,,,P,,,,, with a,,,, 0. Thus if we

H. H. SCHAEFER

466

define a linear mapping p by = a"'(e

—

homomorphism of the (real) algebra .9 of real polynomials on [0, 11 into A which maps the positive cone of .9 into the positive cone of A. This, as we have observed on earlier occasions, implies a unique, continuous extension fi to the continuity of p; thus which is a real spectral measure such that a = ft(1). It follows from Theorem 3 that 0(a) c [0, 11 and the proof is complete. a

COROLLARY 1.

tion c1e

a

In any ordered, semi-complete locally convex algebra, the rela-

c2e (c1 , c2 e R) implies g(a) c [c1, c2].

The essential contents of Theorem 4 can be transformed into a necessary and sufficient condition for an a £ A to be a real spectral element. Let at(r) = — a(r eR), and denote by C(a; r) the (convex) conical extenre + a, a0(r) = sion of (i.e., the smallest cone containing) the set CoRoi.LAU1Y 2.

m, fl C N) U For a to be a real spectral element of A, it is necessary and

sufficient that for at least one constant r >

0,

C(a; r) is a weakly normal cone

is

weakly normal. If K denotes

mA.

PROOF.

Let To > 0 be such that C(a;

To)

the closure C(a; To) in A, then K is weakly normal, an abelian subset of A, and contains a1(r0)° = e. Hence K is the positive cone for an ordering of A such that —Toe a roe. This implies 0 a + roe 2y0e whence it follows

by Theorem 4 that a + roe is a positive and therefore a a real spectral element of A. The necessity of the condition is easily proved with the aid of Theorem 1. A corresponding condition holds for arbitrary (complex) spectral elements: For c A to be a complex spectral element, it is necessary and sufficient that

there exist a representation c = a + ib where a, b commute and such that for some r > 0, C(a; r)C(b; r) is contained in a weakly normal cone. The necessity

of this condition follows from a remark made earlier in this section; the proof of sufficiency proceeds by constructing two spectral measures presenting a, b, respectively, and presenting c by their product (Theorem 2). 3. Spectral algebras. As the preceding section shows, spectral elements of a locally convex algebra can be completely described in terms of convex

cones (or equivalently, order relations) in A; we proceed to show that a similar situation prevails with respect to the spectral subalgebras of A in which spectral elements (by their definition) are imbedded. Our first theorem concerns the topological properties of spectral subalgebras.

Two locally convex algebras are called equivalent if there exists a topo. logical and algebraic isomorphism of one onto the other; we denote, as before, the spectral radius of a e A by r(a). THEOREM 5. Let A be a spectral subalgebra of A in its induced topology.

CONVEX CONES AND SPECTRAL THEORY

Consider the following properties: (a) a —' r(a) is continuous on A;

(b) A is equivalent to an algebra (c) A is closed in A. Then (a) is equivalent to (b), and (a) implies (c). PROOF. (a) (b). Let A be the range of a spectral measure p on X into A; by Lemma 1, we can assume without loss of generality that p is one-toonto A such that, if a = one. Thus p is an algebraic isomorphism of p(f), r(a) = If II (Theorem 3). It follows that a r(a) is a norm on A with into respect to which A is a Banach algebra; since p is continuous on A, this norm topology is finer than the (induced) topology of A. Hence if a —* r(a) is continuous on A, the two topologies are identical and p is a topological isomorphisrn.

then any isomorphism (b) (a). If A is equivalent with some establishing this equivalence is a spectral measure; the remainder follows from Theorem 3. (a)

(c).

complete.

is This follows from (a) (b) and the fact that every The situation that a r(a) is continuous presents itself, in par-

ticular., when A is a normed algebra since r(a)

Ia

in this case.

COROLLARY... Every spectral subalgebra of a normed algebra is complete.

It is an open problem to characterize the closure in A of a spectral subalgebra A on which the spectral radius is not a continuous function of a. It follows from the definition of a spectral algebra and Theorem 1 that each spectral algebra can be ordered in such a way that it becomes the (real or complex, accordingly as A is real or complex) linear hull of its unit interval J = (a: 0 a e}. To within a completeness condition, this property characterizes spectral algebras (and hence algebras of all continuous scalar functions on a compact space). THEOREM 6. Let A be a-commutative, ordered locally convex algebra which is the linear hull of its unit interval J; I is assu,ned to be semi-complete. Then A is a spectral algebra. PROOF. It follows from the hypothesis that A = A1 + iA1 where A1 = Ur n[—e, e] (the interval [—e, e] is equal to 2J — e). For each a e A1 there exists, by Theorem 4 and its second corollary, a spectral measure on c(a) c R which is positive for the given order of A. Denote by X the compact ae A1}. Since the family (Wa} is clearly abelian, Theorem 2 imspace fl plies that the product measure p ® exists on X into A (obviously the required semi-completeness of J can replace the semi-completeness of A in the present circumstances). Since a = for every a e A1, it follows that a = p(f,,) when denotes the projection of X onto X4 which is a continuous real-valued function on X. Thus if a + ib A (a, b A1), it follows that a + ib = + ifb) and the proof is complete. By combining Theorem 6 and the type of argument used in the proof of Theorem 5, one obtains what appears to be a new characterization of the

H. H. SCHAEFER

468

for compact X. Let A be a commutative (real or complex) Banach algebra with THEOREM 7. and only if it is capable of an ordering unit. A is equivalent to some under which it is the linear hull of its unit interval J. Such an ordering is necessarily a lattice ordering'0 of A. PRooF. The necessity of the condition is obvious. If the condition is satisalgebras

fied, Theorem 6 implies that A is a spectral algebra, i.e., there exists a spectral measure p on a compact space X onto A. By Lemma 1, p can be assumed to be an algebraic isomorphism; the continuity of a —o r(a) implies that p is a homeomorphism. From Corollary 1 of Theorem 4 and Theorem

3 it follows now that p is an order isomorphism (for the natural order of and the given order of A) and the proof is complete. REMARK. It is an unsolved question whether in Theorem 7, the word "commutative" can be omitted.* More generally, is a real algebra consisting entirely of real spectral elements necessarily commutative? With the aid of Theorem 6, it is not difficult to verify that every real algebra of Hermitian

operators on a Hilbei-t space H which is closed for the uniform operator similarly, every closed topology, is a real spectral subalgebra of *.algebra of normal operators on H is a (complex) spectral subalgebra of .9'(H). What examples are there other than those classical ones? We wish to quote, without proof, the following result whose details have been carried out in [12, §4]. Let E denote a semi-complete, locally convex vector lattice on which every the algebra of weakly positive linear form is continuous; denote by Continuous endomorphisms of E under the topology of simple

and by / the unit interval of .5t'(E) for the induced order. (These assumpThen tions imply that J is semi-complete, and an abelian subset of the linear hull of J is a spectral subalgebra of .2'(E). If, moreover, T -.-' 0 implies r(T)—.O, then this algebra is a complete normable algebra under the induced topology, and hence closed in Sf(E). For instance, it follows from the preceding result that the closed subalgebra of that contains all continuous endomorphisms of L,(O, 1) which have a diagonal matrix representation with respect to a fixed unconditional basis of (p > 1) is spectral. Moreover, with the aid of [7, (11.3)] it can (in

be shown that the algebras of Hermitian and normal operators mentioned above are contained in spectral algebras of the class considered in the previous paragraph. 4. Extension of spectral measures. Spectral elements of a locally convex

algebra A are defined as elements imbeddable in certain commutative subalgebras which are, essentially, algebras of continuous scalar functions on a compact space X; the spectral simplicity of these algebras, largely due to the absence of a radical, is then reflected in the spectral behavior of their 10

In

the complex case, a lattice ordering of K — K (K the positive cone in A). in proof. B. J. Walsh has answered this question affirmatively.

* Added

469

CONVEX CONES AND SPECTRAL THEORY

elements. For example, if a is a spectral element, then every isolated point

of 0(a) is a pole of the resolvent of order one, and c(a) = {O} implies a = 0. When A is an algebra of operators, spectral elements have spectral adjoints. However, to obtain the full range of re%ults familiar from the spectral de-

composition of Hermitian and normal operators in Hubert space, a somewhat more elaborate description (namely, in terms of projections) of a spectral operator is needed. We discuss in this section briefly the extension of spectral measures that makes such a description possible. Let S denote a normal (bounded) operator on Hilbert space. The spectral behavior of S is completely characterized by its spectral representation S=

AdP(A);

here .9: 8

P(8) is a countably additive and multiplicativeU mapping, with P(C) = I, of the Borel sets of the extended complex plane C into the set of orthogonal projections on H (under its strong topology). .9 is called the reso-

It is an imto take its values among the orthogonal projections of some Hubert space H; all that is needed to reproduce the classical theory of spectral representation is to construct, for a given spectral element of a locally convex algebra A, a resolution of the unit of A with the above properties but taking its values among the idempotents of A. It will be shown that, under comparatively mild completeness assumptions, every spectral measure with values in A gives rise to a resolution of unity. Conversely, if .9 is a resolution of unity defined on the Baire sets of a compact space X and with values in A, then lution of the identity (or spectral measure) associated with S.

portant realization to see that it is entirely unessential for .9

f—i.

(1€

is a spectral measure on X into A such that p(l) = e. Let A be a locally convex algebra, p a spectral measure on a compact space X into A. For greater convenience of presentation, we assume that A is a real algebra and a real spectral measure. Recall that the Baire (Borel) sets in K are the members of the smallest class of subsets of X that contains all closed sets of type C8 (all closed sets) in X and is invariant under the formation of symmetric differences and countable unions. A (finite) real-valued function on X is a Baire (BoreD function if its inverse f' maps the a-algebra of Borel sets in It into the a-algebra of Baire (Borel) subsets of X. Denote by

the algebra of bounded Baire (Borel) functions on X, and

by A the weak completion of A. A is a locally convex space with respect to tha weak topology u(A, A').

Denote by K the positive cone of A for any order of A with respect to which p is positive (Theorem 1, corollary); since K is weakly normal, A' = K' K' where K' is the dual cone of K. If co€ A', then p,(f) (p(f), is a measure on X [1]; denote by its extension in the sense of [11 to a'(X). is multiplicative if P(ô fl e) = P(ö)P(e) for arbitrary 8,

H. H. SCHAEFER

470

with the locally convex topology

We provide

generated by the

semi-norms (ço€K')

(this topology is, in general, not Hausdorif). It is easy to verify that 'e(X) into A for the and that p is continuous on for is dense in and a(A, A'). Thus p has a unique continuous extension to topologies with values in A that is obviously linear. If it is known that maps into A, then it follows from the separate continuity of multiand A (under the topologies presently considered) that plication in into A since p is a homomorphism of is a homomorphism of

Assume now that this is the case and, in addition, that 4a(1) = e. Then the set function o—*p(8) =

a resolution of the identity defined on the Baire (Borel) sets of X into A, is a homomorphism of the and said to be generated by p. It is clear that is

Boolean c-algebra of Baire (Borel) sets in X onto a Boolean cialgebra of idempotents in A, countably additive for c(A, A'). The following theorem into A, and actually maps gives conditions under which countably additive for topologies other than o(A, A'). THEOREM. 8. Let p be a spectral measure on X into A such that p(l) = denote by K the positive cone of A for the finest order with respect to which p is positive, and by J the corresponding unit interval in A. If J is weakly into A, and the maps semi-complete (weakly complete), then under which

resolution

is

the identity generated by p is countably additive for every consistent

topology on A under which K is normal. PROOF. We assume first that J is weakly complete. If we denote by I the fl I is dense in I for the 1) of unit interval {f: 0 then fl I) c J; this implies /1(1) c 3 and, topology Z considered above, and is the linear hull of I, that the range of /1 is contained in A. since

If I is only semi-complete for the weak topology, we proceed as follows. Let If,,) c I be a sequence such that limf,(t) = f(t) for all t X. Since {f,,) {,ü(f,,)} is a weak Cauchy sequence in A and is a Cauchy sequence for c J implies /1(f) EJ. Hence In is closed under the formation of simple limits of sequences. Since /r'(J) contains all continuous functions in I, it follows that it contains the characteristic functions of arbitrary Baire sets in X and, consequently, that is mapped into A by ,a. Finally, the assertion concerning the countable additivity of follows from its countable additivity for the weak topology c(A, A') and from Theorem C of the preliminary section. This completes the proof. Furthermore, Theorem C implies that when J is weakly complete, the limits ji(z,.) = /1(Za) = limG /1(Z9)

exist for every Baire (Borel) set ö c X, with respect to every consistent

CONVEX CONES AND SPECTRAL THEORY

471

stand for the directed topology on A for which K is normal. (Here sets of open (closed) subsets of X containing (contained in) 8.) We remark in conclusion that resolutions of the identity make it possible to integrate essentially unbounded functions with respect to a spectral measure, and thus to discuss spectral operators on locally convex spaces with unbounded spectrum. The details have been carried out in E12]. REFERENCES

1. N. Bourbaki, Integration, Actualités Sd. md. No. 1175, 1244, 1281, Hermann, Paris, 1952, 1956, 1959; Chapters 1•IV, V, VI. 2. F. F. Bonsall, Endomorphisms of a partially ordered vector space without order

unit, J. London Math. Soc. 30 (1955), 3. , Linear operators in corn piete positive cones, Proc. London. Math. Soc. (3) 8 (1958), 53-75.

4. S. Karlin, Positive operators, J. Math. Mech. 8 (1959), 907-937. 5. M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspehi Mat. Nauk 3 (1948), no. 1 (23), 3-95 = Amer. Math. Soc. Transi. No. 26, 1950.

6. H. Schaefer, Haibgeordnete lokolkonvexe Vektorraume, Math. Ann. 135 (1958), 115-141.

7.

,

Haibgeordnete

lokalkonvexe Vektorräume. II, Math. Ann. 138 (1959),

,

Haibgeordnete

iokalkonvexe Vektorrdume. 111, Math. Ann. 141

259-286.

8.

(1960),

113-142.

9. 10.

10

On non-linear positive operators, Pacific J. Math. 9 (1959), 847-860. , Some spectral properties of positive linear operators, Pacific I. Math. (1960). 1009-1019. ,

11. , On the singularities of an analytic function with values in a Banach space, Arch. Math. 11 (1960), 40-43. 12. , Spectral measures in locally convex algebras, Acts. Math. 107 (1962), 125-173. UNIVERSITY OF MICHIGAN

THE DUAL CONE AND HELLY TYPE THEOREMS BY

F. A. VALENTINE 1.

Introduction. A significant contribution to the theory of convex sets

was made by Minkowski when he introduced the support functional [24; 25]. The dual cone, in recent years, has enabled geometers to appreciate in clear spatial form the nature of the support functional of a convex set. The technique used here is similar to that used by Dieudonnê [3], Klee [19) and others in their proofs of the Hahn-Banach theorem. The same technique was used their support functionals by Fenchel in the characterization of convex sets [5]. Also see Hille and Phillips [14]. A more extensive use of this method was made by Sandgren [33] in his study of certain Helly type theorems. It is perhaps significant to mention that we discovered Sandgren's results before reading his paper. Our paper includes refinements of known results, together with a number of further applications. For instance, parts of the proof of Theorem 12 are new, and Theorem 15 is new. Each theorem in 2 and 3 is preceded by a historical acknowledgement. Before developing the main theory, a few well-known elementary facts will be described in essential detail. The reader who is familiar with the papers mentioned may very well by-pass this first section.

