Selected Topics in Convex Geometry

Maria Moszy´nska Selected Topics in Convex Geometry Birkh¨auser Boston • Basel • Berlin Maria Moszy´nska Warsaw Univ...

Author: Maria Moszynska

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Maria Moszy´nska

Selected Topics in Convex Geometry

Birkh¨auser Boston • Basel • Berlin

Maria Moszy´nska Warsaw University Institute of Mathematics 02-097 Warsaw Poland

Cover design by Joseph Sherman. Mathematics Subject Classiﬁcation (2000): 52-01, 52-02, 52A20, 52A30, 52A35, 52A38, 52A40, 52B11, 52B45

ISBN-10 0-8176-4396-6 ISBN-13 978-0-8176-4396-6

eISBN 0-8176-4451-2

Printed on acid-free paper.

c 2006 Birkh¨auser Boston

Based on the original Polish edition, Geometria zbior´ow wypuklych. Zagadnienia wybrane, c Wydawnictwa Naukowo-Techniczne, Warszawa, Poland, 2001. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkh¨auser Boston, c/o Springer Science+Business Media Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identiﬁed as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 987654321 www.birkhauser.com

(TXQ/MP)

Contents

Preface to the Polish Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Preface to the English Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Part I 1

Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Distance of point and set. Generalized balls . . . . . . . . . . . . . . . . . . . 1.2 The Hausdorff metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 6

2

Subsets of Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Minkowski operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Support hyperplane. The width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Convex sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Compact convex sets. Convex bodies . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Hyperplanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 14 18 19 20

3

Basic Properties of Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Convex combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Convex hull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Metric projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Support function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 28 31 34

vi

Contents

4

Transformations of the Space Kn of Compact Convex Sets . . . . . . . . 4.1 Isometries and similarities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Symmetrizations of convex sets. The Steiner symmetrization . . . . 4.3 Other symmetrizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Means of rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 39 41 48 49

5

Rounding Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The ﬁrst rounding theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Applications of the ﬁrst rounding theorem . . . . . . . . . . . . . . . . . . . . 5.3 The second rounding theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Applications of the second rounding theorem . . . . . . . . . . . . . . . . .

53 53 55 56 58

6

Convex Polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Polyhedra and their role in topology . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Convex polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Approximation of convex bodies by polytopes . . . . . . . . . . . . . . . . 6.4 Equivalence by dissection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Spherical polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 64 67 69 71

7

Functionals on the Space Kn. The Steiner Theorem . . . . . . . . . . . . . . 7.1 Functionals on the space Kn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Basic functionals. The Steiner theorem . . . . . . . . . . . . . . . . . . . . . . . 7.3 Consequences of the Steiner theorem . . . . . . . . . . . . . . . . . . . . . . . .

73 73 77 82

8

The Hadwiger Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 8.1 The ﬁrst Hadwiger theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 8.2 The second Hadwiger theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

9

Applications of the Hadwiger Theorems . . . . . . . . . . . . . . . . . . . . . . . 97 9.1 Mean width and mean curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 9.2 The Crofton formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 9.3 The Cauchy formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Part II 10

Curvature and Surface Area Measures . . . . . . . . . . . . . . . . . . . . . . . . 109 10.1 Curvature measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 10.2 Surface area measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 10.3 Curvature and surface area measures for smooth, strictly convex bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

11

Sets with Positive Reach. Convexity Ring . . . . . . . . . . . . . . . . . . . . . . 125 11.1 Sets with positive reach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 11.2 Convexity ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Contents

vii

12

Selectors for Convex Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 12.1 Symmetry centers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 12.2 Selectors and multiselectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 12.3 Centers of gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 12.4 The Steiner point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 12.5 Center of the minimal ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 12.6 Pseudocenter. G-pseudocenters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 12.7 G-quasi-centers. Chebyshev point . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

13

Polarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 13.1 Polar hyperplane of a point with respect to the unit sphere . . . . . . 159 13.2 Polarity for arbitrary subsets of Rn . . . . . . . . . . . . . . . . . . . . . . . . . . 161 13.3 Polarity for convex bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 13.4 Combinatorial duality induced by polarity . . . . . . . . . . . . . . . . . . . . 165 13.5 Santal´o point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 13.6 Self-duality of the center of the minimal ring . . . . . . . . . . . . . . . . . 168 13.7 Metric polarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

Part III 14

Star Sets. Star Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 14.1 Star sets. Radial function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 14.2 Star bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 14.3 Radial metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 14.4 Star metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

15

Intersection Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 15.1 Dual intrinsic volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 15.2 Projection bodies of convex bodies. The Shephard problem . . . . . 186 15.3 Intersection bodies of star bodies. The Busemann–Petty problem . 187 15.4 Star duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

16

Selectors for Star Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 16.1 Radial centers of a star body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 16.2 Radial centers of a convex body . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 16.3 Extended radial centers of a star body . . . . . . . . . . . . . . . . . . . . . . . . 198 Exercises to Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Exercises to Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Exercises to Part III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

viii

Contents

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

Preface to the Polish Edition

This book (more precisely, its Part I and the beginning of Part II) is based on the monographic lecture under a similar title, presented several times, in various versions, at the Department of Mathematics, Computer Science, and Mechanics of Warsaw University. I hope it will be helpful for lectures and seminars. Perhaps it may be of interest for mathematicians working in geometry (in a broad sense) or in other ﬁelds. I will be happy if the book is useful as well for those using mathematics as a tool. I shall be grateful to readers for their critical remarks.1 Many people deserve my thanks. The ﬁrst is Professor Jakub Bodziony. It was due to him that I heard of stereology, which, to some extent, is based on the Hadwiger theorems and the Crofton formulae. These famous results are at the heart of Part I of my book. In turn, my contact with Jakub Bodziony was due to my colleagues Andrzej Palczewski and the late Wiesław Szlenk, who in 1989 organized a seminar on mathematical methods in natural science. It is hard to overestimate the merits of Krzysztof Przesławski, whose remarks made it possible for me to avoid many mistakes and to improve the text. ˙ I wish to thank Tomasz Zukowski and my son, Marcin Moszy´nski, for their help and advice. I am grateful to Adam and Agnieszka Bogdewicz for their cooperation in preparing the ﬁgures. Warszawa, April 2001

1 E-mail: [email protected]

Maria Moszy´nska

Preface to the English Edition

Following a suggestion of Peter Gruber, Erwin Lutwak, and Carla Peri, I decided to translate my book into English. My motivation was obvious: very few people interested in the subject can read a Polish book. Peter Gruber’s advice was to start at least one year after the Polish edition appeared, because usually there are many errors to be found and corrected. He was right: in the ﬁrst year after the appearance of the book (autumn 2001 to autumn 2002), my colleagues, students, and I found many things to be improved. Thus, I used the opportunity to make corrections, to complete or add some proofs, to reorganize parts of the text, and to update some of the results and references. I very much appreciate remarks and suggestions of the referees. I am grateful to everybody who found an error in the Polish edition. My special thanks go to my friend Irmina Herburt; our cooperation has played a very essential role in preparing the new edition. I hope that this new version, though certainly not perfect, is much better than the old one. Warsaw, September 2004

Maria Moszy´nska

Introduction

Although convex geometry has its roots in the middle of the nineteenth century, only in the last few decades has it become one of the most vivid branches of contemporary geometry. The book consists of three parts. In Part I, the Hadwiger theorems on functionals are presented as the main topic, with the classical proof, which was the only existing one until 1995 (compare [64] and [38]). Chapters 1–3 concern basic notions of metric geometry, in particular of Euclidean geometry, and geometry of convex sets. Chapter 4 deals with maps of the family Kn of compact convex subsets of Rn into itself, for instance the Steiner symmetrization, which is used in Chapter 5 in the proof of one of two rounding theorems (in German Kugelungstheoreme, sometimes called also theorems on a snowball). Chapter 6 deals with convex polytopes, their role in the class Kn (approximation theorems), and the equivalence by dissection. Chapter 7 is devoted to functionals on Kn , in particular basic functionals (intrinsic volumes) and the Steiner theorem. The Hadwiger theorems are the subject of Chapter 8. In Chapter 9 (the last one in Part I) one can ﬁnd their applications, in particular the Crofton formulae, which can be considered as the origin of geometric tomography (see [20]). Part II (Chapters 10–13) deals with various generalizations of notions previously considered. First, in Chapter 10, we give a survey (based on [62]) of curvature measures and surface area measures, which are an important tool of convex geometry and may be treated as a “localization” of intrinsic volumes. By the use of curvature measures we show the relationship between the mean curvature considered in Section 9.1 and the integral of the mean curvatures in the sense of differential geometry.

xiv

Introduction

Further, in Chapter 11, various extensions of the class Kn are presented: the class of sets with positive reach and the convexity ring U n (called convex ring in the literature). In Chapter 12 functionals on Kn are replaced by selectors, which select a point from every convex body. Chapter 13 concerns one more operation on the class Kn : polarity. Part III, the shortest one (Chapters 14–16), is devoted to the class of star sets, which recently turned out to be a useful tool of geometric tomography (see [20], a very good monograph being updated by the author in the Internet).2 This domain is too rich to be developed in this book, but because of its essential role it cannot be avoided. The whole book with the exception of Chapter 1 concerns Euclidean spaces. The restriction to Rn (with the Euclidean metric) is justiﬁed by the fact that every n-dimensional Euclidean space is isometric to Rn . We do not follow the tendency to distinguish between the afﬁne space Rn and the corresponding linear space; thus we identify a point x = (x1 , . . . , xn ) ∈ Rn with the vector with coordinates x1 , . . . , xn . Consequently, the unit sphere S n−1 := {x ∈ Rn | x = 1}, which is the boundary of the unit ball B n := {x ∈ Rn | x ≤ 1}, is often treated as the set of vectors with length equal to 1 (compare [64]). In principle, we follow the terminology and notation used in [64]. Thus, B(X ) is the class of Borel subsets of X ; the function Vn : B(Rn ) → R is the n-dimensional volume, i.e., Vn = λn , the Lebesgue measure; σn−1 is the (n − 1)-dimensional spherical Lebesgue measure; and κn := Vn (B n ),