Although we introduce the theory for sets in Euclidean rspace E, many of the results hold equally well in a complete inner product space. In order to describe matters easily, we use the following notations: Notation. In the following, x• u denotes the usual inner product of x e E, u E, where E is the space. The closure, interior, boundary and convex hull of a set S c E are denoted by ci S, mt S, bd S and cony S. respectively. The closed line segment joining x e E and y E E is indicated by xy, whereas L(x, y) denotes the line determined by x and y, if x * y. The interior of a set S c E relative to the minimal flat containing it is denoted by intv S. Set union, intersection and difference are U, (1 and respectively. We let 0 and 0 stand for the origin of E and the empty set, respectively. Finally, x + y denotes vector addition where xe E, ye E, and Ax denotes scalar multiplication, x e E, 2€ being the real field. We assume the set M 0 except for Theorem 5. DEFINITION 1. If M is a convex set in E then the real-valued functional h is the set of all those pairs Eu, h(u)], u E, such that h(u) = supx-u < co.

functional h is called the support functional of M. The support functional satisfies the following well-known properties [1).

The

473

F. A. VALENTINE

474

THEOREM l. The domain of definition D of a supPort functional h given by (1) is a convex cone having the origin 0 as an apex.

The function h is a continuous, subadditive and positively honwgeneous'functional for all u E D, so that h(Au)

(2)

Ah(u)

,

D, u€ D, v€D.

A

h(u + v) < h(u) + h(v),

0, u E

Theorem 1 is an immediate consequence of Definition 1. For instance, if

ueD, veD,

we have

+ v)

XEM

x€.V

sup Ax.u = xEM

+ supx•v < 00, ZEM

reM

so that u ÷ v e D, Au e D, so that (2) holds.

It should be observed that if a point x0e M exists such that h(u) = x0•u, then = h(u) is the equation of a plane of support to M at x0, since (1) implies

x€M. The continuity of h follows from the convexity of It. THEOREM 2.

If M is a bounded closed convex set in E, then for each u e E

there exists a point x0 M such that h(u) = x0 . u.

This is a consequence of the compactness of M. In the infinite dimensional case, the so-called weak compactness implies the corresponding result. REMARK 1. If M has the support functional It defined on D, then a translate M + x1 of M has a support functional It1 defined on D such that

ueD. THEOREM 3. If M is a nonempty closed convex set in E with supPort functwnal It defined on a nonempty domain D then M satisfies the condition

h(u),uGDJ.

To prove Theorem 3, observe that the set H,. [x: x'u h(u), u fixed in DJ is a closed half-space. Since M c H,. for each u D, we have

McflH,,. ,.E 0 there exists a point z H,.) M * 0. Without loss of generality, assume 0 M. Since M is closed, there exists a point w = Az (0 < A < 1) such that w M. Since M is a closed convex set, and since the space E is locally convex [19], there exists a hyperplane H such that H, B ri M = 0. Hence, H strictly separates Mand z, since w = Az, 0 < A < I, and since 0 €M. Suppose

Thus, there exists a vector

whose corresponding segment from 0 is perpendi-

cular to H and intersects H. Let H,' denote the closed half-space bounded

THE DUAL CONE AND HELLY TYPE THEOREMS

475

< so that by H which contains M. Since M c we have SUPXEM e D. However, since z H:, we have z IL, a contradiction.

Hence, (3) holds. DEFINITION 2. A functional h satisfying condition (2) for all u on a convex

cone D c E is called a support functional. We are now in a position to develop the main theme of this paper. 2. The dual cone. As mentioned in the introduction, the principle of the dual cone has been used by Fenchel [5], Sandgren [331, and it plays an increasingly important role in modern functional analysis, Köthe [22]. It was essentially used also by Rademacher and Schoenberg [29] in their simple proofs of two theorems of Kirchberger [18]. It is the opinion of this writer that the principle of duality used here has only begun to be exploited. No other concept has influenced us as much in recent years, and my colleagues have patiently endured my sustained enthusiasm, for which I am indeed grateful. is the real field, let If E is a Euclidean space and E) denote the usual product space with the corresponding product topology. The reader may wish to carry through the corresponding theory when E is a complete inner product space, which can be done with no difficulty. If E = E2, then E) may be chosen to be E3. This model may be useful to the reader unfamiliar with the theory. The axis (z, 0) will be called the vertical z axis

of

E). DEFINITION 3.

Let M be a convex set in a Euclidean space P2, whose support

functional h is defined on D. Points (z, u) e

Then the dual cone C of M is the set of all

E) such that

C=[(z,u):z h(u),ueD]. Although we will be primarily interested in the case when D is a closed

cone, there are instances when this is undesirable. However, if M is bounded and closed, then C has the following preferred form. THEOREM 4.

If M is a closed bounded convex set in E, then the dual cone

E) defined for all u E, and having the (5) is a closed convex cone in E) as its vertex (0 is the origin of P2). The vertical ray origin (0, 0) e [(z, 0), z > 0] C mt C. PROOF. The boundedness of M implies the inequality in (1) holds for all P2, so that h is defined for all u. Since h is a convex functional, the set C is convex. The positive homogeneity of h implies that C is a cone with (0, 0) as vertex. Clearly C is closed in the product topology. Also since u

(I, ø)eintC, clearly (z, ø)eintC, if z>0.

As shown by Fenchel [5], convex sets can be characterized so simply by their support functionals. There is no need to resort to the Gateaux differential [1] as was done in the past. The following proof is due to Fenchel 15]. THEOREM 5.

Let h be a

functional defined

a closed convex cone

F. A. VALENTINE

476

D * 0 having 0 as an apex (see Definition 2). Then the set of points M= [x:x.u h(u), all u€D] convex set.

is a PROOF.

It is not the convexity of M

which

ft=rl,u€D

is difficult to prove, for if

+ Bh(u) = h(u), so that ax + $y e M. Hence the only real difficulty is to prove M is not empty. Consider the dual cone C given by (5). Since C is a closed convex cone containing the point (1, 0), and since (—1, 0) C, a fundamental separation E) exists which contains e theorem [19] implies a closed hyperplane E) being locally convex. The translate (—1, 0) and misses C, the space Hof which contains the vertex (0, 0) of C supports C such that (1, 0) H. E) the hyperplane H has the form Since (1, 0) H, in ah(u)

H=[(z,u):z=xo.u,ueE). Since

H supports

C, we have

The proof where E is a complete inner x0 M, and thus M * 0. product space is essentially the same, with a few modifications similar to the corresponding proof for the Hahn-Banach theorem [191. REMARK 2. Let E" be a closed flat in E of deficiency n. The dual cone of is a cone whose domain of definition is a subspace of dimension n. (The deficiency of E" is the difference of the dimension of E and of E*.) This well-known and obvious result is a consequence of the fact that the condition in (1) holds only for those directions which are perpendicular to and these directions yield a flat of dimension n. In order to derive simple proof s for Helly type theorems, the following significant modifications of the theorem of Carathéodory [21 are given. THEOREM 6. Let S be a set in a linear space L. Choose an arbitrary point v e S. Then x e cony S sf and only if x is contained in a finite dimensional simplex having its vertices in S, and having v as one its vertices. PROOF. The proof follows conventional lines. See [1). THEOREM 7.

Let

(i = 1,

n)

be n convex sets in a linear

space L and

let

Then x E cony

v,xj,xt, ...,x. in Sufficiency.

if and only lix €4, where 4 is a simplex with vertices

and where s

n.

By Theorem 6, there exists a finite dimensional simplex 4 and containing x. If N> n, by a in U= x, , x2, •,

with vertices v,

relabeling x1 E

C1,

of subscripts there exist two vertices, say x1 and such that C1. Since C1 is convex, it is simple to verify that we can replace x1x2 such that XE COflV (v U z U x3 u •.. u XN). By

x1 and x2 by a point z

THE DUAL CONE AND HELLY TYPE THEOREMS

477

induction, the stated conclusion follows. Generalizations of theorems of Carathéodory type have been recently presented by my colleague Professor Motzkin in a colloquium address before the American Mathematical Society [271.

The two following theorems are similar to the theorems of Sandgren [33] and are obtained from Theorems 6 and 7. THEOREM 8. Let {M,, i e A} be a family of convex sets in E. If they all have a point in common then all the dual cones C, corresponding to M., i A, have a common nonvertical hyperplane of support in

ueE].

E) of the form [(z, u): z —

PRooF. Choose x0 M, so that h,(u) for all u eD,, 1€ A. This implies that the hyperplane E(z, u): z = ue El in E) bounds all the cones C given by (5) with h replaced by h,. It is clearly nonvertical in E). When the sets M4 are bounded and closed, it is of particular interest to obtain Sandgren's theorem [33] by a significant application of Theorem 6. THEOREM 9. Let {M1, i A) be a family of bounded closed convex sets in E. They all have a Point in common if and only jf all the open cones, mt C,, i A, have a common nonvertical hyperplane of support [(z, ii): z = x0 U, u E E] in (a', E), where C, is the dual cone of M1. This is equivalent to the condition

(6)

conv(UintC1)

PROOF. A. First, suppose fl, M1 * 0, where, for simplicity of notation, the index i ranges over A. This implies a common nonvertical hyperplane of

support

H= [(z,u),z = xo.u,ueE] to all the cones C , i A, exists. Clearly H also supports each mt C,, i e A, at (0, 0). Conversely, if H supports each mt C, i A, it also supports each C,, i A, since C1 = clint C,, so that •u h(u) for all u E, i A, and xoefl1M1. B.

To prove the necessity of (6), suppose (6) fails. Since (1, Ø)€ mt C1, ie A,

Theorem 6 of Carathéodory implies there exists an s-dimensional simplex 4

having its vertices in U mt C, having (1, 0) as one of its vertices, and such that (0, 0)64. Since each vertex of 4 is in mt C,, and since (1, 0) E mt C1, i e A, each hyperplane in E) through (0, 0) must intersect at least one of the open cones mt C,, 1€ A. However, this violates the result of the preceding paragraph A. Conversely, suppose (6) holds. This implies (0, 0) bd cony (Li1 mt C1). A nonvertical hyperplane H of support to cony (U1 mt C1) through (0, 0) exists. Hence, paragraph A above implies fl1 M, * 0. Again we use the Carathéodory type Theorem 7 to yield very simply the

F. A. VALENTINE

478

following theorem of Lanner [23J. in a recent letter Professor Folke Lanner informed me that Professor F. Riesz (311 in a letter to his brother Professor M. Riesz described a proof essentially the same as that which follows. ., n) be n compact convex sets in a Euclidean TSEOREM 10. Let M, (i = sfrace E, and let h, denote the support function of Mi (i = 1, ., (i = .. n) have Then a necessary and sufficient condition that the sets a common Point is that

(7) holds for each set of points u (i =

•, n) satisfying

1,

Sufficiency. We prove the sufficiency by proving its contrapositive. Suppose M, = 0. Then Theorem 9 implies

E).

=

(ci

E) is contained in a simwhere v = (0, 0), where s n, and C (k = 1,. .•, s). The proof of Theorem 7 implies that we may, without Xk loss of generality, relabel subscripts and choose s so that Xk Ck (k = 1, . , s). Since E), let Xk (z&, E)(k = 1, s). Since

Theorem 7 implies that the point (—1, 0) e

plex 4 having vertices

v, X1,

•, x1

•

(—1, O)€conv(v, x1, there exists constants A,

> 0, such that

0,

(—1,0) =

v1)

where s + 1. v,) = (0, 0) and A, = 0, if (0, 0) as a common vertex, we have U, Ay, dition (9) implies —1

=

0

A,e,

Since C, are cones, having

A,h,(v,) =

Since

h,(v,), con-

E h,(u,)

ru,.

Since (10) violates (7) and (8), conditions (7) and (8) imply proves the sufficiency. Necessity. This is trivial, for x0 M, * 0 implies

M, * 0. This

x0.u,_h,(u,) for all u,€E, whence (8) implies (7). 3.

ilelly type theorem8. We now establish 1-lelly's theorem [121, this time

479

THE DUAL CONE AND HELLY TYPE THEOREMS

appealing to Carathéodory's Theorem 6. (See also Sandgren [33].) THEOREM 11.

Let

be

a family of compact convex sets in Euclidean n-space

having at least n + 1 members. A necessary and sufficient condition that all the members of 5 have a common point is that every n + 1 members of 9- have a common point. is finite, then the word "compact" can be If the number of members of omitted to obtain the same conclusion. PROOF. Let 9- = {M,,,, a A}. To prove the sufficiency of the condition = 0. Theorem 9 when M0 , a A, are compact convex sets, suppose flIEA where C. is the dual cone of M.. Since implies cony (U.€4 mt C0) =

(1, 0) mt c. , a A, Carathéodory's Theorem 6 on simplices implies that X.+I , s n, such (0, 0) €4, where 4 is a simp!ex with vertices v, x1 , x2, that E U.€A mt C.. Without loss of generality, relabel subscripts so that e mt C, (i = 1, . , s + 1). This, together with the fact v (1, ø)e mt C,,, a€A, implies that each hyperplane through (0, must intersect (i = 1, - -, s + 1) and hence at least one of the at least one of the edges -, s + 1). Since s + 1 n + 1, this contradicts the (i = open cones mt fact that the s + 1 cones C (i = •, s + 1) have a common plane of support M, = 0, a contradiction. at (0, 0) in Es), which in turn implies that Since the necessity is obvious, this completes this part of the proof. When the set .F has a finite membership, it is a simple matter [29] to prove the existence of compact convex polyhedera P0, a A, such that P. c M., and such that every n + 1 members of {P., a A} have a common point, thus reducing the situation to the compact case. We now state and prove a theorem which exhibits in fine form the power of duality. The implication (ii) (iii) is due to Horn [16] (see Sandgren [33] for a similar proof using duality). For n = 2, it was first proved by Horn and Valentine [15]. Our proof differs from Sandgren's in that we use an elementary theorem proved by Hanner and Radström [11]. The implications (iii) -- (iv) and (iv) —÷ (ii) are due to Klee [21]. The dual reasoning used here implies these results with remarkable simplicity. The implication (1) —' (ii) (Valentine) has been added as it is significant, and the proof of Klee's result v

- -

- .

-

-

(ii) is simplified. Shapley 117]. (iv) —p

For another proof of

(ii)

(iii) see Karlin and

THEOREM 12. Let ..9 = {M,,, a A} be a family of bounded closed convex sets in E. The following four statements are equivalent, where n is a positive integer, and where contains at least n members.

(I) n— 1 (See

For each set of n members of 9 and for each flat

of deficiency

there exists a translate of E"' which intersects these n members of 9-.

Remark 2.)

(ii)

Every n members of 5 have a point in comnion.

Every flat of deficiency n is contained in a flat of deficiency n — 1 which intersects every member of (iv) Every fiat of deficiency n — 1 has a translate which intersects every (iii)

F. A. VALENTINE

480

member of PRooF THAT (1)

(ii).

Suppose this is false.

This and Theorem 9 implies intC,) = ,n), such that By Theorem 7, there exists an

there exists n members of is the dual cone of E), where having mt s-dimensional simplex 4 with $ n having its vertices in E). (1, 0) v as one of its vertices, and containing the origin (0, 0) of E) which contains denote an n-dimensional subspace of I et denote the subspace of E of deficiency n — 1 whose support 4. Now let is a Hence, the set ((z, u): z = h(u), u function h is defined on say M1 (i =

-

E has a linear subspace of the space (a', E,,_1). A translate support function h1 which also has E._1 as its domain of definition. Since 1€ A, (0, 0) e 4, since each vertex of 4 is an interior point of some cone and since v (1, Ø)eintC0 for all aEA, the (n —1)-dimensional hyperplane must intersect at least one of the open in [(z, u): z = hj(u), u e n, and u): z = cones mt (i = 1, •--, n). Hence some cone, say c,, I j E). Dually, h1(u), u e E,,_11 cannot have a common plane of support in of cannot intersect M,. Since this this implies that the translate holds for each translate of the hypothesis (i) is violated, and hence (i) (ii) has been established. PRooF THAT (ii) (iii). In order to prove this we first state the following theorem of Hanner and Radstrom [11]. Let S be a compact set in an ndimensional Euclidean space Suppose p is a Point of & for which no s points of S exist with s n whose convex hull contains p. Then there exists a hyperplane H through p which does not intersect S. To prove (ii) —' (iii) let E' be a flat of deficiency n in E and having a supbe the nport function h defined on a subspace & of dimension n. Let dimensional hyperplane in E.) defined as follows: (11)

H,,

[(z, u): z = h(u), u E E,j -

To prove that (ii) (iii) it suffices to establish the existence of an (n — dimensional subspace H,,_1 of H,, such that

1)-

(12)

for each M,, 5, where C. is the dual cone of M.. We prove (12) for the case when 9 has finite since the infinite case follows readily. To accomplish this let (13)

C. for which H,, n mt = 0, so let A, denote the subset of A such that H,, n mt C. 0, a e A1. Let the interior of relative to H,, be denoted by Tnt so that we have

aEA,. Firstly, if (14)

/

* H,,

481

THE DUAL CONE AND HELLY TYPE THEOREMS

then any (n — 1)-dimensional hyperplane of support in H,, to B through (0, 0) is the desired H,,_1, since it satisfies (12). Secondly, if

conv(U IntK.) = H,,, .E mt K. * 0 for a A2, there exist closed convex cones a E A1, all having 0 as a common vertex such that mt since

c 0 U mt K.,

0, a G A1, and where K. is as defined in (13). By mapping the such that K. as i —i sets J, on the unit sphere of H,, with center at (0, 0), the theorem of Hanner and Radström [11] preceding condition (11) implies the following: (a) either the origin (0, 0) in H,, is contained in a simplex 4 having at most n vertices belonging to UII€A1 0) and hence belonging to U.€4, mt K or for each 1(1= 1, 2, of H,, such that (b) there exists an (n — 1)-dimensional hyperplane A1. 0) for all H,_1n J,,'= (0,

In case (a), since the vertices of 4 belong to UOIEA1 mt C., and since E) through (0,0) could support (1, O)EintC.,aEA, no hyperplane of those n or fewer cones of the set (C.. a A} which contain the vertices of 4. Dually, this contradicts (ii).

In case (b), since the sets K, are locally compact, the sequence of hyperhas a convergent subsequence which converges to an (n — 1)planes dimensional hyperplane H,,_1 c H,, satisfying (12). Let h1 be the linear function of H,,-2 defined on a subspace E,,_1 of E. Then

E'1

[x: x-u

h1(u), u

E,,_1]

1 which contains E', and which intersects every member of 9—. This completes the proof of (ii) (iii). PROOF THAT (iii) (iv). Let E'1 be a variety of deficiency n — 1 > 0 with support function h defined on E,,_1. Let E,, be any n-dimensional subspace

is a flat of deficiency n —

of E containing E,,_1, and choose x e E,,

sional subspace of space E,,-1, where k = dimensional subspace

E,,)

2,

E,,_1.

Let E,,k denote the n-dimen-

which contains the point (1, 0) + x/k and the

3, . -.. Hypothesis (iii) implies E,,k contains an (n — 1)E,,) which misses each open cone mt Ce,, of

E,,) is finite dimensional, a convergent subsea e A. Since the space quence of exists which converges to a space called Also since ri E,,) is locally compact, the space E,,1_1 also misses each c 00, we have open cone mt C,,, a A. Since Ek,, E,,_1) as k E,,_1).

The linear functional h2 of

is defined on E,,_1, so that

[(z, u): z = h1(u), u

E,,_2]

-

flat E1111 in E having h1 as its support function is a translate of E'1 which intersects every member of 9-. Since (iv) — (i) is trivial, we have completed the proof of the equivalence of the four statements. The

The following theorem of De Santis [34] has a particularly elementary proof

F. A. VALENTINE

482

when examined dually. of convex THEOREM 13. If every k + 1 or fewer members of a finite family members of sets in .E contain a common flat of deficiency k, then all the contain a common flat of deficiency Fe. •, s) denote the member of .5, and let D1 denote PROOF. Let M. (i = 1, of M1 (see Theorem 1). the domain of definition of the support function The dual of the hypothesis in Theorem 13 implies that every Fe + 1 of the cones D1 (i = 1, .. ., s) are all contained in a linear subspace of dimension k. This fact implies that

(JD. is contained contains at most k linearly independent vectors, so that should (In this proof, the domains in a linear subspace of dimension k. The dual of the not be confused with the dual cones C, defined in (5).) contains a 1, . •, s) foregoing implies that every Fe + 1 member of M1 (i = common flat of deficiency Fe which is a translate of a fixed space of deficiency in Ek. Let be the set of Fe, denoted by Ek. Choose a basis ej , e2, ••, is convex. which are parallel to e,. Clearly each all lines belonging to Let Ek be the orthogonal complement of Ek. Applying Helly's theorem to for each fixed .j = 1, , N, we immediately obtain the the sets K1, fl

existence of N independent lines in fl

M1,

whence fl..i M contains a flat

of deficiency k.

We now go on to investigate the existence of common transversals and common intersecting planes. For a recent thorough summary of results on these matters see GrUnbaum [8]. We first present a new proof for a known

result for sets in E, and we then present a new result for E1 which no doubt can be extended to in exactly the same way. The following theorem was discovered by Klee [20] and Grunbaum [7] independently.

Suppose .9 is a family of compact convex sets in the Plane Also suppose there exists a line L c such that for each Pair of members in there exists a translate of L which strictly separates these two sets. If every three members of are intersected by a common line in E,, then all the members of .? are intersected by a common line. PROOF. In order to prove this we use the following concept. DEFINFrI0N 4. If C, is the dual cone for the compact convex set M1, then let C denote the symmetric image of with respect to the apex (0; 0) of C and relabel C1 so that C' U C:, is the two napped cone determined by It should be observed that C,' n C,! is either a point, a line or a plane in THEOREM 14.

E2.

E2).

If M4 and M1 are two distinct members of

then, since M, ii M, = 0,

Theorem 9 implies (15)

cony (mt C1 U mt C,) =

Es).

THE DUAL CONE AND HELLY TYPE THEOREMS

483

and M1, Let L which by hypothesis strictly separates is defined on only a and let the support function of L" be Clearly one-dimensional subspace of E2, denoted by E1,. Since L" is a translate of L for each pair i and j, we have (16)

E1

and E1 is independent of i and j. The set

E

L,

(17)

u E,)

[(z, u): z =

is a line in the vertical plane E1) c (a', E,) through (0, 0). that strictly separates and M1 implies we have (18)

— (0,

0) c (mt

The fact

U (mt

Hence, Definition 4 implies (19)

0) c mt (C n CJ). K. denote the set of all lines in E2) through (0, 0)

L1, — (0,

DEFINITION 5. Let

which miss mt C. To prove Theorem 14, we merely need to prove the dual conclusion

fl K,*(0,O).

(20)

.9

To do this choose a point p E2 — + p. namely,

so that the product space of

and

is a plane parallel to (a', E1) through p. Then the intersection of

with

E1

this plane, namely

+/'), is

a closed connected set lying between two open disjoint convex sets mt C1(p) where

and

Cr(p) Cr fl E1 + P) Let (a = 1,2) be a closed half-line of the vertical line which is in the interior of Cr(p) n C7(p). Similarly let

(a

p)

=

= 1,2). 0) + p

and R.' be two half-lines of the line L1, + p which are in the interiors of C(p) n C(p) and C(p) n respectively, where L4, + p is the translate of L,, passing through p. Let the endpoints of (a, = 1,2, a be denoted by x?f, respectively. Since

cony (R?, u

c Ct(p), c Cf(p),

(a,

we have 1(4 n K1 n (R, E1 + p) c

u

so that the left member of (21) is a compact set.

U

U

= 1,2, a *

484

F. A. VALENTINE

The left member of (21) is also a cell (homotopic to zero in itself) [8]. To x) in the plane prove this, first observe that each vertical line + P)

intersects K1 n K5 in a segment or point. In terms of the usual positive E1 + p), this implies that the boundary orientation of vertical lines of E1 + /') relative to E1 + p) can be decomposed into an of K n K, n upper and lower part, each of which is an arc. Since mt C,'(p) n mt C?(p) = 0 (i = 1, 2), the upper part consists of two convex arcs or points which belong to bd and bd CJ(p), respectively, whereas the lower part belongs to two convex arcs or points belonging to lxi C(p) and CJ(p), respectively. The first two arcs are concave upwardly, and the last two concave downwardly. The upper and lower arcs may intersect, but they do not cross. In any case, it is quite clear that these facts imply K, n K5 n E! + p) is a cell. This

fact may be proved in other ways, and a combinatorial procedure is given in Theorem 15 for higher dimensions. To prove (20) holds, it suffices to prove it for each finite subcollection .9'T of .9. Let S be infinite right circular cylinder having the vertical line p) as its axis of symmetry, and such that it contains all of the compact sets K1 fl K5 fl E1 + p) for M1 e M, e It is obvious that + p) are cells for M. e flL since mt C,'(p) n mt C(P) = 0. K. n S n Also since each line in K1 n K5 through (0, 0) must intersect E1 + p), the hypothesis of the theorem implies

÷p)*0 for each triple

M1, Mk in Since the sets in (21) and K, n S n E+ p) are cells, condition (22) implies, by Helly's generalized [8; 131 on intersecting cells, that (20) holds for .9-i. Compactness then implies

(20) holds for .9-. The dual of (20) implies a line exists in E2 which intersects all the members of This completes the proof. We are now in a position to generalize Theorem 14 to E3, and the method used will extend to E,,. THEOREM 15. Let 5 be a family of compact convex sets in E1. Also suppose there exist three distinct planes P (i = 1, 2, 3) in E3 containing a common line such that for each triple of members M1, M1, M3 in 5 each pair of the triple is strictly separated from the remaining member of the triple by a trans late of either P1 or P2 or P8. and this correspondence is cyclic. If every four members of .9 have a common intersecting plane, then all the members of .9- have a common intersecting plane.

For three sets M1 (i = 1,2, 3) in .9-, without loss of generality, we may rearrange the subscripts on (i 1, 2, 3) so that there exist translates of I'd, denoted by + P1. such that M1 and cony (M2 U M2) are strictly separated by P1 + p1 , M2 and cony (M8 U M1) by P2 + P2, A'!3 and cony U M2) by P8 + p - The permutation of indices is cyclic, and we will indicate the

range (1,2,3) of the indices only when necessary. Since the planes {P3 (i = 1,2,3)) have a common line, the support function h, of P is defined on a line A, and A c E3 (i = 1,2,3), where E8 is a plane

485

THE DUAL CONE AND HELLY TYPE THEOREMS

of

E3 independent of i. Also E2

+ p, in the extended space

the dual cone of

As a consequence,

cony (D1 u D2 u A). E5)

is a vertical half-

plane bounded by the line u D1, a = [(z, u): z = is the support functional of P1 + where In order to prove Theorem 15, it is sufficient to show that (20) holds when EL has been replaced by E3. We will show that (20) holds for each finite (i = 1,2,3). Since, by of the family 9. Choose M, c subfamily agreement, P1 + strictly separates M1 from cony (M1 u Mk) (i, j, k = 1, 2, 3, i L1

,

j * k), we may translate the three planes P, + p1 so that the three lines and L3 in (23) are not coplanar, and still have the separation property

relative to M1, M, and M8. Moreover, there exist translates P1 + P1 (i = 1, 2, 3) of P, which divide into seven parts in such a way that'each of the three unbounded nonadjacent

regions bounded by three faces contains one and only one of the sets M3 in its interior. In the rest of the treatment we assume that the translates P + p1 separate the sets M,, ML, M3 this way. The fact that the from cony (M1 U Mk) implies, in terms of a cyclic plane P1 + p, separates notation of indices, that M1 , M2,

(24)

— (0,

0) C [(mt C) fl (mt Ck)] U (mt C1).

In terms of the notation of Definition 4, condition (24) implies that the line into two relatively open half L1 is divided by the origin (0, 0) of (a = 1,2; i 1,2,3) such that rays

RrcCrnCrnCr,

(25)

j, the vertical axis

i

D1, D2, D3 determine a three-dimensional subspace L1c (26)

Choose a fixed point p e Es plane E, + P, namely,

0) and the three lines E2)

of

E2 so that the product space of

E3).

Also

(i = 1,2,3). with the

(27) E3) parallel to E5). To prove (20) for the sets is a hyperplane in K. with E2 replaced by E3 in Definition 5, consider the sets

K1 n K2n

K1n

+P)

(i,j1,2,3,i*j). Let (a=1,2), where

is defined in Definition 4.