ωn := σn−1 (S n−1 ) = nκn

(compare Proposition 7.3.4). Often, if it does not lead to confusion, we write σ instead of σn−1 . The notions deﬁned for Rk , k ≤ n, invariant under isometries (or more generally, under afﬁne maps) can be transferred in the obvious way to an arbitrary k-dimensional afﬁne subspace of Rn . For any nonempty sets X and X , a function f : X → X is called a surjection if f (X ) = X , an injection if different points have different values, and a bijection if it is both a surjection and an injection. For metric spaces (X, ) and (X , ), a function f : X → X is a Lipschitz map if and only if there exists a λ > 0 satisfying the condition ( f (a), f (b)) ≤ λ(a, b)

for every a, b ∈ X ;

(0.1)

the inﬁmum of the set of positive numbers satisfying (0.1) is called the Lipschitz constant of f . A function f with the Lipschitz constant λ ≤ 1 is called a weak contraction, and a function with the Lipschitz constant λ < 1 is called a contraction. Evidently, every Lipschitz function is uniformly continuous. 2 http://www.ac.wwu.edu/∼gardner/

Introduction

xv

A map r : X → r (X ) ⊂ X is a retraction of a space (X, ) provided that r |r (X ) = idr (X ) . A subset X 0 of the space X is a retract of X if there exists a retraction r : X → X 0 (compare [12]). Let us notice that there is a difference between our terminology concerning isometries of arbitrary metric spaces and the terminology used by P. Gruber ([25]). We require an isometry to be a surjection, while any map preserving distances is referred to as isometric embedding (Deﬁnition 1.1.10, Theorem 1.1.11, Example 4.1.2). The symbol card denotes cardinality. The symbol P∞ k=1 denotes the countable Cartesian product of a sequence of sets: P∞ k=1 X k := {(x k )k∈N | x k ∈ X k

for every k}.

The symbols cl , int , bd denote, respectively, closure, interior, boundary in the space (X, ). Usually we omit the subscript if it does not lead to confusion. We use the notation dist(A, B) := inf{(x, y) | x ∈ A, y ∈ B}.

(0.2)

The symbols lin, aff, and pos are used for subsets of Rn : linA is the smallest linear subspace containing A, affA the smallest afﬁne subspace containing A; for every x ∈ Rn \ {0}, posx := {λx | λ ≥ 0}, and for every X ⊂ Rn , posX :=

{posx | X ∩ posx = ∅}.

The symbols relintA and relbdA denote, respectively, the relative interior and the relative boundary of a subset A of Rn , i.e., the interior or the boundary of A with respect to affA. Further, Gkn is the set of k-dimensional linear subspaces of Rn ; for every E ∈ Gkn , the set E ⊥ is the (n − k)-dimensional linear subspace orthogonal to E; for every nonzero vector v we admit v ⊥ := (linv)⊥ . For every k-dimensional afﬁne subspace E in Rn and x ∈ Rn , the set E ⊥ (x) is the (n − k)-dimensional afﬁne subspace passing through x and perpendicular to E; the function π E : Rn → E is the orthogonal projection onto E, and σ E is the symmetry with respect to E. The usual scalar product in Rn is denoted by ◦. G L(n), O(n), and S O(n) are, respectively, the group of linear automorphisms, the group of linear isometries, and the group of proper linear isometries of Rn ; ﬁnally, Tr is the group of translations. The symbol ≡G denotes the congruence of sets with respect to a group G of transformations. w The symbol → denotes the weak convergence of measures on a metric space, here on Rn :

xvi

Introduction w

µi → µ :⇐⇒ ∀X ∈ B(Rn ) (µ(bdX ) = 0 ⇒ lim µi (X ) = µ(X )) i→∞

(compare [4]). Hm is the m-dimensional Hausdorff measure (see [16]), which for subsets of m R coincides with the m-dimensional Lebesgue measure λm , and for subsets of S m with the spherical measure σm (see [64]). j The symbol δi denotes 0 or 1, respectively, for i = j or i = j. We use quantiﬁers: ∀ (for every), ∃ (there exists), and ∃1 (there exists exactly one). In deﬁnitions the word “if” is always understood as “if and only if.” At the end of the book the reader will ﬁnd exercises. Some of them (but not all) are quite elementary.

Part I

1 Metric Spaces

Let (X, ) be a metric space. Thus, X is a nonempty set and the function : X × X → R+ , referred to as a metric, satisﬁes the conditions () (x, y) = 0 if and only if x = y, () (x, y) = (y, x), ( ) (x, y) + (y, z) ≥ (x, z). (We omitted the universal quantiﬁers at the beginning of these sentences. In what follows, we shall often do this when it does not lead to confusion.)

1.1 Distance of point and set. Generalized balls 1.1.1. DEFINITION. For every nonempty subset A of X and x ∈ X let (x, A) := inf{(x, a) | a ∈ A}. The number (x, A) is the distance between the point x and the set A. Let us notice that 1.1.2. The function (·, A) : X → R+ is continuous. Proof. Let x = lim xk . By the triangle inequality (condition ( ) above) and properties of upper bound, −(xk , x) + (x, A) ≤ (xk , A) ≤ (xk , x) + (x, A). Thus (x, A) = lim (xk , A).

4

1. Metric Spaces

Moreover, the function (·, A) is a weak contraction (Exercise 1.1). 1.1.3. (x, A) = (x, clA). Proof. Since A ⊂ clA, it follows that (x, A) ≥ (x, clA). Hence, it sufﬁces to prove that for every a ∈ clA, (x, A) ≤ (x, a).

(1.1)

If a ∈ clA, then a = lim ak for some sequence (ak )k∈N in A; thus (x, A) ≤ (x, ak ) for every k. Passing to the limit for k → ∞, by 1.1.2 (for a singleton) we obtain (1.1). Using 1.1.2 and 1.1.3, it is easy to prove 1.1.4. (x, A) = 0

⇔

x ∈ clA.

1.1.5. DEFINITION. For every A ⊂ X and ε > 0, let (A)ε := {x ∈ X | (x, A) ≤ ε}. The set (A)ε is called ε-hull of A or generalized ball of A. Evidently, 1.1.6. For every nonempty A, B ⊂ X and δ, ε > 0, A ⊂ B ⇒ (A)ε ⊂ (B)ε and ((A)δ )ε ⊂ (A)δ+ε . (Compare Exercise 1.7.) It is also easy to prove that 1.1.7. If A is compact, then (A)ε =

{a}ε .

a∈A

(Compactness is essential for the inclusion ⊂, while ⊃ is true for arbitrary A.) The set {a}ε is the ball with center a and radius ε; it is denoted by the symbol B(a, ε) (or B X (a, ε) if it is not obvious that the ball is taken in (X, )). 1.1.8. DEFINITION. The set A ⊂ X is bounded if there exists an upper bound of the set {(x, y) | x, y ∈ A}. This upper bound is called the diameter of A (in symbols diamA . It is easy to check that

1.1 Distance of point and set. Generalized balls

5

1.1.9. For every A ⊂ X the following conditions are equivalent: (i) A is bounded, (ii) ∃ε > 0 ∃x ∈ X A ⊂ B X (x, ε). 1.1.10. DEFINITION. For arbitrary metric spaces (X, ) and (X , ), a function f : X → X is an isometric embedding (with respect to and ) if ∀x, y ∈ X ( f (x), f (y)) = (x, y);

(1.2)

a surjective isometric embedding is called an isometry. More generally, a surjection f is a similarity with ratio λ > 0 if ∀x, y ∈ X ( f (x), f (y)) = λ(x, y). The spaces (X, ) and (X , ) are isometric (similar) if there exists an isometry (a similarity) f : X → X . It is well known that Rn is not isometric with any of its proper subsets. We give here a complete proof of this assertion.1 1.1.11. THEOREM. Every isometric embedding f : Rn → Rn is an isometry. Proof. Let X = f (Rn ). Of course, f treated as a function of Rn onto X is an isometry, whence it preserves completeness and connectedness; thus X is a closed connected subset of Rn . Obviously, X is not compact, since f is a homeomorphism of Rn on X . Let n = 1. Suppose that X = R. Then X is a closed half-line with an endpoint a. The point a does not disconnect X (i.e., X \{a} is connected), while every point of R, in particular f −1 (a), disconnects R. But this is impossible because f is a homeomorphism. Now let n ≥ 2. Let us notice that the image f (L) of any line L ⊂ Rn is again a line. Indeed, let φ : R → L be an isometry of R on L; then the function ψ : R → f (L) deﬁned by ψ(x) := f φ(x) is an isometry of R on f (L). For any p ∈ X , the set Rn is a union of the family L of all the lines passing through p: Rn = L. (1.3) For every L ∈ L, f −1 (X ∩ L) = f −1 (X ) ∩ f −1 (L) = Rn ∩ f −1 (L) = f −1 (L), whence X ∩ L = L; thus L ⊂ X . Consequently, Rn ⊂ X , by (1.3). Hence X = Rn . As we shall see in Chapter 4, Theorem 1.1.11 cannot be generalized on arbitrary metric spaces (Example 4.1.2). 1 It is a consequence of theorem on perfect metric homogeneity of Rn [11].

6

1. Metric Spaces

1.2 The Hausdorff metric Let C(X ) be the class of nonempty, closed, bounded subsets of a metric space (X, ). For any A, B ∈ C(X ), let H (A, B) := inf{ε > 0 | A ⊂ (B)ε and B ⊂ (A)ε }.

(1.4)

(The lower bound exists, since the sets A and B are bounded.) We shall prove that 1.2.1. The function H : C(X ) × C(X ) → R is a metric. Proof. Obviously, H ≥ 0. Let A, B ∈ C(X ). Since A and B are closed in X , by 1.1.4 it follows that (A)ε = A and (B)ε = B. ε>0

ε>0

Hence H (A, B) = 0 ⇔ ∀ε > 0 (A ⊂ (B)ε and B ⊂ (A)ε ) ⇔ A ⊂ B and B ⊂ A ⇔ A = B. This proves condition (). Evidently H satisﬁes ( ). It remains to verify ( ). Let A, B, C ∈ C(X ); in view of , we may assume that A, B, C are pairwise distinct. Let ε0 := H (A, B) and δ0 := H (B, C). It is easy to see that the set {ε > 0 | A ⊂ (B)ε and B ⊂ (A)ε } is closed, whence its lower bound H (A, B) belongs to it, i.e., A ⊂ (B)ε0 and B ⊂ (A)ε0 . Similarly, B ⊂ (C)δ0 and C ⊂ (B)δ0 . Hence in view of 1.1.6, A ⊂ (C)ε0 +δ0 and C ⊂ (A)ε0 +δ0 . Therefore, H (A, C) ≤ ε0 + δ0 = H (A, B) + H (B, C).

The metric H is called the Hausdorff metric; the limit in the space (C(X ), H ) is called the Hausdorff limit: A = limH An ⇔ lim H (A, An ) = 0. The following formula (1.5) is often given as a deﬁnition of the Hausdorff metric.

1.2 The Hausdorff metric

7

1.2.2. THEOREM. For every A, B ∈ C(X ), H (A, B) = max{sup (a, B), sup (b, A)}. a∈A

(1.5)

b∈B

Proof. Since for every connected S1 , S2 ⊂ R+ with nonempty intersection inf(S1 ∩ S2 ) = max{inf S1 , inf S2 }, by the symmetry of condition (1.5) with respect to A and B it sufﬁces to prove that sup (a, B) = inf{ε > 0 | A ⊂ (B)ε }. a∈A

Let α := sup (a, B) and β := inf{ε > 0 | A ⊂ (B)ε }. a∈A

Then (a, B) ≤ α for every a ∈ A, and thus A ⊂ (B)α ; hence α ≥ β. Suppose α > β; then ∃ε ∈ (0; α) A ⊂ (B)ε , whence supa∈A (a, B) ≤ ε < α, contrary to the assumption.

The properties of the space (C(X ), H ) obviously depend on those of (X, ) (see Theorem 1.2.6). 1.2.3. DEFINITION. A space (X, ) is ﬁnitely compact if every closed, bounded subset of (X, ) is compact. Let us note 1.2.4. PROPOSITION. For every metric space (X, ) the following conditions are equivalent: (i) (X, ) is ﬁnitely compact; (ii) every ball in (X, ) is compact; (iii) every bounded sequence in (X, ) has a convergent subsequence. Evidently, every compact space is ﬁnitely compact. The space Rn is ﬁnitely compact but not compact. This example might suggest that completeness implies ﬁnite compactness. But this implication is false; for instance, the plane R2 with the “railway metric” ˜ deﬁned by x − y if 0 ∈ aff(x, y) (x, ˜ y) = x + y if 0 ∈ / aff(x, y) is complete but is not ﬁnitely compact, since the balls with center (0, 0) are not compact. Similarly, the space l 2 , i.e., the space of real sequences with convergent series of squares, with the metric deﬁned by the formula ((xi )i∈N , (yi )i∈N ) = (

∞ i=1

is complete but is not ﬁnitely compact.

1

(xi − yi )2 ) 2 ,

8

1. Metric Spaces

Since the balls in every metric space are closed and bounded, it follows that every ﬁnitely compact space is locally compact. We shall need the following 1.2.5. LEMMA. If a space (X, ) is ﬁnitely compact, then for every descending sequence (An )n∈N in C(X ), ∞

An = limH An .

n=1

Proof. Let A = ∞ n=1 An . In view of the ﬁnite compactness of (X, ), by the Cantor theorem, the set A is nonempty. Since A ⊂ An for every n, it follows that moreover, ∀ε > 0 ∀n A ⊂ (An )ε . It remains to prove that ∀ε > 0 ∃n 0 ∀n > n 0 An ⊂ (A)ε . Suppose, to the contrary, that there exist ε > 0 and an increasing sequence (kn )n∈N such that (1.6) Akn ⊂ (A)ε . ∞ Let X n := Akn \ int(A)ε for every n and X 0 := n=1 X n . Evidently, (X n ) is a descending sequence of compact sets; by (1.6), they are nonempty. Hence by the Cantor theorem, X 0 = ∅. (1.7) ∞ On the other hand, X 0 ∩ A = A \ int(A)ε = ∅ and X 0 ⊂ n=1 Akn = A (the last equality holds because the sequence (kn )n∈N is increasing), whence X 0 = X 0 ∩ A = ∅, contrary to (1.7). We are now ready to prove 1.2.6. THEOREM (compare [64], Th.1.8.2). If (X, ) is ﬁnitely compact, then (C(X ), H ) is complete. Proof. Let (Cn )n∈N be a Cauchy sequence in (C(X ), H ). Then ∀ε > 0 ∃n 0 ∀n 1 , n 2 ≥ n 0 Cn 1 ⊂ (Cn 2 ) 2ε ; thus in particular, Cn ⊂ (Cn 0 ) 2ε and Cn 0 ⊂ (Cn ) 2ε for n > n 0 .

(1.8)

n 0 Hence the set ∞ n=1 C n is bounded, since it is a subset of (C n 0 )ε ∪ n=1 C n . For every m ∈ N, let ∞ ∞ Am := cl Cn and A := Am . (1.9) n=m

m=1

1.2 The Hausdorff metric

9

The set Am is closed and bounded; thus it is compact, because (X, ) is ﬁnitely compact; obviously, Am+1 ⊂ Am for every m ∈ N. Hence by Lemma 1.2.5, A = limH Am , which together with (1.9) implies that there exists n 1 such that Cn ⊂ cl

∞

Ci

⊂ (A)ε for n > n 1 .

i=n

By (1.8) and (1.9), A ⊂ cl

∞

Ci ⊂ (Cn )ε for n > n 0 .

i=n

Thus A = limH Cn ; hence (Cn )n∈N is convergent.

1.2.7. COROLLARY. The space C(Rn ) is complete. Finally, we mention without proof the following 1.2.8. THEOREM (compare [64], 1.8.3 or 1.8.4). The space C(Rn ) is ﬁnitely compact. Let us observe that Theorem 1.2.8 is stronger than 1.2.7 (compare with Exercise 1.5).

2 Subsets of Euclidean Space

In what follows we are concerned with subsets of Rn for n ≥ 1, or more precisely, of the metric space (Rn , ), where is the Euclidean metric, i.e., the metric induced by the Euclidean norm:

n

(x, y) := x − y = (xi − yi )2 i=1

for x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ). We use the notation C n := C(Rn ). Since Rn is ﬁnitely compact, it follows that C n is the class of compact, nonempty subsets of Rn .

2.1 The Minkowski operations For subsets of Rn , addition and multiplication by a scalar are deﬁned: 2.1.1. DEFINITION. (i) For any A, B ⊂ Rn , A + B := {a + b | a ∈ A, b ∈ B}. The set A + B is the Minkowski sum of A and B.

12

2. Subsets of Euclidean Space

(ii) For any A ⊂ Rn and t ∈ R, t A := {ta | a ∈ A}. The set t A is the product of A by t. It is clear that t A is the image of A under the homothety with center 0 and ratio t. As direct consequences of 2.1.1, we obtain the following two simple statements. 2.1.2. (i) The singleton {0} is the neutral element of Minkowski addition; (ii) addition is associative and commutative; (iii) multiplication by a scalar is distributive with respect to addition. 2.1.3. Minkowski addition and multiplication by a scalar preserve inclusion: Ai ⊂ Bi for i = 1, 2 ⇒ A1 + A2 ⊂ B1 + B2 , A ⊂ B ⇒ t A ⊂ t B. It is easy to see that 2.1.4. The Minkowski operations preserve compactness. Hence + is a function from C n × C n to C n , and multiplication by t is a function from C n to C n . Adding a ball with radius ε to a compact set, we obtain its ε-hull: 2.1.5. For any A ∈ C n and ε > 0, (A)ε = A + ε B n . Proof. Let us ﬁx an x ∈ Rn . Since the metric, and so also its restriction |{x} × A, is continuous, by the compactness of A it follows that x ∈ (A)ε ⇐⇒ ∃a ∈ A x − a ≤ ε ⇐⇒ x ∈ A + ε B n .

The following proposition, which describes the Minkowski operations on generalized balls, is based on 2.1.2 and 2.1.5. 2.1.6. PROPOSITION. (i) For any A, B ∈ C n and α, β > 0, (A)α + (B)β = (A + B)α+β ; (ii) for any A ∈ C n , t > 0, and ε > 0, t (A)ε = (t A)tε . Proof. (i): Obviously,

2.1 The Minkowski operations

α B n + β B n = (α + β)B n .

13

(2.1)

Thus (A)α + (B)β = A + B + (α + β)B n = (A + B)α+β . (ii): t (A)ε = t (A + ε B n ) = t A + tε B n = (t A)tε .

We shall now prove 2.1.7. THEOREM. (i) Minkowski addition is continuous on C n × C n . (ii) Multiplication by a nonnegative scalar is continuous on C n . Proof. (i): As is well known, convergence in a Cartesian product is equivalent to convergence “by coordinates” independently of the choice of a product metric1 .2 Hence, it sufﬁces to prove that ∀ε > 0 ∃δ > 0 ∀A1 , A2 , B1 , B2 ∈ C n H (Ai , Bi ) ≤ δ for i = 1, 2 ⇒ H (A1 + A2 , B1 + B2 ) ≤ ε. Let ε > 0 and let δ :=

ε 2.