We will prove that

in (28) is a cell (homotopic to zero in itself). The

F. A. VALENTINE

lies between the two disjoint unbounded convex sets mt Cj1(P) n E3 + and mt C.8(p). The translates L, + p and Rf ÷ p of L, and Rf, respectively, E3). Since Cf are cones, conditions E2 + p) of lie in the hyperplane set

(25) imply that the one-dimensional sets n mt n mt Nf mt

n (R? + P)

are half-lines.

Since

each of the unbounded nonadjacent regions determined by P, +

P,

2, 3) which is bounded by three faces contains one and only one of the U u N) is intersected by the vertical ray sets M1, M,, M3, the cony (i = 1,

[(z,p): z > 0].

Hence we may choose points

e Nf sufficiently far out on Nf so that cony U U xb and cony in parallel planes, and so that cony (xi' U x' U x) c mt Cf U mt Cf U mt Cf. conv(xr U

xu

intCf n intCf n

U

U

lie

intCf *0.

The following set U U U U U Q cony U is an octahedron each of whose faces is a triangle, and the faces cony and cony U U U are parallel. Moreover, we may choose x' so that the faces are congruent and parallel. The condition x? e Nf (see (31)) and the convexity of mt Cf imply that (33)

cony (x? U

c mt C(p)

u

where i,j = 1,2,3,1 * j; a, = 1,2, a *

Hence the set K13 given by (28) is in the interior of the octahedron as given by (33). Therefore K122 is compact, can be since it is closed and bounded. Incidentally the compactness of shown in another way very easily. We will demonstrate that K123 is a cell (houiotopic to zero). In the simplest case, when C11(p) n Cr(p) = 0 (i = 1, 2, 3), K123 is a cell, having six curved

faces and eight vertices, so that it is topologically like a cube as far as the incidence of faces, edges arid vertices go. However, our proof will also cover the special cases in which the cube-like character of K123 assumes a simple but degenerate form. are disjoint open convex sets in To prove this, since mt and mt E2 + p) which separates mt C,1(p) E, + there exists a plane H, in and mt C(P). Moreover, the form of Q in (33), conditions (32) and (34), imply that H1 n H2 n H8

where

the planes

is a line in

E2 + /) n Q,

The space E + P) is divided by E2 + (i = 1,2,3) into eight octants, so that we may regard the planes

THE DUAL CONE AND HELLY TYPE THEOREMS

+ having q as a common point. Since as coordinate planes in C(p))u ri mt C7(p) = 0, i * j, and since bd Qc L, ri mt Ct(/') = 0, (mt C,'(p))], we have Q ri L•, n bdC (35)

i*j *k.

Consider, for example, the octant A containing the points course, q. Let the three faces of the octant A be denoted by H1, ,

and, of H13, H23

so that The following sets J112 n

(36)

,

H,., ri KI,,. ,

1112

ri

are each cells. For instance, H33 n K323 is bounded by either a simple closed curve consisting of the segments an arc in the bd C and an arc of the Ext

or it may be bounded by two such arcs and a segment, or it may

reduce to a segment or to a point. In any case, it is a cell. The intersection

of the octant A with the set K323 can be decomposed into a set of parallel directed segments and points with initial endpoint on one of the cells in (36) and parallel to the half-line N,1 given in (31), since c mt C(p) Hence the intersection of the octant A with K123 is also a cell in E, + P). Since this holds for the intersection of each of the eight octants, determined (i = 1, 2, 3), with K133 it follows immediately that K12, is itself a cell. by There are other ways to prove that K12, is a cell, and the reader may wish to form the proof in the same manner that was used in Theorem 14. Finally, let S be an infinite right solid cylinder in E, + P) having the vertical line p) as its axis of symmetry, and a radius such that all of

the sets (28) are contained in S for each triple of sets M1, M3, M, in the finite family sets

c 5. It is a relatively simple matter to verify that the

Sn K, n K, n (W, E, +P) Sri +p) are cells for every M and M, in Since K1 n K, n Kb n (0, 0) for i j k, the hypothesis in our theorem implies

E,) =

(37)

for every four members M, M,, in Hence, this, together with the fact that the sets (28), (36.1) are cells, implies, by Helly's generalized theorem on intersecting cells [8; 13], that

fl holds. Since (38) holds for each finite subcollection

implies that

of 5, compactness

488

F. A. VALENTINE

fl.9' * (0, 0). The dual of this condition implies there exists a plane in E, which intersects This completes the proof. all the members of It is of interest to express Rademacher's and Schoenberg's simple proof of Kirchberger's theorem t291 in terms of our notation, since it is so simple. The following is Kirchberger's first theorem [18]. THEOREM 16. Let P = {p} and Q = (q} be two compact collections of points in Euclidean n-space Then P and Q can be strictly separated by a hyperplane if and only if for each subset T of n + 2 or fewer points of P U Q there exists a hyperplane H( T) which strictly separates T n P from T n Q. Psoor. We will prove the sufficiency when P and Q are finite collections. To do this let h, and denote the support functionals for p E P, q E Q, respectively, so that (39)

h,(u)=u.p, hq(u)=u.q,

qeQ.

The dual cone C, of p and the closure of the complement of the dual cone of q, denoted by c:, have the form [(z, is), z h,(u),peP] (40) [(z,u),zhq(u),qeQJ, and (39) implies these are all closed half-spaces. Let T be n + 2 points chosen (ium P U Q. The existence of a hyperplane strictly separating T n P from T n Q implies dually that a relatively open half-line with endpoint (0, 0) in exists which is in the interiors of the cones C,, for all p T n P, q e T n Q. Belly's Theorem 11 applied to the interiors of the cones (halfspaces in fact) in (40) in the (n + 1)-dimensional space &) implies there exists a relatively open half-line in with endpoint (0, 0) which is in the interiors of all the cones C, , P P, C, q e Q. Dually, this is equivalent to the existence of a hyperplane in which strictly separates P from Q. This completes the proof when P and Q are finite collections. A simple argument involving compactness can be given to prove the case when P U Q is not finite. We will leave this to the reader. The following corollary is of some interest. The reader may prefer to restate it in less picturesque style. Cq

CoRoLL4uiY 1. Let B be a finite collection of black sheep and let W be a finite collection of white sheep in If for each set T of n + 2 or fewer sheep chosen from B U W there exists a hyperplane H( T) which strictly separates the black sheep in T from the white sheep in T, then there exists, a hyperplane which strictly separates all the black sheep in B from all the white sheep in W. PROOF. This is an immediate consequence of Theorem 16 in which the collection P consists of all the points belonging to the sheep in B, and Q consists of all the points belonging to the sheep in W.

THE DUAL CONE AND HELLY TYPE THEOREMS

In Corollary 1 has an interesting dual for sets on the sphere in order to state it, we require the following definition. denote the n-dimensional surface of the unit sphere in DEFINITION 6. Let E,,) with center at (0, 0). The the (n + 1)-dimensional Euclidean space with an open The intersection of Point (1, 0) is called the north pole of half-space is called a cap. If a cap contains the point (1, 0) and if it lies on a hemisphere of 5 it is called a convex polar cap. The complement of the closure of a convex Polar cap is called a polar cocap. It should be observed that the set of half-lines having (0, 0) as a common endpoint and intersecting a given convex polar cap is a convex cone which contains the positive z-axis [(z, 0), a > 0].

of the THEOREM 17. Suppose .9- is a finite family of sets on the surface unit sphere in Es). Suppose each member of is a convex polar cap or a polar cocap, and suppose that every n + 2 or fewer members of .9- have have a point in common. a point in common. Then all of the members of (It should be observed that by Definition 6, the members of 9 are open relative. to Se.) - •, s) be the members of .9 which are polar caps, denote the members of .9 which are polar cocaps. Let C, denote the cone formed by the set of half-lines having (0, 0) as a common endpoint and intersecting cl K,. Attach a corresponding meaning to PROOF.

Let K, (1

and let K (j C}.

1,

-

1,

-

-, t)

Since

[(a, Ø):z > 0]c intC1 [(a, 0): z < 0] c mt C,'

(j

= 1, - . -, t)

let

(i1,--.,s),

z,=h,(u) =

(j = 1, ---, t) be the equations of the surfaces of the cones C, and C,', respectively, in Es).

Hence C, C,'

[(a, u): a [(a, u): z

h,(u)J, h(u)] -

The sets M, and M,' in EM, defined as follows, Md

M,'

Lx: x

h,(u), u e E,,J,

h(u),u€EM]

are compact convex sets in EM - Now let B = i = 1, - - -, s} and W = ---,t}. Let T beany set of n+2 members of Bu W. The dual of the hypothesis of Theorem 17 implies there exists a hyperplane H(T) in EM which separates T fl B from T 11 W. Hence, Corollary 1 implies a hyper-

plane H exists which separates B from W. The dual of this statement imthat a relatively open half-line in plies in EM) exists having

F. A. VALENTINE

(0, 0) as endpoint, and lying in all the cones mt (i = 1, . . ., s) and mt C,' (j = -••, I). This, in turn, implies that all the members of 5 have a point in common. This completes the proof. Kirchberger theorem [181 is included for the sake The following of completeness. Our proof reformulates that of Rademacher and Schoenberg [29) in terms of our notation. If for and Q = {q} be two compact sets in THEoREM 18. Let P = each set T of 2n + 2 or fewer points of P U Q there exists a hyperplane H(T) which separates T n P from T n Q, then there exists a hyperplane which separates P from Q (not necessarily strictly). PRooF. As in the proof of Theorem 16, let C, and c be the closed halfspaces defined in (40). By hypothesis every 2n + 2 or fewer of these closed half-spaces, for p P. q EQ. have a ray in common with endpoint 0) in E,.). By a theorem of Steinitz [37] or Dines and McCoy [41 or Gustin with endpoint (0, 0) which (101 this implies the existence of a ray in

lies in all the half-spaces C,, p P and Ce', q Q. The dual of this implies that a hyperplane exists in E% which separates P from Q.

E) and the dual cone 4. ConcludIng remarb. The extended space often enable one to obtain an insight into the situation. The dual cones usually center the difficulties in the neighborhoods of the origin (0, 0) of R). The story is still an incomplete one since there remain theorems whose duals have not been studied. To mention one, the duals of the theorems of Helly type on the sphere, and the recent result of Grunbaum [9] on "the dimension of intersecting sets" have a close connection. The contents of Grilnbaum's paper [91 includes the following new result. THEOREM 19.

Suppose

is a finite family of convex sets in Euclidean n-

space E1. If the intersection of each n + d or fewer members of

contains a convex

set of deficiency d, where d is an integer with 0
Theorem 19 has an interesting interpretation in terms of duality. Just as in the proof of Helly's theorems, we may investigate the case when the members of

are compact.

Since each member M of 5 contains an (n — d)-dimensional sphere, the dual cone for M has a special form. The following concept was also used by Sandgren [33] for other purposes. Also see Gale [6] and Gerstenhaber [61. DEFINITION 7. A closed convex cone K in E with apex at 0 is called a lunar d-cone if the maximal number of linearly independent lines which Pass through

0 and which lie in K is d. For brevity a lunar d-cone will be also called a d-cone.

In

a right circular cone with apex at 0 is a 0-cone; a convex cone

bounded by two nonparallel half-planes each containing 0 is a 1-cone; a half-

THE DUAL CONE AND HELLY TYPE THEOREMS

491

space bounded by a plane through 0 is a 2-cone, and the whole space is a 3-cone.

Theorem 19 implies that each set of n + d of the dual cones corresponding are contained in a lunar d-cone in to the members of having ((z, 0): z > 0] as an interior ray. For E,,) the dual of Theorem 19 implies that all the dual cones are contained in a lunar d-cone. However, since M * 0, the following form of the dual of Theorem 19 for can be used instead. THEOREM 20. Let .5Y' = (K,, a = 1, -••, m} be a finite family of closed convex cones is E having the origin 0 as a common apex. If the union of every n + d or fewer members of is contained in a lunar d-cone having 0 as an apex, where 0
We will show that for d = 1, the proof is relatively simple, although the corresponding case for Theorem 19 is not so obvious. Since for 0 < d < n — 1, the whole space E is a lunar n-cone containing K, there exists a cone C containing and having 0 as an apex, which has minimal lunacy r. If r d, then the entire Theorem 20 follows. We will now prove the theorem for d = 1. Suppose r 2. The linear subspace of dimension r of E, contained in the cone C of minimal lunacy r is denoted by L,. Since contains a finite number of members, L ,is the convex hull of points belonging to U. Kg,, and we have by the theorem of Steinitz [37] that, relative to Lr, 0 is in the interior of a simplex 4 whose vertices xi, - - -, x. (s 2r) belong to K. Since n — 1 > 1, we have n 3, and the only case needing proof is that for which contains at least four members. If there exist two sets of diametrically opposite pairs of vertices of 4 relative to 0, then clearly these four points of U. K. cannot be contained in a lunar 1-cone. Hence, there exists a vertex of 4, say x1, such that the line L(x1, 0) intersects (xa, . .., x) -

only in x1. Let

L(x1, 0) n bd4

x1 uy1 -

is not a vertex of 4, y1 is contained in the relative interior of a 4 where has at most n vertices, and at least two vertices. Hence 0 is in the relative interior of Since

simplex

4,

cony (x1 U di).

Since 4, has at least three vertices, and at most n + 1 vertices, the vertices

of 4, are points of U. K. which are not contained in a lunar 1-cone having 0 as an apex. Hence, we have a contradiction, and Theorem 19 has been proved for d = 1. This proof should not be confused with the corresponding theorem where d = n — 1 (see Grilnbaum [9]) which is also elementary. The

proof for d = 2 is also fairly easy to construct from the initial fact that three sets of diametrically opposite pairs of vertices of 4 relative to 0 cannot exist. The corresponding proof for d> 1 should follow the same lines, and it is quite clear that an appropriate theorem of Carathéodory type will yield the

F. A. VALENTINE

492

desired result. This should be done. Also Theorem 19 and the theorems of Robinson 132] on "spherical theorems of Helly type" have an intimate connection. This is a matter which invites further investigation. The lunar dcones will play an important role. In conclusion, since it is a rare coincidence for the proofs of a theorem and its dual to be of equal difficulty, there is a double reason to investigate the dual. One may gain either a simpler proof or a less obvious theorem. 1. T. Bonnesen and W. Fenchel, Theorie der konvexen Korper, Springer, Berlin, 1934. 2. C. Carathéodory, tJber den Variabiiitifitsbereich der Koeffizienten von Potentreihen, die gegebene Werte nicht annahnten, Math. Ann. 64 (1907), 95-115. 3. J. Dieudonné, Sur théorè',ne de Hahn-Banach, Rev. Sci. 79 (1941), 642-643. 4. L. L. Dines and N. H. McCoy, On linear inequalities, Trans. Roy. Soc. Canada Sect. LII 27 (1933), 37-70. 5. W. Fenchel, Convex cones, sets and

functions, Mimeographed lecture notes, Prince-

ton, 1953. (Also see Bonnesen.) 6. D. Gale, Convex polyhedral cones and linear inequalities, and M. Gerstenhaber, Theory of convex polyhedral cones, Activity Analysis of Production, and Allocation, Wiley,

New York, 1951. 7. Branko Grünbaum, On a theorem of L. A. Santalo, Pacific. J. Math. 5

(1955),

351-359.