(2.2)

If H (Ai , Bi ) ≤ δ for i = 1, 2, then

Ai ⊂ (Bi )δ and Bi ⊂ (Ai )δ , whence by 2.1.3 and 2.1.6(i), A1 + A2 ⊂ (B1 + B2 )ε and B1 + B2 ⊂ (A1 + A2 )ε , i.e., H (A1 + A2 , B1 + B2 ) ≤ ε. (ii): If t = 0, then t A = {0} for every A ∈ C n ; thus the continuity of multiplication by 0 is evident. Let t > 0. For any ε > 0 let δ := εt . If H (A, B) ≤ δ, then A ⊂ (B)δ and B ⊂ (A)δ , whence by 2.1.3 and 2.1.6(ii), t A ⊂ (t B)ε and t B ⊂ (t A)ε . Thus H (t A, t B) ≤ ε.

2.1.8. REMARK. In fact, we have proved the following stronger assertion (compare with [15], p. 253): (i) If ˆ is an arbitrary product metric in C n × C n that satisﬁes for every Ai , Bi , the condition ˆ H (Ai , Bi ) ≤ ((A 1 , A2 ), (B1 , B2 )) for i = 1, 2, then Minkowski addition is uniformly continuous with respect to the metrics ˆ and H .3 1 That means a metric that induces the product topology. 2 Product metrics are studied extensively in [34]. 3 See condition (2.2).

14

2. Subsets of Euclidean Space

(ii) Multiplication by an arbitrary nonnegative t, A → t A, is uniformly continuous (with respect to H ).

2.2 Support hyperplane. The width If E is a closed half-space in Rn and H = bdE, then for any nonzero vector v orthogonal to H either H + v ⊂ intE or (H + v) ∩ E = ∅. In the second case v is an outer normal vector of E. 2.2.1. DEFINITION. Let A be a closed, nonempty subset of Rn . A closed halfspace E is a support half-space of A if A ⊂ E and A ∩ bdE = ∅. Then the hyperplane H := bdE is a support hyperplane of A, the set A ∩ H is a support set, every point of this set is a support point, and an outer normal vector v of the half-space E is an outer normal vector of H . We shall prove the following. 2.2.2. THEOREM. For every A ∈ C n and every v = 0 there is a unique support hyperplane of A with outer normal vector v. Proof. Let H be the set of hyperplanes orthogonal to v and let L := linv. For every H1 , H2 ∈ H, dist(H1 , H2 ) = x1 − x2 , where {xi } = L ∩ Hi for i = 1, 2 (compare (0.2)). Obviously, dist|H × H is a metric in H, and the function πˆ : H → L that assigns to any H ∈ H its orthogonal projection π L (H ) is an isometry. Since A is compact and nonempty, it follows that so is π L (A). Let a ∈ π L (A) and t0 := sup{t ∈ R | a + t · v ∈ π L (A)}. Then evidently, πˆ −1 (a + t0 · v) is the unique support hyperplane of A with outer normal vector v. In view of Theorem 2.2.2, we may use the symbols H (A, v) and E(A, v), respectively, for the support hyperplane and the support half-space of a compact set A with v being an outer normal vector, and the symbol A(v) for the support set A ∩ H (A, v) (Figure 2.1). Applying 2.1.3, one can easily prove the following. 2.2.3. For every A1 , A2 ∈ C n and v = 0, H (A1 + A2 , v) = H (A1 , v) + H (A2 , v) (A1 + A2 )(v) = A1 (v) + A2 (v).

2.2 Support hyperplane. The width

Figure 2.1.

2.2.4. DEFINITION. For A ∈ C n and a nonzero vector v, let b(A, v) := dist(H (A, v), H (A, −v)). Then b(A, v) is the width of A in the direction of v (Figure 2.2).

Figure 2.2.

Directly from 2.2.4 it follows that 2.2.5. For every A, B ∈ C n , A ⊂ B ⇒ ∀v b(A, v) ≤ b(B, v). 2.2.6. DEFINITION. For any A ∈ C n , let d(A) := inf b(A, v). v

Then d(A) is the minimal width of A. Let us notice that the diameter of a set is its maximal width: 2.2.7. For every A ∈ C n ,

15

16

2. Subsets of Euclidean Space

diamA = sup b(A, v). v

Proof. If A is a singleton, then the equality is obvious. Let card A ≥ 2. Since for every x, y ∈ A, x = y ⇒ (x, y) ≤ dist(H (A, y − x), H (A, x − y)) = b(A, x − y), it follows that diamA ≤ sup b(A, v). v

Let v = 0. If a and b are the support points for H (A, v) and H (A, −v), respectively, then b(A, v) ≤ (a, b) ≤ diamA. Hence sup b(A, v) ≤ diamA. v

2.2.8. DEFINITION. The mean value of the function b | C n × S n−1 is referred to as the mean width: for every A ∈ C n , 1 ¯b(A) := b(A, u) dσ (u). σ (S n−1 ) S n−1 We shall prove that the minimal width, diameter, and mean width are continuous; moreover, they are Lipschitz functions (see 2.2.10). 2.2.9. LEMMA. For every A ∈ C n , v = 0, and δ > 0, b((A)δ , v) = b(A, v) + 2δ. Proof. Let x and y be support points for H ((A)δ , v) and H ((A)δ , −v). By 1.1.7, there exist a, b ∈ A such that x ∈ {a}δ and y ∈ {b}δ . Moreover, since x, y ∈ bd(A)δ , it follows that x − a = δ = y − b. It is easy to see that H ((A)δ , v) is also a support hyperplane (and thus a tangent hyperplane) of the ball {a}δ at x, and a is a support point for H (A, v). The situation is similar for the vector −v and the point b. Thus the vectors x − a and y − b are parallel to v, and b((A)δ , v) = dist(H ((A)δ , v), H ((A)δ , −v)) = dist(H (A, v), H (A, −v)) + 2δ = b(A, v) + 2δ.

2.2.10. THEOREM. The functions d, diam, b¯ : C n → R are Lipschitz continuous.

2.2 Support hyperplane. The width

17

Proof. We may assume that A = B. Let δ = H (A, B). Then A ⊂ (B)δ and B ⊂ (A)δ , whence by 2.2.5 and 2.2.9, b(A, v) ≤ b(B, v) + 2δ and b(B, v) ≤ b(A, v) + 2δ. By 2.2.6–2.2.8, passing to the lower and the upper bound for v ∈ S n−1 , we obtain |d(A) − d(B)| ≤ 2 H (A, B)

and |diamA − diamB| ≤ 2 H (A, B);

Integrating over S n−1 , we obtain ¯ ¯ |b(A) − b(B)| ≤ 2 H (A, B). We close this section by proving simple theorems on two other real functions on C n : the volume and the radius of the circumscribed ball with center 0. 2.2.11. THEOREM. Let A, Ai ∈ C n , i ∈ N, A = lim H Ai . Then lim sup Vn (Ai ) ≤ Vn (A).

Proof. Since

(A)ε = A,

ε>0

by the properties of any measure, lim Vn ((A)ε ) = Vn (A).

ε→0

(2.3)

By the assumption and by the monotonicity of measure, we obtain ∀ε > 0 ∃i 0 ∀i > i 0 Vn (Ai ) ≤ Vn ((A)ε ); Hence ∀ε > 0 lim sup Vn (Ai ) ≤ Vn ((A)ε ), which, together with (2.3), completes the proof. 2.2.12. DEFINITION. For every A ∈

Cn ,

let

r0 (A) := inf{α > 0 | A ⊂ α · B n }. 2.2.13. THEOREM. The function r0 : C n → R is a weak contraction. Proof. Let us notice that by 2.2.12, for every X ∈ C n , X ⊂ r0 (X )B n . Thus, as can be easily checked, for every ε > 0 (X )ε ⊂ r0 (X )B n + ε B n = (r0 (X ) + ε)B n . Hence r0 ((X )ε ) ≤ r0 (X ) + ε.

(2.4)

Let ε := H (A, B); then A ⊂ (B)ε and B ⊂ (A)ε , whence by (2.4), r0 (A) ≤ r0 (B) + ε and r0 (B) ≤ r0 (A) + ε. Therefore, |r0 (A) − r0 (B)| ≤ H (A, B).

18

2. Subsets of Euclidean Space

2.3 Convex sets We are now concerned with convex subsets of Rn . 2.3.1. DEFINITION. A set A ⊂ Rn is convex if for every pair of its points, {a, b}, the segment (a, b) is contained in A. The following statement is a direct consequence of 2.3.1. 2.3.2. An afﬁne image of a convex set is convex. The Minkowski operations preserve convexity: 2.3.3. (i) If A1 and A2 are convex, then A1 + A2 is convex. (ii) If A is convex, then for any t ∈ R the set t A is convex. Proof. (i): Let x, y ∈ A1 + A2 . Then x = x1 + x2 and y = y1 + y2 for some xi , yi ∈ Ai , i = 1, 2. Since Ai is convex, it follows that (xi , yi ) ⊂ Ai . Thus (x, y) =

{(1 − t)x + t y | t ∈ [0, 1]}

⊂

{(1 − t)x1 + t y1 | t ∈ [0, 1]} + {(1 − t)x2 + t y2 | t ∈ [0, 1]} (x1 , y1 ) + (x2 , y2 ) ⊂ A1 + A2 .

⊂

(ii) follows directly from 2.3.2.

Let us give a few simple examples: 2.3.4. EXAMPLES. (a) Every segment (with two endpoints, one endpoint, or without any endpoints) as well as any afﬁne subspace of dimension k ∈ {0, . . . , n} is convex. In view of 2.3.2, it sufﬁces to consider the segment (0, e1 ) on the axis x1 and a subspace spanned by k axes of the coordinate system (for k = 0). (b) Every ball is convex. Indeed, in view of 2.3.2, it sufﬁces to show that B n is convex. Let a, b ∈ B n , i.e., a ≤ 1 and b ≤ 1. Then ∀t ∈ [0, 1] (1 − t)a + tb ≤ (1 − t)a + tb ≤ 1 − t + t = 1, whence (a, b) ⊂ B n .