8.

,

Common

transversals for families of sets, J. London Math. Soc. 35

(1960), 408-416.

9.

,

The

dimension of intersections of convex sets, Pacific J. Math.

12 (1962), 197-202. 10. William Gustin, On the interwr of the convex hull of a Euclidean Set, Bull. Amer. Math. Soc. 53 (1947), 299-301. 11. Olof Hanner and Hans Radstrom, A generalization of a theorem of Fenchel, Proc. Amer. Math. Soc. 2 (1951), 589-593. 12. E. Helly, Uber Men gen konvexer Kor per mit gemeinschaftlicher Punkten, Jber. Deutsch. Math. Verein. 32 (1923), 175-176. 13. , Uber Systeme aigebschlossener Mengen mit gemeinschaftiichen Punkten, Monatsh. Math. 37 (1930), 281-302. 14. E. Hille and R. Phillips, Functional analysis and semigroups, Amer. Math. Soc. Colloq. Vol. 31, Amer. Math. Soc., Providence, R. 1., 1957. 15. A. Horn and F. A. Valentine, Some properties of L-sets in the plane, Duke

Math. J. 16 (1949), 131-140. 16. A. Horn, Some generalizations of Hefly's theorem on convex sets, Bull. Amer. Math. Soc. 55 (1949), 923-929. 17. S. Karlin and L. S. Shapley, Some applications of a theorem on convex functions,

Ann. of Math. (2) 52 (1950), 148-153. 18. P. Kirchberger, Uber Tschebyschefsche Annaherungs Methoden, Math. Ann. 57 (1903), 509-540.

19. V. L. Klee, Convex sets in linear spaces, Duke Math. J. 18 (1951), 443-466. , Common secants for plane convex set8, Proc. Amer. Math. Soc. 5 (1954),

20.

639-641.

21. (1951), 272-275.

,

On

certain intersection properties of convex sets, Canad. J. Math. 3

THE DUAL CONE AND HELLY TYPE THEOREMS

493

22. G. Kôthe, Topologische linear Raüme. I, Springer, Berlin, 1960. 23. Folke Lanner, On. convex bodies with at least one point in common, Kungi. Fysiogr. Salisk. i Lund Förh. 13 (1943), 41-50. 24. H. Minkowski, Theorie der konvexen Korper, insbesondere Begrundung ihrea Ober.flächenbegriffs, Ges. Abh. 2 (1911), 131-229.

25. , Volumen und Oberftäche, Math. Ann. 57 (1903), 447-495. 26. N. H. McCoy. (See Dines.) 27. Theodore S. Motzkin, Convex sets in analysis, Bull. Amer. Math. Soc. (to appear). 28. R. Phillips. (See Hille.) 29. H. Rademacher and I. J. Schoenberg, Hefly's theorems on convex domains and Tchebycheff's approximation problem, Canad. J. Math. 2 (1950), 245-256. 30. Hans Radstrom. (See Hanner.) 31. M. Riesa and F. Riesz. (See comment preceding Theorem 10 of this paper.) Also see Sandgren's paper, p. 23. 32. C. V. Robinson, Spherical theorems of Helly type and congruence indices of spherical caps, Amer. J. Math. 64 (1942), 260-272. 33. Len'iart Sandgren, On convex cones, Math. Scand. 2 (1954), 19-28. 34. Richard De Santis, A generalization of HeUy's theorem, Proc. Amer. Math. Soc. 8 (1957), 336-340.

35. 1. J. Schoenberg. (See Rademacher.) 36. L. S. Shapley. (See Karlin.) 37. E. Steinitz, Bedingt konvergente Reichen und konvexe Systems. I-Il.!!!, J. Reine Angew. Math. 143 (1913), 128-175; 144 (1914), 1-40; 146 (1916), 1-52. 38. F. A. Valentine. (See Horn.) 39. P. Vincensini, Figures convexes et variétés linéaires de l'es paces euclidean a n. dimensions, Bull. Sci. Math. 59 (1935), 163-174. UNIVERSITY OF CALIFORNIA, Los ANGELES

UNSOLVED PROBLEMS Like elementary number theory, the subject of convexity lends itself readily

to the statement of interesting unsolved problems. Many of these can be appreciated on an intuitive level and may be accessible to anyone with a bright idea, for the subject (on the whole) is one of many ideas and specific approaches but little machinery. The discussion of unsolved problems was an important part of the Symposium, both informally and in a special session

devoted to them; several of the papers published here originated in such discussion. Unsolved problems are found in many of the papers, and the list below contains other problems stated during the Symposium or sent later to the editor.

V.K. W. CHENEY

M a linear subspace of R', and denote the pth-power norm in Let II for each x e let ir,x be the (unique) point of M which is nearest to x with (M, x fixed) as respect to II I,. What can be said about the behavior of

H. S. M. COXETER

If the edges of a convex polyhedron all touch a sphere of unit radius so as to form a "crate" from which the sphere cannot escape, prove or disprove that their total length is at least 9 i/T. (Cf. [1] for discussion of the related problem in which the requirement of touching is omitted.) [1J A. S. Besicovitch, A cage to hold a unit-sphere, these Proceedings, pp. 19-20. L. DANZER

Given a convex body (i.e., a convex, compact point-set with nonempty interior)

C in R'. Say its (euclidean) width (minimal distance between two parallel supporting hyperplanes) is d(C). Define its k-dimensional width dk(C) to be the maximal width attained by any intersection of C with a k-dimensional flat. Clearly d1(C) = diam (C) = d(C). I ask for the numbers q(k; n): = inf {dk(C)/d(C): C a convex body in R" (1

iii particular, for q(2; 3).

It is trivial that 495

k

UNSOLVED PROBLEMS

496

q(k+ 1;n+

but is it true, that also q(k; n + 1) Note that q(2; 3) < 1, as shown in [1).

q(k;

n)?

The example there could be simplified

(using a regular tetrahedron instead of a cube), but certainly that method is not good to prove q(2; 3) < .995 nor will it yield lower bounds for q(k; n). Of course one may ask the same questions for any Minkowskian metric over R' instead of the euclidean one. [I] L. Danzer, Uber die maximale Dicke der ebenen Schnitte eines konvexen Körpers, Arch. Math. 8 (1957), 314-316.

C. DAVIS

In n-space (n > 2) it is natural to consider, along with the diameter D and width d of a convex body K, intermediate measures. In particular, let Dk(K) = mm D(PkK), where Pk means projection to an (n — k + 1)-flat and the minimum is over all Pk; and d4(K) = max where means section by an 1flat and the maximum is over all For ellipsoids, one proves from the Fischer-Courant principle that = dk, the length of the kth principal axis. In general, of course, = D1 D and = d1 = d; however, the extension of the ellipsoid case can not go far, even for centrally symmetric K. PROBLEM. In 3-space, find the possible range of variation of d2/D2. Perhaps even the dependence of this range on dID could be found; see Besicovitch's problem. It is clear that d2/D2 1, and it is equal to 1 not only for ellipsoids but for a variety of other bodies including all those with rotational symmetry. It is equal to i/)374 for the regular tetrahedron. The smallest value I know is L/T/2, attained for a class of centrally symmetric octahedra including the regular, and including others of arbitrarily small dID. For n > 3 other possibilities arise: let Dk1k2(K) = max provided n; similarly k1 + k, I do not claim to see hope of provDkikaka, --S. ing anything about these compound quantities for general K. (Cf. Danzer's problem.) A. DVORETZKY

For a Minkowski space E

and

vi(E) = max

uIx1U=i

mm

positive integer k, ii

± x1 ±

± ...

define

±

)

j,

the minimum is over all 2k possible choices of + and — signs and the maximum is over all k-tuples x1, •- •, of unit vectors. What can be said about the numbers vk(E)? where

UNSOLVED PROBLEMS

K. FAN PROBLEM.

Are the following two rotundity properties of a Banach space

X equivalent? (1) Every sequence (2) Every sequence

in X with in X with

U (x,, +

U

+

= =

1

1

is convergent. and having no

weak cluster point of norm <1 is convergent. PROBLEM. What can be said of the structure of the lattice of all closed bounded convex sets in a normed linear space? (For results of this nature, see [1; 2].) El] E. E. Floyd and V. L. Klee, A characterization of reflexivity by the lattice of closed subspaces, Proc. Amer. Math. Soc. 5 (1954), 655-661.

G. W. Mackey, Isomorphisms of normed linear spaces, Ann. of Math.

(21

(2) 43 (1942), 244-260.

B. GRUNBAUM

Characterizations of circles and spheres Besicovitch was the first to establish [1] the conjecture of V. Mizel that the circle is the only 'closed convex curve C in the, plane with the property (i) Whenever three vertices of a rectangle belong to C, the fourth vertex also belongs to C. A simpler proof of Besicovitch's result was found by Danzer [21. In view of this result, the following questions seem to arise rather naturally. 1. Is the circle the only closed convex curve (resp. simple closed curve) C with the property: (ii) Whenever three edges of a rhombus R support C, the fourth edge of R also supports C? 2. Is the circle the only convex Curve of constant width with the property: (iii) Each point of C is the vertex of a square, all of whose vertices belong to C?

It is well known that the circle is not the only convex curve with property (iii), even if all the squares are required to be of the same size. 3. Is the (n — 1).dimensional sphere the only surface S of constant width in with the property: (iv) Every point of S is the vertex of a regular n-dimensional octahedron, all of whose vertices belong to S? For n 3 it is even conceivable that property (iv) alone characterizes spheres among all convex surfaces. (1]

A. S. Besicovitch, A problem on a circle, J. London Math. Soc. 36 (1961),

241—244. [2] L.

99-100.

W. Danzer, A characterization of the circle, these Proceedings, pp.

UNSOLVED PROBLEMS

B. GRUNBAUM—T. S. MOTZKIN

A graph is k-polyhedral if its nodes and edges can be identified with the

vertices and edges of a k-dimensional convex polyhedron. (See [1] for references and for some properties of polyhedral graphs.) For n 5, the complete graph with n nodes is known to be 4-polyhedral. Conjecture: For k 4, every kpolyhedral graph is 4-polyhedral. (1] Branko Grunbaum and Theodore S. Motzkin, On polyhedral graphs, these Proceedings, pp. 285-290.

P. C. HAMMER 1. REFLECTION OVER A CONVEX CURVE IN THE PLANE (B. H. NEUMANN). Let C be a closed convex curve in the plane with no line segments in its boundary.

From a point p exterior to C choose that line of support through p which has C on its left (looking from p). Let q = jp be the reflection of P through the point of contact on the line of support. PROBLEM. Is there a simple closed curve B (other than C) such that ID = B? REMARKS (HAMMER).

It may be shown that there are n-point sets X for

each n 3 such that fX = X and that the union of all of these is unbounded in the plane. Moreover if Y is an open "annulus" bordering C, then X = —00 < n < oo} is an open set and its boundary is fixed under f. However, is X bounded and if so is its boundary a simple closed curve for some

Y? That there is a large class of convex curves C with solution curves B may be seen as follows:

Let B be a closed convex curve and in each family of parallel chords of B take the two which cut a fixed smaller area a from B (a < 4/9 area B). Then the intersection C of all closed strips between such parallel chords determine a convex curve C such that fB = B.

On the other hand it is easy to construct a convex curve C such that lf'pln 1, for certain points p converges to a point on C. Whether or not is unbounded for some p we have one may find a curve C such that not settled.

Note that the transformation f is area preserving and that the problem is actually affine. 2.

THE X-iuy PROBLEMS (HAMMER).

Suppose there is a convex hole in an

otherwise homogeneous solid and that X-ray pictures taken are so sharp that the "darkness" at each point determines the length of a chord in C along an X-ray line. (No diffusion, please.) How many pictures must be taken to permit exact reconstruction of the body if: a. The X-rays issue from a finite point source? b. The X-rays are assumed parallel? For the planar counterpart, we have shown that two perpendicular directions are insufficient for (a) and we conjecture that 3 directions are sufficient, although

UNSOLVED PROBLEMS

whether or not such directions must be strategically chosen is also open. 3. SELF-CIRCUMFERENCE (HAMMER). Let C be a closed convex curve in the plane. Then, as Minkowski first proved, to each interior point p of C there is determined an (asymmetric) metric with C as the unit circle, i.e., d(p, q) = 1 for q e C. Hence to each p interior to C there is determined two circumferand a_(p) of C. ences PROBLEM. What are the properties of the set of points such that a+(p) = a_(p)? In particular, if Po is the point such that the minimum value of the set mm [a+(p), a_1(p)J is achieved at p° is a+(Po) = a_(P0)? Moreover are mm a+(p)

and mm a_(p) achieved at the same point? REMARKS. If C is symmetrical with respect to a point then a+( p) = a-( p) for each p interior to C. The minimum value of circumference might be

called the self-circumference of C. 4.

These circumferences are affine invariant.

SELF-CIRCUMFERENCE (G0LAB VIA B. GRUNBAUM).

Let the situation be as

in (3). What is the maximum self-circumference for all convex curves C? REMARKS (HAMMER).

For the triangle the self-circumference 9 is achieved

and Po is the centroid. In this case a+(p0) = a_1(p0). Laugwitz showed that the regular hexagon has minimal self-circumference 6 and the square maximal self -circumference 8 among convex curves with a center. Presumably 6 is the absolute minimum self-circumference. This is one of the few cases in which the circle does not appear as an extreme solution. V. KLEE

Suppose K is a compact convex subset of a Hausdorff linear space. Must

the topology of K be locally convex; i.e., is it true that for each x K and each neighborhood U of x there exists a neighborhood V of x relative to K such that V is convex and V c U? Must K have the fixed point property? Must K have an extreme point? A.

For 0 < n < m and 0

k n — 1, an rn-dimensional compactum in E will be called (n, k)-convex provided its intersection with each n-dimensional affine subspace is k-acyclic. Then (1, 0)-convexity is (by definition) equivalent to ordinary convexity, and (n, n — 1)-convexity is also known to be equivalent to ordinary convexity. There exists a simple geometric characterization of (2, 0)-convex sets in E3. PROBLEM.

Find a geometric characterization of (n, k)-convexity. T. S. MOTZKIN

Find an intrinsic characterization of those n-tuples (k0, . . -,

such that

500

UNSOLVED PROBLEMS

there exists an n-dimensional convex polyhedron having (for 0 faces of dimension i. exactly

I

n

— 1)

R. R. PHELPS

Suppose K is a compact convex set in a locally convex Hausdorff linear Must K include a point x such that for each y e K— {x), supfK = fx > fy for some continuous linear functional f? space.

H. H. SCHAEFER I. Let E denote a (Hausdorif) locally convex vector space over R, K a convex cone of vertex 0 in E, and K' the dual cone (of linear forms nonnegative on K) in the topological dual E' of E. Does there always exist

KcE such that E=K—Kand E'=K'—K'? (In a normed space, thecone K spanned by a ball of radius r >

0

and with center at a distance > r from

the origin, answers the question affirmatively.) II. Denote by A an algebra over R, provided with a locally convex vector space topology under which multiplication is separately continuous. A spectral element of A is an element contained in a subalgebra which is the continuous homomorphic image of some (X compact). Is a subalgebra of A, consisting entirely of spectral elements, necessarily commutative? (This question arises in connection with Part II, §3 of [1].) [1] Helmut H. Schaefer, Convex cones and spectral theory, these Proceedings, pp. 451-471.

F. A. VALENTINE—E. G. STRAUS

Does there exist a nonempty compact set S in R' such that 2 m(x) for all x e S, where m(x) is the number of convex subsets of S which are maximal relative to being convex, including x, and having dimension

n

— 1?

INDEX OF UNSOLVED PROBLEMS Besicovitch, A. S. 15, 19 Bishop, E. and Phelps, R. R. 34 Corson, H. and Klee, V. L. 37, 38, 42, 48, 49 Coxeter, H. S. M. 60, 61, 67 Danzer, L. W. 100 Danzer, L. W., Grünbaum, B. and Klee, V. L. 144—146, 148-155, 159—161, 163

Davis, C. 181, 185, 197, 198 Dvoretzky, A. 209 Grtinbaum, B. 235-264, 271-281, 290 Hammer, P. C. 302, 314, 315 Klee, V. L. 349, 352-354, 357 Phelps, R. R. 393 Valentine, F. A. 490—492 Unsolved Problems 495-500

501

109, 117—120, 122-132, 137—142,

AUTHOR INDEX Italic numbers refer to pages on which a complete reference to a work by the author is given.

Roman numbers refer to pages on which a reference is made to a work of the author. For example, under Minkowski would be the page on which a statement like the following occurs: "This theorem was proved earlier by Minkowski 17, § 2] in the following

manner... Boldface nuabera indicate the first page of the articles in this volume. Abe, Y., 116, 16* Ahiezer, N. 1., 82, 93 Alaoglu, L., 74, 93 Alexandroff, p., 125, 168 Allen, J. E., 160, 183 Amemiya, 1., 75, 83, 98 Anderson, K. W., 82-84, 86, 88, 90, 91, 93 Anderson, R. D., 21, 28, 273, 281 Ando, T., 75, 83, 86-88, 91, 92, 98

Besicovitch, A. S., 13, 15, 19, 10, 21, 24, 99, 100, 141, 150, 164, 233, 242, 254-256, 265

Bessaga, C., 37, 45, 50 Bieberbach, L., 257, 165 Bielecki, A., 152, 164, 237, 254, 265 Bing, R. H., 161, 164 Birch, B. J., 117, 164, 247, 252, 265 Birkhoff, G., 74, 93, 215, 219 Bishop, E., 27, 35, '92, 94, 401 Blanc, B., 160, 164 Blaschke, W., 141, 164, 235, 244, 251, 258,

Arms, R. J., 98 Aronszajn, N., 143, 161, 183, 200, 355-357, 858

262, 285

Ascoli, G., 94 Asplund, E., 152, 168, 235, 242, 248, 253,

Blichfeldt, H. F., 60, 70 Blumenthal, L. M. 114, 132, 138, 139, 158,

258, 264

159, 161, 164

Atsuji, M., 48, 49, 50 Auerbach, H. 261, 264

Boas, R. P., Jr., 84, 94 Bochner, S., 84, 94 Bohm, J., 70 Bohme, W., 242, 243, 185 Bohnenblust, H. F., 50, 75. 94, 108, 115,

Bagemihi, F., 151, 163 Balinski, M. L., 287, 290

Banach, S., 49, 50, 76, 85, 94, 153, 154,

123, 134, 165, 281, 303, 355, 858 Boland, J. C., 104, 127, 172 Boltyanskil, V. G., 104, 114, 115, 132, 137,

168, 450

Barlow, R. B., 847 Barlow, W., 54, 70 Barthel, W., 802 Bartle, R. G., 40, 50, 450 Baston, V. I. D., 151, 164

165, 177, 242, 244, 247, 249, 251, 252, 256, 170, 274, 284

Bonnesen, T., 14, 74, 94, 104, 116, 132, 165, 210, 222, 228, 241, 249, 251, 258,

Bauer, H., 143, 157, 163, 164, 211, 218, 219

265, 272, 281, 808, 316, 492 Bonnice, W., 116, 117, 185 Bonsall, F. F. 451, 471 Borel, A., 126, 165 Borsuk, K., 126. 137. 165, 271, 279, 280,

Seale, E. M. L., 326, 327 Behrend, F., 139, 184, 241, 164 Belgodère, p., 802 Bellman, R., 1, .11 Bendat, J., 191, 200 Bender, C., 54, 70 Benzer, S., 127, 164 Serge, C., 123, 156, 164, 354, 858 Berkes, 3., 251, 265 Berstein, J., 164

281

R. C., 241, 249, 265 Bourbaki, N., 35, 94, 215, 219, 349, 358, Bose,

461, 471

Bredon, G., 248, 264 Bremermann, H. 3., 162, 163, 165 503

AUTHOR INDEX

Bruckner. M., 290 de Bruijn, N. G., 280, 281 Brunn, H., 116, 130, 165 Buck, E. F., 241, 260, 265 Buck, R. C., 94, 241, 260, 265 Bunt, L. H. N., 117, 165 Busemann, H., 73, 92, 93, 94, 161, 165, 303 Calm, A. S., 817 Carathéodory, C., 115, 118, 151, 165, 225, 282, 279, 881, 285, 290, 486, 476, 492

Charnes, A., 327 Choquet, G., 143, 165, 168 Clarkson, J. A., 50, 74, 77, 84, 85, 94 Comfort, W. W., 133, 186 Cooper, W. W., 327 Corson, H. H., 37, 351, 353, 354, 358

EeHs, J., Jr., 73, 94 Efimov, N. V., 81, 92, 95 V. A., 46, 50 Eggkston, H. G., 13, 14, 20, 104, 108, 114, 115, 132, 135, 141, 150, 166, 233, 235, 142, 252, 255, 256, 258, 260, 266, 272-274, 276, 881, 808 Ehrhart, E., 132, 167, 247—249, 251, 252, 255, 266

Ellis, J. W., 156, 167 Erdos, P., 104, 119, 137, 149, 167, 169, 278, 280, 281, 282

Estermann, T., 237, 247, 256, 258, 266 Ewald, G., 73, 95 Fan, K., 1, 11, 77-19, 82-84, 86, 87, 91, 92, 95, 156, 167, 211

Coxeter, H. S. M., 19, 20,53,53,54,58,60-62, 64, 66, 69. 70, 71, 149, 150, 166, 225, 232

Fáry, 1., 156, 167, 239, 242, 254, 256, 257,

Croft, H. T., 278, 281 Cudia, D. F., 73, 74, 90, 94 Czipszer, J., 153, 154, 166

Fejes Tóth, L., 54, 62, 69, 70, 71, 104, 128, 144, 149, 167, 169, 263, 866, 275, 282 Fenchel, W., 14, 74, 94, 104, 116, 117, 132,

Daleckil, Yu. L., 190, 191, 200 Danzer, L. W., 99, 101, 131. 139. 144, 147, 149, 161, 162, 166, 241, 247, 252, 285, 272, 275-278, 281, 349, 358

Davenport, H., 61, 71 David, N., 303 Davis, C., 181, 187, 200 Day, M. M., 74-76, 78, 82-89, 92, 94, 210, 219, 258, 265, 352, 356, 358, 401, 450 Debrunner, H., 104, 114, 122-124, 126, 127, 129-132, 144, 152, 169, 277, 279, 280, 282 Derman, C., 3*7 De Santis, R., 119, 166, 481, 492 Dieudonné, J., 352, 353k 358, 450, 473, 492

Dines, L. L., 116, 166, 490, 492 Diaghas, A., 30* Dirichiet, G. L., 54, 71 Dixmier, J., 91, 94 Drandell, M., 156, 166 Dresher, M., 130, 166 Dudley, U. W., 486 Dukor, 1. G., 109, 166 Dunford, N., 35 Dvoretzky, A., 109, 166, 203, 210, 263, 265

Eberlein, W. A., 352, 358, 450 Edeistein, M., 46, 50, 114, 166

*66

158, 159, 165, 167, 110, 222, 223, 240, 241, 249, 251, 258, 285, 272, 181, 808, 316, 349, 359, 473, 475, 492

Few, L., 149, 166 Firey, W. 3., 244, 266 Fisher, E., 436 Floyd, B. B., 353, 859 Foguel, S. R., 90, 95

Fortet, R., 95 Franklin, S. P., 160, 167 Fréchet, M., •49, 50, 827

Frink, 0., Jr., 215, 219 Frucht, R., 129, 167 Fullerton, R. B., 358, 859, 458 Fulton, C. M., 236, 237, 266 Funk, P., 251, 262, 266 Gaddum, J. W., 159, 160, 167, 170, 3*7 Gale, D., 11, 116, 118, 120, 136, 137, 151, 154, 167, 221, 225, 2*2, 244, 266, 274, 275,

182, 285, 290, 350, 859, 490, 492 Ganapathi, P., 866 Gantmacher, F. R., 11, 340, 347 Gehér, L., 153, 154, 166, 355, 359 Gericke, H., 141, 168, 239, 262, 268, 303 Gerriets, C. 3., 264 Gerstenhaber, M., 490, 492 Ghika, A., 156, 162, 168

AUTHOR INDEX

Ghouila-Houri, A., 123, 168, 354, 359 Gillespie, D. C., 437, 450 Glicksberg, L, 77-79,82-84, 86,87, 91, 92, 95 Godbersen, C., 258, 266 Gohberg, 1. C., 137, 168 Goldman, A. J., 327 Coodner, D. B., 357, .159 Gordon, H., 133, 166

Gorin, E. A., 48, 50 Graves, I.. M., 40, 50 Green, J. W., 160, 168 Groemer, H., 150, 168, 244, 258, 267 de Groot, J., 159, 168 Gross, W., 237, 267 Grosswald, E., 253, 264 Grothendieck, A., 357, 358, 859, 450 Grotzsch, H., 152, 168 Grünbaum, B., 101, 115, 118, 120, 123, 124, 126, 129—133, 135, 137, 140, 142-144, 146, 147, 149, 150, 152-154, 158, 159, 168, 168, 168, 169, 210, 233, 236, 240, 242-244, 247254, 260, 264-267, 271, 274-279, 281, 282, 285, 290, SOS, 349, 355-357, 358, 359, 482,

490, 491, 492

Guinand, A. P., 56, 57, 71 Günter, S., 54, 71 Gustin, W., 114. 116, 132, 160, 168, 169, 492

Hadwiger, H., 104, 109, 114, 122424, 126, 127, 129-132,

137, 144, 149,

150, 152,

155, 159, 169, 237, 267, 272, 275, 277-280, 282

Hajós, G., 61, 71

Halberg, C. J. A., Jr., 150, 170 Hales, S., 53, 71 Hall, M., 816 Hammer. P. C., 156, 170, 247, 251, 253, 254, 264, 267, 291, 291, 808, 305, 315, 316, 350, 351, 859 128, 135, 142, 170, 277, 282, 357, 358, 859, 479, 480, 492

Hanner, 0., 117,

Hare, W. R. Jr., 159, 160, 170 Harrop, R., 129, 131, 170, 354, 859 Helly, E., 101, 101, 102, 103, 106, 109, 124— 126, 160, 170, 478, 492

Henriksen, M., 357, 859 Heppes, A., 137, 170, 274, 278, 282 Hubert, D., 170 Hille, E., 473, 492 Hirakawa, .1., 262, 267

505

Hirsch, W. M., 327 Hirschfeld, R. A., 95 Hirschman, 1. 1., 347

Hjelmslev, J., 116, 170 Hlawka, E., 150, 170 Hoffman, A. J., 317, 327 Hurwitz, W. A., 437, 450 Hopf, E., 170 Hopf, H., 125, 163, 279, 283 Horn, A., 121, 158, 159, 170, 479, 492

Isbell, J., 356, 359 Jacobs, W., 326, 827 James, R. C., 32, 85, 49, 50, 85, 86, 92, 93, 95, 353, 859

Jaworowski, J. W., 280, 283 Jerison, M., 211, 218, 219 John, F., 140, 170, 241, 248, 258, 267 Johnson, S. M., .127 Jung, H. W. E., 114, 132, 158, 170, 276, 283 Jussila, 0. K., 170 0 Juza, M., 355, 359

Kadec, M. 1., 45, 46, 49, 50, 86, 87. 95 Kadison, R. V., 75, 95 Kakutani, S., 95, 133, 170, 308 Kalisch, G. K., 161, 171 Karlin, S.. 75, 94, 104. 108, 114, 115, 120, 121, 123, 158, 165, 171, 329, 847, 350, 359, 451, 459, 471, 479, 492

Kato, T., 200 Keller, 0. H., 37, 50 Kelley, J. L., 143, 171, 215, 219, 357, 360 Kelly, L. M., 73, 95, 138, 139, 171 Kelly, P. J., 242, 267, 303 Kendall, D. G., 143, 171 Kepler, J., 53, 71 Kijne, D., 129, 171 Kirchberger P., 114, 116, 171, 475, 488, 490, 492

Kirszbraun, M. D., 153, 154. 171 Klee, V. L., 21, 23, 24, 25, 27, 35, 37, 37, 45, 50, 74, 79, 83, 85, 87-90, 92, 93, 95, 101, 104, 106, 114, 116, 117, 120, 121, 123, 124, 131, 133, 156, 165, 167, 169, 171, 247, 252, 265, 267, 273, 276-278, 280, 281, 283, 305, 307, 308, 315, 349, 349-353, 858-360,

393. 401, 473, 479, 482, 492

AUTHOR INDEX

506

Klein, M., 827 Knaster, B., 123, 171, 275, 280. 283 Kneser, H., 156, 171 Kneser, M., 155, 171 Knothe, H., 237, 267 Konig, D., 103, 106, 171 Konstantinesku, F., 92, 96 Korányi, A., 188, 200, 201 A., 156, 171, 237, 249, 254, 267 Kothe, G., 353, 360, 438, 450, 475, 498 Kovetz, Y., 264 Kozinec, B. N., 255, 267 S. N., 82, 96 Krasnosselsky,