By 2.1.5, 2.3.3, and 2.3.4(b) we infer that 2.3.5. If A is convex, then for every ε > 0 its ε-hull (A)ε is convex. It is evident that the Minkowski operations restricted to convex sets share all the properties described in Section 2.1 for any subsets of Rn . Moreover, multiplication of a convex set by nonnegative scalars is distributive with respect to addition of scalars, i.e., in (2.1) the ball B n can be replaced by an arbitrary convex set A: 2.3.6. PROPOSITION. For every convex A and every α, β ≥ 0, α A + β A = (α + β)A.

(2.5)

2.4 Compact convex sets. Convex bodies

19

Proof. If α = 0 = β, then (2.5) has the form {0} = {0}. Let us assume that α + β > 0. The inclusion ⊃ in (2.5) is true for arbitrary A. β α ⊂ : Let t := α+β . Then t ∈ [0, 1] and 1 − t = α+β ; thus for every a1 , a2 ∈ A, αa1 + βa2 = (α + β)((1 − t)a1 + ta2 ) ∈ (α + β)(a1 , a2 ). Hence α A + β A ⊂ (α + β)A, because (a1 , a2 ) ⊂ A.

The following example proves that in 2.3.6 the assumption that A is convex is essential. 2.3.7. EXAMPLE. Let A = S n−1 . Then 0 ∈ A + A, but 0 ∈ 2A, because 2A is the sphere with radius 2 and center 0. Thus A + A ⊂ 2A. It is easy to check that 2.3.8. The closure of any convex set is convex. We shall prove the following. 2.3.9. PROPOSITION. For every closed subset X of Rn the following conditions are equivalent: (i) X is convex, (ii) ∀a, b ∈ X 12 (a + b) ∈ X. Proof. The implication (i) ⇒ (ii) is evident. (ii) ⇒ (i): Let a, b ∈ X . From (ii) it follows that the set X ∩ (a, b) is dense in (a, b), whence (a, b) ⊂ clX = X,

which completes the proof.

2.3.10. DEFINITION. Let X ⊂ A function f : X → R is convex if its epigraph, {(x, t) ∈ X × R : t ≥ f (x)}, Rn .

is convex. A function f is concave if − f is convex. Proof of the following statement is left to the reader (Exercise 2.5): 2.3.11. PROPOSITION. Let X be a convex subset of Rn . For every continuous function f : X → R the following conditions are equivalent: (i) f is convex, (ii) ∀x, y ∈ X ∀t ∈ [0, 1] f ((1 − t)x + t y) ≤ (1 − t) f (x) + t f (y), (iii) ∀x, y ∈ X f ( 12 (x + y)) ≤ 12 ( f (x) + f (y)).

2.4 Compact convex sets. Convex bodies Let Kn be the class of nonempty compact convex subsets of Rn , and let K0n be the class of convex bodies, i.e., compact convex subsets of Rn with nonempty interior.

20

2. Subsets of Euclidean Space

Since afﬁne maps of Rn into itself are continuous and afﬁne automorphisms (i.e., afﬁne bijections) of Rn are homeomorphisms, from 2.3.2 we deduce the following. 2.4.1. COROLLARY. The class Kn is invariant under afﬁne maps and K0n is invariant under afﬁne automorphisms of Rn . We shall prove 2.4.2. PROPOSITION. (i) Kn is closed under the Minkowski operations. (ii) K0n is closed under Minkowski addition; moreover, if A1 ∈ K0n and A2 ∈ Kn , then A1 + A2 ∈ K0n . (iii) multiplication by any nonzero scalar preserves K0n . Proof. (i): By 2.1.7, the Minkowski operations preserve compactness, and by 2.3.3 they preserve convexity. (ii): Let A1 ∈ K0n and A2 ∈ Kn . By (i), it sufﬁces to verify int(A1 + A2 ) = ∅.

(2.6)

Since intA1 = ∅, it follows that int(A1 + x) = ∅ for every x (because the translations are homeomorphisms). This together with the equality A1 + A2 = {A1 + x | x ∈ A2 } yields (2.6). (iii) also follows from (i) and topological invariance of interior (because the homotheties are homeomorphisms). As a direct consequence of 2.3.5 and 2.4.2(ii),(iii), we obtain the following. 2.4.3. A ∈ Kn ⇒ (A)ε ∈ K0n . Let us note that K0n is not closed in Kn (so all the more in C n ): e.g., a sequence of concentric balls Bk with radii k1 is Hausdorff convergent to a singleton. In the next chapter we shall prove that Kn is closed in C n (Corollary 3.2.13).

2.5 Hyperplanes This section is of auxiliary character. Let E n be the set of all the hyperplanes in Rn . It can be parametrized by means of the function φ : S n−1 × R+ → E n deﬁned by the formula φ(v, t) := {x ∈ Rn | x ◦ v = t}. Evidently, 2.5.1. (a) If E = φ(v, t), then t = dist(0, E) and tv = π E (0). (Figure 2.3).

(2.7)

2.5 Hyperplanes

21

(b) The function φ is a surjection, φ|(S n−1 × (0, ∞)) is an injection, and ∀v ∈ S n−1 φ(v, 0) = φ(−v, 0).

Figure 2.3.

We introduce a topology in E n , deﬁning the limit (Exercise 2.6): 2.5.2. DEFINITION. For any sequence of hyperplanes (E k )k∈N , and for any E ∈ En, E = lim E k ⇐⇒ k

∃v, (vk )k∈N , t, (tk )k∈N v = lim vk , t = lim tk , E = φ(v, t), E k = φ(vk , tk ). k

k

Directly from 2.5.2 it follows that (compare with Exercise 2.7) 2.5.3. For every isometry f : Rm → Rm and arbitrary sequence (E k )k∈N in

Em,

E = lim E k ⇒ f (E) = lim f (E k ).

Since every hyperplane in Rn is isometric to Rn−1 , the assertion 2.5.3 allows one to extend Deﬁnition 2.5.2: the set E n can be replaced by the set of (n − 2)-dimensional afﬁne subspaces of some hyperplane. 2.5.4. PROPOSITION. Let E = limk E k for some E, E k ∈ E n . Then (i) (xk )k∈N ∈ P∞ k=1 E k , x = limk x k ⇒ x ∈ E. (ii) ∀x ∈ E ∃(xk )k∈N ∈ P∞ k=1 E k , x = lim x k . Proof. Let E k = φ(vk , tk ), E = φ(v, t), v = lim vk , t = lim tk . (i): If xk ∈ E k , then by (2.7), xk ◦ vk = tk ; thus x ◦ v = t by the continuity of scalar product. Hence x ∈ E. (ii): Let x ∈ E. Then x ◦ v = t. In view of 2.5.3 we may assume that x = 0 ∈ E, whence t = 0. Let xk := π E k (0) for every k. Then xk = sk vk ∈ E k for some sk ≥ 0; thus tk = xk ◦ vk = sk , and hence xk = tk vk for every k. Passing to the limit, we obtain lim xk = 0, which completes the proof. Proof of the next proposition is left to the reader (Exercise 2.8):

22

2. Subsets of Euclidean Space

2.5.5. PROPOSITION. If H, E, E k ∈ E n for k ∈ N and H ∩ E = H = H ∩ E k for every k, then E = lim E k ⇒ E ∩ H = lim(E k ∩ H ). We shall now prove two theorems on a sequence of intersections of convex bodies by hyperplanes. 2.5.6. THEOREM. Let A ∈ Kn . If E k ∈ E n and E k ∩ intA = ∅ for k = 0, 1, . . . , then E 0 = lim E k ⇒ E 0 ∩ A = lim(E k ∩ A). H

Proof. Let E 0 = lim E k . Then (a) ∀(xik )k∈N ∈ P∞ k=1 (E i k ∩ A) x 0 = lim x k ⇒ x 0 ∈ E 0 ∩ A, and (b) ∀x0 ∈ E 0 ∩ A ∃(xk )k∈N ∈ P∞ k=1 (E k ∩ A) x 0 = lim x k . Indeed, since A is closed, condition (a) follows directly from 2.5.4 (i); let us prove (b) (Figure 2.4). Obviously, we may assume that E 0 , E 1 , . . . are pairwise distinct.

Figure 2.4.

Let x0 ∈ E 0 ∩ A, x1 ∈ E 1 ∩ intA \ {x0 }, and let xk be the intersection point of E k and aff{x0 , x1 }. Then xk ∈ A. Let vk ⊥ E k , v0 = lim vk , and let βk = ((x1 − x0 ), vk ) for k = 0, 1, . . . . It is easy to see that β = lim βk and xk − x0 = dist(x0 , E k ) ·

1 . sin βk

Hence, in view of 2.5.1 and 2.5.2, x0 = lim xk . This proves (b). By Theorem 1.8.7 in [64] (compare Exercise 1.2), from (a) and (b) it follows that E 0 ∩ A = lim(E k ∩ A). H

This completes the proof.

2.5 Hyperplanes

23

2.5.7. THEOREM.4 Let Ak ∈ Kn for k ∈ N∪{0} and E ∈ E n . If E ∩intAk = ∅ for k = 0, 1, . . . , then A0 = lim Ak ⇒ A0 ∩ E = lim(Ak ∩ E). H

H

Proof. It sufﬁces to prove that for any Y ∈

Kn ,

∀ε > 0 ∃δ > 0 (E ∩ intY = ∅ ⇒ (Y )δ ∩ E ⊂ (Y ∩ E)ε ).

(2.8)

Indeed, if Y satisﬁes (2.8), then for every X ∈ Kn , X ⊂ (Y )δ ⇒ X ∩ E ⊂ (Y )δ ∩ E ⇒ X ∩ E ⊂ (Y ∩ E)ε ; thus for X := Ak and Y := A0 , since ∀δ > 0 ∃k0 ∀k > k0 Ak ⊂ (A0 )δ , it follows that ∀ε > 0 ∃k0 ∀k > k0 Ak ∩ E ⊂ ((A0 )δ ∩ E)ε . For X := A0 and Y := Ak the proof is analogous. Thus (2.8) yields the conclusion.