M. A., 109, 114, 171, 172

Kraus, F., 200 M. G., 82, 92, 93, 219, 340, 347, 451, 471

Krzyz, J., 347 Kubota, T., 116, 163, 238, 268 Kuhn, H. W., 11, 156, 172 Kuiper, N. H., 129, 131, ]59, 172 Kuratowski, C., 123, 171, 285, 288, 289, 290 Lagrange, R., 132, 172

Land, A. H., 327 Landau, 454 Lanner, F., 108, 172, 478, 498

Laugwitz, D., 73, 96,

139,

166, 241, 242,

280, 283 Macheath, A. M., 236, 237, 268 MacCoil, L. A., 304

MacDuffee, C. C., 200 Mairhuber, 5. C., 115, 172 Marchaud, A., 159, 172 Marcus, M., 200 Markus, A. S., 137, 168 Matumura, S., 261, 268 Mayer, A. E., 160, 172 Mazur, S., 42, 51, 82, 96 Mazurkiewicz, S., 123, 171 McCoy, N. H., 116, 166, 490, 492

McMinn, T. J., 24, 25 McShane, E. 5., 96, 153, 154, 172 Meizak, Z. A., 132, 172 Menger, K., 138, 160, 172 Meschkowski, H., 61, 71 Michaei, E., 161, 172, 353, 860 Mickle, E. 5., 153, 154, 172 Milman, D. p., 90, 96, 211, 218, 219 Mimes, M. W., 75, 96 Milnor, 5., 129, 178 Minkowski, H., 60, 71, 74, 98, 115, 147, 178, 233, 246, 247, 268,

265, 268, 303

Lavrent'ev, M. A., 25!, 254, 268 Leech, J., 54, 71 Leichtweiss, K., 135, 141, 172, 241, 247, 248, 258, 268, 276, 283, 303

Lelek, A., 161, 172 Lekkerkerker, C. G., 104, 127, 172 Lenz, H., 139, 166, 241, 258, 265, 268, 272, 275, 276, 283 Leray, 5., 126, 172

Levi, F. W., 109, 123, 137, 138, 159, 172, 236, 237, 255, 256, 288,

Lusternik, L. See Lyusternik, L. A. Luxemburg, W. A. J., 75, 98 Lyusternik, L. A., 251, 254, 268, 271, 279,

277,

278, 283

Levin, E., 150, 170

Lighthall, H., Jr., 827 Lindenstrauss, J., 355, 357, 358, 360 Linis, V., 256, 268 Lippmann, H., 304

Loewner, C., 200 Long, R. G., 51

Lorch, E. R., 73, 96 Lovaglia, A. R., 77, 82, 83, 86, 90, 92. 96 Lowner, K. See Loewner, C.

304, 493

Minty, G. 5., 154, 173 Miyatake, 0., 262, 268 Molnár, 3., 114, 125, 158, 173 Monge, G., 317, 327 Monna, A. F., 156, 173 Morton, G., 827 Moser, L. 280, 283 Moser, W., 280, 288

Motzkin, 'F. S., 115, 118, 124, 151, 156, 158-160, 169, 173, 247, 268, 279, 282, 285, 285, 290, 305, 350, 360, 361, 389, 392, 436, 477, 493

Nachbin, L., 135, 142, 143, 173, 277 283, 355-357, 360

Nakamura, M., 201 Nakano, H., 75, 76, 78, 82, 91, 96 Nasu, V., 304 Nef, W., 156, 178 Neumann, B. H., 115, 178, 247, 249-252, 268

AUTHOR INDEX

Newman, D. J., 242, 251, 252, 268 Nijenhuis, A., 161, 173 Nitka, W., 161, 172 Nohi, W., 264 Nordlander, G., 96 Ohmann, D., 804 Ostrowski, A., 201

j. F., 240, 268, 272, 274, 288 Panitchpakdi, P., 143, 161, 168, 355-357,

P11,

358

Pasqualini, L., 160, 173 Pauc, C., 161, 174 A., 45, 49, 50, 96 Perkal, J., 272, 273, 283

Pettis, B. J., 90, 96 Petty, C. M., 237, 268, 308, 304 Phelps, R. R., 27, 35, 81, 86, 90, 92, 93,

Rényi, C., 151, 175 Yu. G., 242, 269, 304 Révész, P., 278, 282 Riesz, F., 97, 478, 498 Riesz, M., 478, 498

Ringrose, J. R., 92, 97 Rio, S. T., 305, 316 Robinson, C. V., 116, 138, 158, 175, 492, 498 Rogers, C. A., 60, 62. 70, 71, 137, 143, 147, 149, 166, 167, 175, 210, 257, 258, 269 Rolewicz, S., 210, 210 Roy, S. N., 240, 265

Ruben, H., 56, 71, 347 Rudin, M. E., 353, 360 Rund, 1-1., 73, 97

Ruston, A. F., 82, 97 Rutman, M. A., 211, 218, 219, 451, 471 Rutovitz, D., 260, 269

94, 96, 393, 401 Phillips, R.. 473, 492

Salkowski, E., 262; 269

Pleijel, A., 269 Plunkett, R. L, 161, 174, 357, 860 Pólya, G., 264, 333, 847 Poritsky, H., 403, 486 Poulsen, E. 'F., 155, 174 Prager, W., 326, 327 Prenowitz, W., 156, 174 Proschan, F., 347 Proskuryakov, I. V., 109, 174 Ptlk, V., 115, 174, 353, 360, 437, 450 Pucci, C., 242, 269

Sandgren, L., 108, 109, 120, 175, 349, 860, 473, 475, 477, 479, 490, 493 Santalô, L. A., 114, 121, 124, 129, 130, 133, 138, 158, 160, 175, 272, 283 Sasaki, M., 75, 83, 93 Schaefer, H. H., 451, 471 Schläfti, L., 54, 57, 58, 71 L. G. Schnirelmann, L. See Schoenberg, 1. J., 108, 114-116, 130, 154, 174, 175, 347, 486, 475, 488, 490, 493 Schopp, J., 144, 175, 251, 269, 277, 283 Schur, 1., 201 Schutte, K., 54, 71 Schwartz, J., 35 Schweppe, E. .1., 159, 175

Rabin, M.,

109, 174

Rademacher, H., 108, 114-116. 130, 174, 258, 269, 475, 488, 490, 493 120, 129, 131, Rado, R., 109, 115, 117, 248, 252, 150, 152, 156, 160, 170, 269, 350, 360 Radon, J., 103, 107, 115, 147, 159, 174, 246, 247, 262, 264, 269 RSdstrom, H., 117, 157, 162, 170, 174, 479, 480, 492

Radziszewski, K., 237, 254, 265, 269 Ramsey, F. P., 350, 360 Rankin, R. A., 61, 70, 71, 149, 174 Rebassoo, H. L., 291 Rédei, L., 239, 254, 257, 266 Rémès, E., 115, 174 Rényi, A., 151, 175

Samuel,

P, 215, 219

I. E., 73, 97 Selfridge, J. L., 275, 283 Semadeni, Z., 356, 359 Shapley, L. S., 108, 114, 115, 120, 121, 123, 158, 165, 171, 350, 359, 479, 492

Shephard, G. C., 143, 147, 175, 236, 257, 258, 269

Sherman, S., 191, 194. 200 Shimogaki, T., 92, 97 Shimrat, M., 114, 175 Shkliarsky, D., 280, 284 Sholander, M., 260, 269 Sierpiñski, W., 51, 355, 360 Skornyakov, L. A., 156, 175

AUTHOR INDEX

508

Smith, T. J., 295, 297 Smith, W. E., 827 Smulian, V. L., 75, 77-79, 83, 89-91, 97, 352, 860, 450

Taylor, A. E., 90, 97, 258, 270 Tietze, H., 150, 176 Toeplitz, 0., 436 Tucker, A. W., 11, 827

Snirel'man, L. G., 114, 115, 175, 271, 279, Umegaki, H., 201 Ungar, P., 181, 251, 262, 270

280, 283

Sobczyk, A., 247, 267, 291, 303 Sokolowky, D., 827 Solian, P. S., 137, 175

Valentine, F. A., 104, 108, 109, 114, 120,

Soos, G., 176

121, 130, 153, 154, 156, 160, 170, 176, 177,

Ste&in,

304, 349, 860, 473, 479, 492

S. B., 81, 92, 95

Steenrod, N., 129, 176 Stem, S. K., 233, 236, 968,

237,

249, 254,

255,

269

Steiner, J., 238, 269

Steinhagen, P., 141, 152, 176, 272, 284 Steinhaus, H., 249, 250, 261, 269 Steinitz, E., 115, 159, 176, 285, 290, 490, 491, 493 Sternbach, L., 51

Stewart, B. M., 255, 270 Stieltjes, T.-J., 160, 176 Stinespring, W. F., 201 Stoelinga, T. G. D., 117, 176 Stoker, J. J., 37, 51 Stone, M. H., 156, 178, 211, 219 Straszewicz, S., 116, 132, 176, 278, 284 Straus, E. G., 115, 150, 161, 170, 171, 178, 389

Veblen, 0., 159, 177 Verblunsky, S., 114, 132, 177 Viet, U., 136, 177, 249, 270 Vigodsky, M., 158, 177 Vincensini, P., 114, 129, 131, 159, 177, 493

Vinogradov, A. A., 82, 98 Volkov, V. 1., 135, 177, 276, 884 Voronoi, 6., 54, 71 de Vries, H., 159, 162

Wada, J., 86, 88, 89, 97 van der Waerden, B. L., 54, 71 Wahlin, G. E., 114, 132, 164 Wallinan, H., 211, 215, 219 Well, A., 126, 177, 210 Weston, J. D., 354, 359 Whitehead, 5. H. C., 177 Widder, D. V., 847

Su, B., 238, 270

Sundaresan, K., 97 Surányi, J., 151, 175 Süss, W., 136, 176, 237, 247, 251, 258, 270 S., 151, 176 Swinnerton-Dyer, H. P. F., 150, 176 Sz.-Nagy, B., 97, 135, 142, 176, 201, 248,

Yaglom, I. M., 104, 114, 115, 132, 174, 242, 244, 247, 249, 251, 252, 256, 270, 274, .684

Yamamuro, S., 75, 83, 97 Yang, C.-T., 280, 284 Yoneguchi, H., 116, 168 Young, J. W., 159, 177

270, 277, 284

Szego, G., 264, 847 Szekeres, G., 119, 167

Taft, R. G., 285, 290 Takesaki, M., 801

Zaguskin, V. L., 139, 177, 241, 270 Zalcwasser, Z., 437, 450 Zarankiewicz, K., 264 Zaustinsky, E. M., 133, 177 Zindler, K., 242, 261, 270

SUBJECT INDEX Autothety, 362 centers of, 376 Autotranslation, 362 Axioms of convexity, 109, 155-160

2-center, 363 4-hull, 363 between, 216 convex, 215 convex hull, 216 extreme. 216 supporting subset, 217

Base, 310 Basic dimension, 365 Basis, 229

substitutive, 37, 41-45 Bauer's minimum principle, 211, 218 BdS, 473 Bernoulli numbers, 57 Binary intersection property, 356 Binormals, 293, 415 Borsuk's conjecture, 137 problem, 271 Borsuk-Ulam theorem, 221, 222 Boundary points, linear combinations of,

inductive, 211 stable,

U

211

inductive, 212 stable, 211

Acentricity, 389 Additive functions, 309 semigroup, 385 Adjacent sets, 147, 149-151

389

Boundary, outer, 389 Breadth, equivalent, 296

Admissible

n-tuple, 389 side, 403 Affine center, 382, 384 Algebra locally convex, 454 spectral, 454

Cap, 489 Carathéodory's theorem, 103, 115-117, 313 Centers halfcone, 377 of endothety, 375, 376 of symmetry, 375 Central symmetry, 373 Centroid, 240 Chain, 46-48 Characteristic cone, 38-40 Chebyshev set, 81, 90, 92, 93, 401 Circle

Ambiconvex hull, 378

pair, 378 polyhedral sets, 385 sets, 378, 384 Ambiguity, 286 Analytical representation, 291 Antimonotone, 365 Antipodes, 222 Antitonic functions, 310 Approximation theory, 110-115 Arc length function, 299, 300 Asphericity, 203 Associated complex spectral measure, 464 Associated convex bodies, 292, 294, 296 Associated modular, 91 Asymmetry, measure of, 233 Asymptotic upper bound, 70 Autothetic sets, 375

a characterization of, 99 involute of, 419 Circular helix, 418 Circumscribed bodies, 112, 134-140 circle, 221 Classes complete, 366 gapless, 366 Classification, 362 Closed multiplicative semigroups, 373-375 Closure, convex, 306 509

SUBJECT INDEX

510

Closure functions, 309 Kuratowski, 310 CIS, 473 Combinatorial topology, 109, 125, 126, 129 Combinatorially equivalent 227 isomorphic, 227 Compactness, weak, 352, 353 Complemental, 362 Complementary semiflats, 378 Complete classes, 366 endovector sets, 364 families, 365 hull, 365 Compressed, 364 Complex convexity, 157, 162-163 Computation, 67 Cone

centers, 376 characteristic, 38-40 convex, 37-43, 48-50 dual, 475 generating, 456 lunar, 490 normal, 452 positive, 37, 41—43, 49, 452

support, 28 Cones, 371

Conical singular point, 23 Conics, 139 Conjugate modular, 91 Consistent, 1 Constant projection, 356

retraction, 356 width, 141 Constant breadth, 291, 293, 294 convex curves, representation of, 291 curves, 291

relative to

298

surfaces, 291

with respect to

301

Continuity, 315 properties, 379 uniform, 38, 48 Continuous, 86 functions (spaces of), 45, 46 norm, 86 Contractive functions, 309

Cony S, 473

Convergence is almost uniform, 442 Convergence is uniform in the mean, 442 Convex bodies, 37-41, 45-47, 73, 74, 221 associated, 292, 294, 296 Convex closure, 306 cone, 28, 107, 129, 158 curves, 181

curves and rectangles, 99 function, 123 hull, 103, 115-119, 159-160, 162, 225, 306, 308

polygons, 225, 230, 302 polyhedra, 383

region R, 403 set, analytic, 222 sets, 368 totally, 355 Convexity axioms of, 109, 155-160 complex, 157, 162, 163 connected, 117, 118 generalized, 155—163

metric, 161, 162 preserving properties, 342 projective, 159 spherical, 157-159 Core, 362 Cover minimal, 274

universal, 273 Covering, 111-113, 133-140, 145-150, 354 Critical point, 246 set, 246 Cross-polytope, 61 Cubic close-packing, 53 Cuboctahedron, 53