Figure 2.5.

Let Y ∈ Kn . We prove (2.8). For every x ∈ relbd(Y ∩ E) we choose an outer normal vector u x ∈ S n−1 for Y at the point x. Let vx ∈ S n−1 be the normal vector of the hyperplane E for which 4 See [41], Theorem 14.3, which is a little stronger: the assumption that E has a nonempty intersec-

tion with the interior of Ak is made only for k = 0.

24

2. Subsets of Euclidean Space

αx := (vx , u x ) ≤

π . 2

Let us note that αx depends not only on x, but also on the choice of u x ; however, since E ∩ intY = ∅, it follows that αx = 0 independently of the choice of u x . Now let ε > 0. We deﬁne: δ :=

inf

inf ε sin αx .

x∈relbd(Y ∩E) u x

The set relbd(Y ∩ E) is compact; thus, if δ = 0, then αx = 0 for some x. Hence δ > 0. It can be seen that Yδ ∩ E ⊂ (Y ∩ E)ε (compare with Figure 2.5), which proves (2.8).

3 Basic Properties of Convex Sets

3.1 Convex combinations n with coefﬁIn afﬁne geometry the afﬁne combination of points a1 , . . . , ak ∈ R

k k ti ai . cients t1 , . . . , tk ∈ R satisfying i=1 ti = 1 is deﬁned as the point i=1 Further, the afﬁne subspace affA generated by a nonempty subset A of Rn is understood as the set of afﬁne combinations of points from A. It is also called the afﬁne hull of A. We shall deal with counterparts of those notions in convex geometry: the convex combination and the convex hull.

k 3.1.1. DEFINITION. (i) Let t1 , . . . , tk ∈ [0, 1] and i=1 ti = 1. The point

c(a1 , . . . , ak ; t1 , . . . , tk ) :=

k

ti ai

i=1

is the convex combination of points a1 , . . . , ak with coefﬁcients t1 , . . . , tk . (ii) For any A ⊂ Rn C(A) := {c(a1 , . . . , ak ; t1 , . . . , tk ) | a1 , . . . , ak ∈ A, t1 , . . . , tk ∈ [0, 1], k ∈ N}. Thus, evidently, C(A) ⊂ affA. If {a1 , . . . , ak } is afﬁne independent, then the set C({a1 , . . . , ak }) is a simplex, and the points a1 , . . . , ak are its vertices. A simplex with vertices a1 , . . . , ak will be denoted by (a1 , . . . , ak ).

26

3. Basic Properties of Convex Sets

Hence (in accordance with the given notation), a simplex (a1 , a2 ) is the segment with endpoints a1 , a2 . A simplex (a1 , a2 , a3 ) is the triangle with vertices a1 , a2 , a3 . It is easy to verify that 3.1.2. For every A = ∅ the set C(A) is convex. The following simple statement gives a parametric description of a convex set. 3.1.3. PROPOSITION. For every nonempty subset A of Rn the following conditions are equivalent: (i) A is convex, (ii) C(A) = A. Proof. The implication (ii) ⇒ (i) follows directly from 3.1.2. (i) ⇒ (ii): Let us note that for every A, A ⊂ C(A). Assume that A is convex. It sufﬁces to show that for any k ∈ N, a1 , . . . , ak ∈ A, t1 , . . . , tk ∈ [0, 1], and ti = 1 ⇒ c(a1 , . . . , ak ; t1 , . . . , tk ) ∈ A.

(3.1)

(3.2)

For k = 1 condition (3.2) is a tautology: ∀a1 ∈ A a1 ∈ A. Let k ≥ 2. that (3.2) is true for k − 1; let a1 , . . . , ak ∈ A, t1 , . . . , tk ∈

Assume k ti = 1. [0, 1] , and i=1 If tk = 1, then c(a1 , . . . , ak ; t1 , . . . , tk ) = ak ∈ A. Let tk < 1 and ti = Then

k−1

i=1 ti

ti for i = 1, . . . , k − 1. 1 − tk

= 1 and

c(a1 , . . . , ak ; t1 , . . . , tk ) = (1 − tk )c(a1 , . . . , ak−1 , t1 , . . . , tk−1 ) + tk ak ∈ A

by the inductive assumption. Thus (3.2) is true for k; this completes the proof. The function C is Minkowski additive: 3.1.4. For any nonempty A, B, C(A + B) = C(A) + C(B). Proof.1 ⊂: Let x ∈ C(A + B); thus x is a convex combination of some points in A + B, i.e., there exist t1 , . . . , tk ∈ [0, 1], a1 , . . . , ak ∈ A, and b1 , . . . , bk ∈ B such that i ti = 1 and 1 Compare [64].

3.1 Convex combinations

x=

k

27

ti (ai + bi ).

i=1

Hence x ∈ C(A) + C(B). ⊃: Let now x ∈ C(A)

+ C(B); then there exist t1 , . . . , tk , s1 , . . . , sl ∈ [0, 1] such that ti = 1 = s j and x= ti ai + sjbj.

Of course, we may assume that k = l. Let a := ti ai and b := s j b j . Then x =( s j )a + ( ti )b = ti s j (ai + b j ). i, j

Since

i, j

ti s j =

i

ti (

sj) = (

ti )(

s j ) = 1,

j

it follows that x ∈ C(A + B).

The following Carath´eodory theorem says that the set C(A) is generated by afﬁne independent subsets of A; thus C(A) is the union of simplices with vertices in A: 3.1.5. THEOREM. Let A ⊂ Rn . For every x ∈ C(A) there exists an afﬁne independent subset {a1 , . . . , ak } ⊂ A such that x ∈ (a1 , . . . , ak ). Proof.2 Let k := min{m ∈ N | x ∈ C({x1 , . . . , xm }), xi ∈ A, i = 1, . . . , m}. Then there exist a1 , . . . , ak ∈ A such that x ∈ C({a1 , . . . , ak }), whence there exist t1 , . . . , tk ∈ [0, 1] such that

ti = 1 and x =

k

ti ai .

i=1

Suppose the set {a1 , . . . , ak } is afﬁne dependent, i.e., one of its points is an afﬁne combination there exists (s1 , . . . , sk ) = (0, . . . , 0),

kof the others. Equivalently,

k such that i=1 si ai = 0 and i=1 si = 0. It is evident that at least one of the numbers s1 , . . . , sk is positive; thus the set { stii | i ∈ {1, . . . , k}, si > 0} is nonempty. Let stmm be the smallest number of this set and let αi := ti − stmm ·si for i =

k 1, . . . , k. Notice that x = i=1 αi ai , all the coefﬁcients in this combination are nonnegative, their sum is equal to 1, and αm = 0. Thus x is a convex combination of k − 1 points, contrary to the assumption that k is minimal. 2 Compare [64], p. 3.

28

3. Basic Properties of Convex Sets

Another important result, with various applications, is the following Helly theorem (for a proof see [64], Theorem 1.1.6). 3.1.6. THEOREM. Let Xbe a ﬁnite family of convex subsets of Rn . If for every n+1 A1 , . . . , An+1 ∈ X the set i=1 Ai is nonempty, then

X = ∅.

From Theorem 3.1.6 we derive its different version. It concerns a family of arbitrary cardinality, but the elements of this family are assumed to be compact. 3.1.7. THEOREM. Let X ⊂ Kn . If for every A1 , . . . , An+1 ∈ X the set n+1 i=1 Ai is nonempty, then X = ∅. Proof. In view of 3.1.6, every ﬁnite subfamily of X has a nonempty intersection; i.e., the family X has the ﬁnite intersection property (compare with [15], p. 123). Let A0 ∈ X and let X0 := {A0 ∩ A | A ∈ X }. Then X0 ⊂ Kn and X0 = A0 , whence X0 consists of closed subsets of the compact space A0 . The family X0 has the ﬁnite intersection property as well, because for every A1 , . . . , Ak ∈ X , k

(A0 ∩ Ai ) =

i=1

k

Ai = ∅.

i=0

Hence by Theorem 3.1.1in [15],the family X0 has nonempty intersection, and thus X = ∅, because X0 ⊂ X .

3.2 Convex hull The following statement is a direct consequence of Deﬁnition 2.3.1. 3.2.1. PROPOSITION. The intersection of an arbitrary family of convex sets is convex. 3.2.2. DEFINITION. For any subset A of Rn , let F(A) be the family of all convex sets containing A. The intersection of F(A) is called the convex hull of A: convA :=

F(A).

In view of 3.2.1, convA is convex; it is the minimal (with respect to inclusion) convex subset of Rn containing A (see Figure 3.1).

3.2 Convex hull

29

Figure 3.1.

3.2.3. PROPOSITION. For every A, B ⊂ Rn , A ⊂ B ⇒ convA ⊂ convB. Proof. It sufﬁces to observe that A ⊂ B ⇒ F(B) ⊂ F(A).

We shall now prove that the convex hull coincides with the set of convex combinations: 3.2.4. THEOREM. For every A = ∅ convA = C(A). Proof. By (3.1), A is a subset of C(A). Since by 3.1.2, the set C(A) is convex, it follows that convA ⊂ C(A). The operation C is increasing with respect to inclusion, whence C(A) ⊂ C(convA). Thus, by the implication (i) ⇒ (ii) in 3.1.3, C(A) ⊂ convA, because convA is convex.

From 3.2.4 combined with 3.1.3 we obtain the following characterization of convex sets. 3.2.5. COROLLARY. A subset A of Rn is convex if and only if A = convA. The following statement is a direct consequence of 3.1.4 combined with 3.2.4. 3.2.6. COROLLARY. For every A, B ⊂ Rn , conv(A + B) = convA + convB. From 3.2.6 we derive 3.2.7. PROPOSITION. conv ((A)ε ) = (convA)ε . Proof. We apply in turn 2.1.5, 3.2.6, 2.3.4 (b), and 2.1.5: conv ((A)ε ) = conv(A + ε B n ) = convA + conv(ε B n ) = convA + ε B n = (convA)ε .

30

3. Basic Properties of Convex Sets

Let us note that the convex hull of a closed set need not be closed. For example, let A = {(x1 , 0) ∈ R2 | 0 ≤ x1 ≤ 1} ∪ {(0, x2 ) ∈ R2 | x2 ≥ 0}; then convA = {(x1 , x2 ) ∈ R2 | 0 ≤ x1 < 1 and x2 ≥ 0} ∪ {(1, 0)}, whence convA is not closed in R2 , though A is closed. However, the following holds. 3.2.8. THEOREM. A ∈ C n ⇒ convA ∈ C n . Proof. Let, as above, F(A) be the family of convex sets in Rn containing A and let F0 (A) := F(A) ∩ C n . On the one hand, by 3.2.1, convA ⊂

F0 (A).

(3.3)

On the other hand, for every X ∈ F(A) there exists a set X 0 ∈ F0 (A) contained in X . For example, let X 0 = B ∩ clX , where B is a ball containing A (then X 0 is compact, since it is a closed and bounded subset of Rn , and is convex by 2.3.4, 2.3.8, and 3.2.1). Hence convA ⊃ F0 (A). (3.4) By (3.3) and (3.4), the set convA is the intersection of compact subsets of Rn , whence it is compact. In view of Theorem 3.2.8, the function conv may be considered as a function of C n into itself, or as a function from C n into Kn . By 3.2.5, it maps C n onto Kn : 3.2.9. THEOREM. The function conv : C n → Kn is surjective. We shall prove 3.2.10. THEOREM. The function conv : C n → Kn is a weak contraction. Proof. Let A1 , A2 ∈ C n . We have to prove that H (convA1 , convA2 ) ≤ H (A1 , A2 ).

(3.5)

Since condition (3.5) is symmetric with respect to A1 , A2 , it sufﬁces to prove the inequality inf{δ > 0 | convA1 ⊂ (convA2 )δ } ≤ inf{δ > 0 | A1 ⊂ (A2 )δ }.

(3.6)

Let us note that in view of 3.2.3 and 3.2.7, A1 ⊂ (A2 )δ ⇒ convA1 ⊂ (convA2 )δ , whence the set on the left-hand side of (3.6) contains the set on the right-hand side. This implies (3.6). As a direct consequence of 3.2.10, we obtain 3.2.11. COROLLARY. The function conv : C n → Kn is continuous.

3.3 Metric projection

31

In view of 3.2.5 and 3.2.11, the function conv is a continuous map of the space C n onto its subset Kn , and convA = A for every A ∈ Kn . We can formulate it as follows: 3.2.12. COROLLARY. The function conv is a retraction of C n on Kn . Since a retract of any space is closed in this space, from 3.2.12 we deduce the following. 3.2.13. COROLLARY. The family Kn is closed in C n . Evidently, every closed subspace of a ﬁnitely compact metric space is ﬁnitely compact. Therefore, as a consequence of Corollary 3.2.13 combined with Theorem 1.2.8, one obtains the following theorem, which is well known as the Blaschke Selection Theorem (compare with[64], p.!50). 3.2.14. THEOREM. The space Kn is ﬁnitely compact.

3.3 Metric projection In Section 3.2 we characterized convex sets in terms of the operation conv (Corollary 3.2.5). The following theorem gives a characterization of closed convex sets in terms of distance between a point and a set.3 3.3.1. THEOREM. ([54]) For every subset A of Rn the following conditions are equivalent: (i) A is convex, nonempty, and closed in Rn , (ii) for every x ∈ Rn there exists a unique nearest point in A, i.e., a unique a ∈ A with (x, a) = (x, A). Proof. (i) ⇒ (ii): Let x ∈ Rn . If x ∈ A, then of course, x is a unique point of A nearest to x. Let x ∈ A. Since A is closed and nonempty, it follows that there is at least one a ∈ A with (x, a) = (x, A). Suppose there are two points a1 and a2 such that (x, ai ) = (x, A) for i = 1, 2, and let c = 12 (a1 + a2 ). Then c is the midpoint of the base (a1 , a2 ) of the isosceles triangle (a1 , x, a2 ), whence (x, c) < (x, A). But this is impossible, because c ∈ A by the convexity assumption. (ii) ⇒ (i): From (ii) it follows that A = ∅. It also follows that A is closed; indeed, otherwise there exists x ∈ clA \ A, and by (ii), there is an a ∈ A such that (x, a) = (x, A); hence a = x, since (x, A) = 0 by 1.1.4. 3 See footnote on page 11 of [64].

32

3. Basic Properties of Convex Sets

Up to now we have used only a part of condition (ii): the existence of a nearest point in A for every point x. The uniqueness of the nearest point is needed to prove that A is convex (we roughly follow the argument in [64], Theorem1.2.4). Suppose that A is not convex. Then there exist x1 , x2 ∈ A such that (x1 , x2 )∩ (R2 \ A) = ∅. For i = 1, 2, let ai be the point in (x1 , x2 ) ∩ bdA most distant 2 from xi . Then (a1 , a2 ) ∩ intA = ∅. Let B0 be a ball with center p0 := a1 +a 2 , disjoint from A: B0 = B( p0 , r0 ). The set of radii of the balls containing B0 and disjoint from A is bounded; let r1 be its upper bound and let B1 = B( p1 , r1 ). Of course, B1 ∩ A is a singleton { p}, where p is the unique point of A nearest to p1 . If bdB0 ∩ bdB1 = ∅, then these two spheres have exactly one point in common, p, which does not belong to A, because r1 > r0 . Since p0 ∈ ( p1 , p), the sphere bdB1 has at least two points in common with bdA, i.e. p1 has more than one nearest point in A, contrary to the uniqueness part of (ii). Thus bdB1 ∩ bdB2 = ∅, whence B1 is not a maximal ball containing B0 and disjoint from intA, because translating B1 by ε · ( p0 − p1 ) for a sufﬁciently small ε, we can obtain a ball disjoint from A with a bigger radius; thus radius r1 is not maximal, contrary to the assumption. In view of 3.3.1 (more precisely, in view of the implication (i) ⇒ (ii)), for every nonempty, closed, and convex subset A of Rn there exists a function assigning to a point x ∈ Rn the point of A nearest to x. 3.3.2. DEFINITION. Let A be a nonempty, closed, and convex subset of Rn . The function ξ A : Rn → A deﬁned by the condition ξ A (x) = a ⇐⇒ (x, a) = (x, A) is the metric projection onto A (Figure 3.2). We shall prove the following. 3.3.3. LEMMA. Let A be a nonempty, convex, and closed subset of Rn ; let x ∈ Rn \ A and a = ξ A (x). If a hyperplane H0 satisﬁes the conditions a ∈ H0

and

u = x − a ⊥ H0 ,

then H0 = H (A, u). Proof. Since a ∈ A ∩ H0 , by deﬁnition of support hyperplane (Deﬁnition 2.2.1) it sufﬁces to prove that the set A lies on the other side of H0 from the point x. Suppose the opposite; then there exists a point b ∈ A \ {a} such that

3.3 Metric projection

33

Figure 3.2.

0 < (u, b − a) <

π . 2

Since by the assumption, x − b > x − a, it follows that

(a − b, x − b) <

π , 2

whence the height of the triangle (x, a, b) passing through the vertex x intersects (a, b) at some point c. Obviously, c ∈ A and x − c < x − a, contrary to the assumption on a. 3.3.4. THEOREM. For any nonempty, closed, convex subset A of Rn , the metric projection ξ A is a weak contraction. Proof. Let x, y ∈ Rn , a := ξ A (x), and b := ξ A (y). We have to prove that b − a ≤ x − y.

(3.7)

Of course, (3.7) is satisﬁed if a = b. Assume that a = b. Let v := a − b and let Ha and Hb be the hyperplanes orthogonal to v such that a ∈ Ha and b ∈ Hb . If x = a and y = b, then from Lemma 3.3.3 it follows that a and b are support points for H (A, x − a) and H (A, y − b), respectively. Hence v ◦ (x − a) ≥ 0

and

v ◦ (y − b) ≤ 0.

(3.8)

It is obvious that (3.8) is also satisﬁed if x = a or y = b. By (3.8), the points x, y lie on different sides of the open strip bounded by the parallel hyperplanes Ha and Hb ; this proves (3.7). As a consequence of Theorem 3.3.4 and Lemma 3.3.3, we obtain the following. 3.3.5. THEOREM. If A is a nonempty, closed, and convex subset of Rn , then ξ A (Rn \ A) = bdA. Proof. The inclusion ⊂ is obvious.

34

3. Basic Properties of Convex Sets

⊃: Let a ∈ bdA. Then there exists a sequence (xk )k∈N in Rn \ A convergent to a. Let ak := ξ A (xk ) for every k. By 3.3.4, a = lim ak , because 0 ≤ (a, ak ) = (ξ A (a), ξ A (xk )) ≤ (a, xk ). For some α > 0 all the members of (ak )k∈N belong to B(a, α). Let L k := ak +pos(xk −ak ) and yk ∈ L k ∩bdB(a, α); then evidently, ξ A (yk ) = ak for every k. The sequence (yk )k∈N has a subsequence (yik )k∈N convergent to a y ∈ bdB(a, α). Since the metric projections are continuous (compare 3.3.4), it follows that ξ A (y) = lim ξ A (yik ) = lim aik = a.