Curvature, 73, 416 centroid, 241 Curve envelope, 405

of regression, 413 osculating, 415 Curves, constant breadth, 291 Curves, convex, 181

of constant Minkowski breadth, representation of, 297 Cusps, 411

SUBJECT INDEX Deficiency, 476

Densest packing of equal spheres, 60 Density, 54

221, 291 affine, 254

Diameter,

Diametral

chords, 292, 293, 296 lines, 291, 294, 296 line family, 294, 295, 298 line family, essential, 294, 304

Differentiability of the norm, 89 Differential equivalence, 291, 294 Dimensionally ambiguous, 286 Direct product, 382, 384 Dirichlet region, 54 Distance-functions, 235 D-localized, 79, 83

Dodecahedron rhombic, 54 trapezo-rhombic, 54 Domain bounded, 311, 313 finite, 311

Essential diametral line, 296 Euclidean breadth, 292 Exact, 356 Exactness of various constants, 355 Existence set, 81, 92, 93 Expansion constant, 355 Expansion function, 309, 311, 313 Expansive function, 310 Exposed

310

140, 158, 349

Edge, 225

Element, quasi-interior, 458 Ellipsoid,

Euclidean cells, 129-139, 141, 144—

152, 155 Endo-A, 363

Endomorphism, positive, 456 Endoring,

375

Endothetic

sets, 373

Endotranslation,

363

sets, 364 Endovectors, 361 if. Envelope, 292 curve, 405 Endovector

of family of straight lines, 406 Equivalence of breadth, 292 Equivalent, 86 in

75, 82

I-

additive functions, 309 cone, 475 properties, 89 Duality, 74, 88-90, 108,

point,

Extended strength, 381 Extended topology, 305, 310, 313 Extended topological system, 310 Extension property, 142, 357 Extreme point, 82, 215, 308, 357, 358 relative, 398, 400 Extreme subset, 215 Euclidean geometry, 73 Even, 80, 83, 86 modular, 88 closure function, 310 contractive function, 310 expansive function, 310 interior function, 310 limit function, 310 limit point, 310 primitive function, 310

finite function, 311, 313 finite closure function, 312 finite expansive function, 312 Double limit condition, 448 Dual,

51L

breadth, 292, 293, 299

Fexposed point, 75, 82 hyperplane, 74 nonsupport point, 75 rotund, 75 rotundity, 75, 82 smooth, 75 support point, 75, 82 hull, 211 kernel, 211 Face, 227

Face-centered cubic lattice, 53 Family A is said to be quasi -equicontinuous, 449

Family

of lines, 297 of straight lines, 406 Families, Complete, 365 main, 365 maximal, 366 translative, 366

SUBJECT INDEX

512

Families of sets homothets, 131, 132, 134—136, 146-149, 151, 152 disjoint, 130—132, 151—153

intersectional, 123-125 translates, 131—138, 144—147, 149, 150, 155

Fan, 41, 42 Filter, 214 Finite, 83 Finsler spaces, 73 First norm, 76, 91, 92 Flatness, 83 Flats, 365 osculating, 405 Fourier series, 299 Fréchet differentiable, 78, 83, 89 Fresnet equations, 416 Full,

452

g-neigbborhoods, 310

Gallai problems, 128, 144, 152, 153 Gapless classes, 366 hull, 366 Gateaux differentiable, 78, 82, 83, 89 Gauge, 41, 42 Generalizations of convexity, 155-163 of Helly's theorem, 119-128 Generating cone, 456 Generator, single, 368 Geometry metric, 73 Minkowskian, 73 Riemannian, 73 Graphs, 118, 127, 152 Grids, 367 Group, 312

Halfcone centers, 377 Half cones, 375 Halfspace, 111, 113, 140—142, 156, 161, 162

Harmonic function, positive, 406 Helicord, involute, 418 Helix, circular, 418 Helly, Eduard, 101, 102 Helly

problems and numbers, 124, 127, 128 type theorems, 478

Helly's theorems, 101—104, 106-109

Hereditary, 212 Hermitian, 2 matriceS, 1 Herpolhode, 422 Hexagonal close-packing, 54 Hexagonal points, 242 Homothets, 105, 106 Homothety, 362 Hull, 362 ambiconvex, 378 complete, 365 gapless, 366 Hyperconvex, 356, 357 Hyperplane, 74, 226, 227 Idempotent, 309

Identical arc length functions, 299 Inclusion preserving, 309 Indecomposable polyhedra, 244 Inequalities, linear, 1 Infinite dimensional Finsler spaces, 73 Infinitely increasing, 83, 86 Inscribed circle, 221 Integration, 68 Interior functions, 309, 310 Internal associated convex bodies, 296

Internal f-primitive functions, 310 Intersection basis, 307, 308 pattern, 126, 127 theorems, 101—163

Intersectional, 362 Intv S, 473

Invariant points and sets, 238 Inverse

star centers, 377 stars, 374, 377 overstar center, 378 overstars, 374 strength, 378 Involute helicoid, 418 Involute of circle, 419 Involutes, spherical, 424 Irreducible sets, 244 subbody, 296 Isolated point, 308 Isomorphic, 84, 85 Isomorphism, 73, 85, 87, 88

513

SUBJECT INDEX

Isotonic, 309

join, 362 Jung's theorem and its relatives, 112, 113, 131-137, 140, 145, 146, 154

k rotund, 77, 82-84 Kernel, 211

Krein-Milman theorem, 211, 215, 218

Krein's theorem on the convex extension of a weakly compact set, 440 Kuratowski closure function, 310 Lattice, 160 packing, 60 Level sets, 382, 384 Limit functions, 309, 310, 313 point, 310 Lindelöf property, 351, 353, 354 Line families, 294-298 outwardly simple, 291, 296, 297, 299,302 Linear combinations of boundary points, 389 if. homeomorphisms, 73 manifolds, 312, 314 programming, 317 space, 312, 313

Linear inequalities, 1 in Hermitian matrix variables, 1 in the n variables, 403 Linear'y closed, 377 hoineomorphic, 86 Lines, 297 Lipschitzian, 221, 297 transformations, 48, 152-155

Local uniform smoothness, 73 Locally Banach, 73

convex algebra, 454 convex vector lattice, 459 uniformly rotund, .77, 82-84, 86, 87 uniformly smooth, 78, 82 Löwner's ellipsoid, 241 Lunar d-cone, 490 Main families, 365 Map, quasi-interior, 458 Matrix-convex, 187

Matrix-monotone, 187 Maximal families, 366 sets, 314 Measure of asymmetry, 233 of

24

spectral, 454 Measurement, quantum-mechanical, 197 Measures of symmetry, 233 Metric cells, 355 convexity, 161, 162 geometry, 73 projection, 93 spaces, 139, 143, 144, 153-155 Midpoint locally uniformly rotund, 77, 82-84, 88 smooth, 78, 83 Midpoint local uniform smoothness, 73 Milman-Rutman's theorem, 211, 218 Minimal base, 314

base of neighborhoods, 305 cover, 274 families, 366 intersection basis, 312, 314 neighborhood, 308 Ø.supporting subsets, 217 union basis, 312 Minkowski spaces, 73, 133-131 Minkowskian geometry, 73 metric, constant breadth, 291, 293, 294 metrics, 291, 297 Modular, 75, 79, 92 norm, 76, 87, 92 Modulared linear lattices, 85 Modulated vector lattices, 75, 79, 83, 8690, 92 Modulars, 81 Modulus of rotundity, 85 Monge sequence, 317-319, 325 Multiplicative semigroup, 364, 370 Nearideals, 385 Neighborhood, 310, 312, 314 base, 312 Neighborhoods, 305, 308 Neighborly polyhedra, 118, 151

514

SUBJECT INDEX

Neighbors, 225 Non-atomic, 83 Non-Euclidean space, 69 Non-lattice packing, 61 Non-support point, 75 Normal cone, 452 Normality, 353, 354 Norm-closed, 377 Normed linear spaces, 133, 143, 152-155

Ordered locally convex algebra, 454 Ordered topological vector space, 452 Osculating curve, 413 flats, 405 plane, 413 Osculatory, 79, 83

o. s. line family, 297 Outer boundary, 389 Outwardly simple, 297 Overstar center, 378 Overstars, 375 inverse, 374, 378

Packing, 54 Packings, 147,. 149, 150 Pair, ambiconvex, 378 Pairproduct, 378, 384 Paracompactness, 353 Parallel, 296 Parallelotopes, 127, 129-131, 137, 142-145, 150-152, 154

Pedal function, 292, 294, 297, 299, 302 Period, 367 Permutationally symmetric, 364 Plane, osculating, 413 Point locally uniformly totund, 78, 82, 83,

Polyhedra, 106, 118, 119, 129, 150-152 conveX, 383

Polyhedral graphs, 285 dimension of, 285 Polyhedron, regular, 383 Polytopes 225-227, 230 Positive cone, 37, 41-43, 49, 75, 452 definite, 1 endomorphisms, 456

functions, totally, 337 harmonic function, 406 spectral element, 465 spectral measure, 461 Positivity, total, 329 if. Primary f-limit point, 310 f-self-dense function, 310 self-dense, 314

Primitive functions, 309 Principal filters, 214 normal, 415 Product direct, 382, 384 of Banach spaces, 83 spaces, 83, 87 Projection constant, 356 operator, 76, 80 Projective convexity, 159 functions, 309 Proper, 452 ambiconvex set, 379 dimension, 372 endothety, 385

86

Point of F-smoothness, 75 Point symmetrization, 293 Points

of rotundity, 74 of smoothness, 74 Polar cap, 489 Polar body, 74 Polarity, 349 Polhode, 422 Polygons convex, 225, 230 regular, 230

Quasi-interior element, 458 map, 458 Quasipencil, 297 Quotient spaces, 83, 84 Quasiconvex, 46-48 r-neighborhood, 310, 312, 314 Radon's theorem, 103, 107-109, 117, 118, 159

Rectangles, convex, 99 Reflexive, 91

SUBJECT INDEX

Reflexivity, 91 Regression, curve of, 413 Regular 230 n-gon 600-cell, 64 polygons, 230 polyhedron, 383 Relative extreme point, 398, 400 Relatively constant breadth curves, finite constructions, 300 Representative spherical triangle, 421 Resolvent, 454 Retraction constant, 356 Reuleaux triangle, 13 Rhombic dodecahedron, 54 Riemannian geometry, 73 Ring, 312 Rolling without slipping, 422 Rotation total, 408 vector, 420 Rotund, 74, 77, 82-86, 91-93 Rotundability, 87 Rotundable, 87, 89 Rotundity, 73 if. Ruled surface, 413

Saturated, 453 Scalar product, 228 Schläfli function, 56 Schur-convex, 198 Second norm, 76, 83, 91, 92 Segment, 225 Self-dense function, 310 Semiflats, complementary, 378 Semigroup, 312 additive, 385 closed multiplicative, 375 connected multiplicative, 373, 374 multiplicative, 364, 370 Semiregularity, 91 Semiregular polyhedron, 383 Semiring, 368 Semispaces, 156, 305, 306, 314, 350, 351 Separation by hyperplanes, 106, 107, 109, 111, 130 theorem, 27 Set of constant width, 15 Sets, 365 ambiconvex, 384

515

ambiconvex polyhedral, 385 autothetic, 373, 375 Chebyshev, 401 complete endovector, 364 convex, 368 endothetic, 373 level, 382, 384 symmetric, 373 Simple, 15 orderings, 306 Simplex, 112, 113, 123, 136-138, 140, 141, 143, 151

Single generator, 368 Single integral generator, 369 Six-partite point, 241 Smooth, 74, 75, 78, 82-87, 91-93 Smoothability, 87 Smooth conjugate, 92 Smoothness, 73, 74, 85, 90, 93 Space, Minkowski, 73 Spectral algebra, 454 element, 454 element, positive, 465 Spectral measure, 454 associated complex, 464 product, 461, 463 support, 454 Spectrum, 454 Sphere, 105, 152 Spherical convexity, 157—159 424 Spherical

Spherical to within Stability, 1 Star centers, 377 Stars, 374, 377, 380 inverse, 377, 381

203

Starshaped sets, ill Steinitz's theorem, 115, 116 Stone's theorem, 211, 214 Strength, 379, 381, 389 Strict convexity, 79, 80 Strict 'B-cone, 453 Strictly convex, 80, 83, 86, 91, 222 Strictly positive linear form, 459 Strip, 292 Strong cones, 371 stars,

374, 377

halfcones, 375 overstars, 375

SUBJECT INDEX

516

Strongly ambiguous, 286 Subpencil, 300, 302 Subspaces, 83, 84 Substitutive basis, 37, 41—45 Superminimality condition, 243 Support cone, 28, 394ff. functional, 23ff., 393, 473, 475 point, 27-35, 75, 82, 393 spectral measure, 454 theorem, 27 Supporting hyperplane, 225 strips, 292-294 Suppression, 364 Surface-area centroid, 240 Surface, toroidal, 424 Surfaces, constant breadth. 291 Symmetric body, 222 convex bodies, 129, 134-137, 140, 141, 145-147, 149, 150

permutationally, 364 sets, 373 Symmetroid, 293-295 Symmetrization, central, 292, 293 Symmetry, centers of, 375 central, 373 measures of, 233 Topological classification, 37 Topology, extended, 305, 310, 313 weak, 352, 353 Toroidal surface, 424 Torsion, 416

Total rotation, 408 Totally convex, 355 Totally positive functions, 337 Translative endovector sets, 367 families, 366 Translativity, 366 Transportation problem, 317, 318 Transversals, 110, 111, 114, 121, 129-132 Trapezo-rhombic dodecahedron, 54 Tyhonov cube, 81

Ultimate root, 315

Ultra filter, 214 Umbra! notation, 56 Unambiguous, 286 Uniform

continuity, 38, 48 flatness, 83 rotundity, 74, 87 Uniformly

convex, 80, 83, 9L 92 even, 80, 83, 91, 92 Fréchet differentiable, 75, 78, 83, 89 Gateaux differentiable, 78, 83, 89 increasing, 91 rotund, 74, 77, 82-85, 87, 91 smooth, 78, 82-85, 91 Unique extension property, 81, 93 Unional, 362 families, 371 Unit ceLl, 37, 39-41, 45, 49, 50

tangent vector, 416 Univalent, 221 U'iiversal cover, 273 Universally continuous, 90

v-inverse star, 381 v-star, 380 Variation diminishing property, 336, 337 Vector

lattice locally convex, 459 rotation, 420 Virtual envelope, 297, 300 Visible, 111 Voronoi polyhedron, 54

Wallman's theorem, 211, 214 Warehouse problem, 317, 319, 322 Weak compactness, 352, 353, 437 topology, 352, 353 Weak*

locally uniformly rotund, 77, 82 uniformly rotund, 79 Weakly ambiguous, 286 k rotund, 77, 82 locally uniformly rotund, 77, 82, 83, 86

uniformly rotund, 79 Width, 140, 141, 154, 221, 222