Let us note the following consequence of Theorem 3.3.5 combined with Lemma 3.3.3. 3.3.6. COROLLARY. If A is a nonempty, convex, and closed subset of Rn , then for every a ∈ bdA there exists a support hyperplane of A passing through a. Moreover, the existence of a support hyperplane of A at every point of bdA is equivalent to the assertion that A is nonempty, closed, and convex (compare Exercise 3.2). Thus we have another characterization of this class of sets. Now we shall again make use of Lemma 3.3.3. 3.3.7. THEOREM. For every nonempty closed subset A of Rn , the following conditions are equivalent: (i) A is convex, (ii) A is the intersection of all its support half-spaces. Proof. The implication (ii)⇒ (i) is evident. (i) ⇒ (ii): Let E(A) be the family of support half-spaces of A. Obviously, A⊂ E(A). Suppose x ∈ A; then by 3.3.3, the hyperplane H0 passing through the point a = ξ A (x) and perpendicular to x − a is a support hyperplane for A. Thus x does not belong to the support half-space with boundary H0 , whence x ∈ E(A). 3.3.8. COROLLARY. For any pair of disjoint sets A1 , A2 ∈ Kn , there exist disjoint closed half-spaces E 1 , E 2 such that Ai ⊂ E i for i = 1, 2. Disjoint closed half-spaces E 1 , E 2 whose existence is assured by 3.3.8 are said to separate the sets A1 , A2 .

3.4 Support function The notion of support function is an important tool of convex geometry.

3.4 Support function

35

3.4.1. DEFINITION. Let A ∈ Kn . The function h A : Rn → R deﬁned by the formula h A (x) := sup{a ◦ x | a ∈ A} is the support function of A. (We shall often write h(A, ·) instead of h A (·).) The ﬁrst part of the next theorem is a direct consequence of 3.4.1. It says that for every A ∈ Kn the support function of A is sublinear. For various proofs of the second part, about the existence and uniqueness of a compact convex A with h A = f for a given sublinear function f : Rn → R, see [64], Theorem1.7.1. 3.4.2. THEOREM. (i) For every A ∈ Kn , every x, y ∈ Rn , and α, β > 0, h A (αx + βy) ≤ αh A (x) + βh A (y). (ii) For every sublinear function f : Rn → R there is a unique A ∈ Kn with hA = f . Let us prove 3.4.3. PROPOSITION. Let A ∈ Kn and u ∈ S n−1 . If a ∈ A ∩ H (A, u), then h A (u) = a ◦ u. Proof. Let E 0 be the support half-space of A, with outer normal unit vector u ∈ S n−1 , E 0 = E(A, u), and let a ∈ A ∩ H (A, u). Obviously, for every x ∈ E 0 , (x − a) ◦ u ≤ 0, with equality for x ∈ H (A, u). Since A ⊂ E 0 , it follows that x ◦ u ≤ a ◦ u for every x ∈ A and h A (u) = sup{x ◦ u | x ∈ A} = a ◦ u.

As we shall see, the support function restricted to S n−1 is positively linear with respect to the Minkowski operations: 3.4.4. THEOREM. For any A1 , A2 ∈ Kn , t1 , t2 ≥ 0 , and u ∈ S n−1 , h(t1 A1 + t2 A2 , u) = t1 h(A1 , u) + t2 h(A2 , u). Proof. Directly from 3.1.1, it follows that ht A = t · h A for every A ∈ Kn and t ≥ 0. Thus it sufﬁces to prove that h is additive with respect to the ﬁrst variable: for every u ∈ S n−1 ,

36

3. Basic Properties of Convex Sets

h(A1 + A2 , u) = h(A1 , u) + h(A2 , u).

(3.9)

To this end, let us note that for any support points a1 , a2 of A1 , A2 , respectively, in the direction of the vector u, their sum a1 + a2 is a support point of the set A1 + A2 in the same direction (compare 2.2.3). Hence, (3.9) follows from 3.4.3. The restriction of a support function to the unit sphere has the following geometric interpretation. 3.4.5. THEOREM. For every u ∈ S n−1 , dist(0, H (A, u)) h A (u) = −dist(0, H (A, u))

if 0 ∈ E(A, u) if 0 ∈ E(A, u).

Proof. Let a ∈ A ∩ H (A, u). Then H (A, u) = {x ∈ Rn | (x − a) ◦ u = 0}, whence dist(0, H (A, u)) = |a ◦ u|. Thus, by 3.4.3, the assertion follows.

3.4.6. COROLLARY. h A |S n−1 ≤ h B |S n−1 ⇐⇒ A ⊂ B. Proof. From Theorem 3.4.5 it follows that for every u ∈ S n−1 , h A (u) ≤ h B (u) ⇐⇒ E(A, u) ⊂ E(B, u). It now sufﬁces to apply 3.3.7.

We shall use 3.4.6 and 3.4.4 in the proof of the following cancellation law for Minkowski addition of compact convex subsets of Rn : 3.4.7. THEOREM. If A1 , A2 , B ∈ Kn , then A1 + B ⊂ A2 + B ⇒ A1 ⊂ A2 . Proof. Let A1 + B ⊂ A2 + B. Then by 3.4.6 and 3.4.4, for every u ∈ S n−1 , h(A1 , u) + h(B, u) ≤ h(A2 , u) + h(B, u), whence h A1 |S n−1 ≤ h A2 |S n−1 . Thus, A1 ⊂ A2 by Corollary 3.4.6.

The following example shows that the convexity assumption in 3.4.7 is essential: 3.4.8. EXAMPLE. Let A1 = B = B n and A2 = S n−1 . Then A1 + B = A2 + B = 2B, while A1 ⊂ A2 . From 2.2.4 combined with 3.4.5 we deduce the following statement concerning width:

3.4 Support function

37

3.4.9. PROPOSITION. For every A ∈ K n and u ∈ S n−1 b(A, u) = h(A, u) + h(A, −u). Finally, let us note a useful relationship between the support function and the Hausdorff metric (compare with [64] Theorem 1.8.11). 3.4.10. THEOREM. For every A1 , A2 ∈ Kn , H (A1 , A2 ) = sup |h(A1 , u) − h(A2 , u)|. u∈S n−1

Proof. By (1.4), H (A1 , A2 ) = inf{α > 0 | A1 ⊂ A2 + α B n and A2 ⊂ A1 + α B n }. Hence, by 3.4.4 combined with 3.4.6, H (A1 , A2 ) = inf α > 0 | h A1 | S n−1 ≤ h A2 | S n−1 + α and h A2 | S n−1 ≤ h A1 |S n−1 + α = inf α > 0 | sup |h(A1 , u) − h(A2 , u)| ≤ α u∈S n−1

= sup |h(A1 , u) − h(A2 , u)|. u∈S n−1

4 Transformations of the Space Kn of Compact Convex Sets

Any continuous function f : Rn → Rn induces the function f ∗ : C n → C n that assigns to an X ∈ C n its image f (X ) ∈ C n : f ∗ (X ) := f (X ). There is an extensive literature concerning relationships between the properties of a function f and those of the induced function f ∗ (compare [37]).

4.1 Isometries and similarities 4.1.1. THEOREM. For every f : Rn → Rn and λ > 0 the following conditions are equivalent: (i) f is a similarity with ratio λ of the space Rn ; (ii) f ∗ is a similarity with ratio λ of the space C n (with the Hausdorff metric). Proof. (i) ⇒ (ii): Let f be a similarity with ratio λ. Obviously, for every x ∈ Rn and Y ∈ C n , ( f (x), f (Y )) = λ(x, Y ). Thus, by Theorem 1.2.2, H ( f (A), f (B)) = max sup ( f (a), f (B)), sup ( f (b), f (A)) = λ H (A,B). a∈A

b∈B

It remains to show that f ∗ is surjective. Let Y ∈ C n and X := f −1 (Y ). Then of course, f ∗ (X ) = Y and X ∈ C n (because f −1 is a similarity).

40

4. Transformations of the Space Kn of Compact Convex Sets

(ii) ⇒ (i): Assume f ∗ to be a similarity with ratio λ > 0; i.e., it is surjective and for every A, B ∈ C n , H ( f ∗ (A), f ∗ (B)) = λ H (A, B). Hence f ∗ is injective, and for A = {a} and B = {b}, f (a) − f (b) = H ( f (A), f (B)) = λ H (A, B) = λa − b. It remains to show that f is surjective. Let y ∈ Rn ; since {y} = f ∗ ({x}) for some x ∈ Rn and f ∗ ({x}) = { f (x)}; it follows that y = f (x). There exist isometric embeddings of the space Kn into Kn that are not induced by any transformations of Rn : 4.1.2. EXAMPLE. Let A ∈ Kn . The map X → X + A is an analogue of a translation of Rn ; we shall call it translation (of Kn by A). From (1.4), 2.1.3, and 3.4.7 it follows that the restriction of this map to Kn is an isometric embedding of Kn into Kn . Indeed, for every δ, X + A ⊂ (Y + A) + δ B n ⇐⇒ X ⊂ Y + δ B n , whence H (X + A, Y + A) = H (X, Y ). However, if A is not a singleton, then the translation by A is not surjective, and thus is not an isometry of Kn . Moreover, it is not induced by any transformation of Rn , since it does not preserve the family of singletons. The following two interesting theorems together with Theorem 4.1.1 give a complete characterization of the isometries and the isometric embeddings of C n and Kn . 4.1.3. THEOREM. ([25]). Every isometry of C n onto C n is induced by an isometry of Rn .1 4.1.4. THEOREM. ([26]). Every isometric embedding F of Kn into Kn satisﬁes the condition F(X ) = f ∗ (X ) + A for some isometry f : Rn → Rn and some A ∈ Kn . In the next section we shall consider a useful example of a continuous map of the space K0n into K0n that is not induced by any transformation of Rn (Theorems 4.2.14 and 4.2.16). We shall need the following statement, which is part of a theorem on characterization of induced maps due to the Charatoniks ([13]).2 1 For convex sets this theorem was proved by R. Schneider in [61]. 2 Their theorem was proved for a class of sets that is slightly different from C n .

4.2 Symmetrizations of convex sets. The Steiner symmetrization

41

4.1.5. THEOREM. Let X be a family of nonempty compact subsets of Rn that contains the set of singletons. If F = f ∗ : X → X for some f : Rn → Rn and F : X → X preserves the set of singletons, preserves inclusion, and F (A) ⊂ F(A) for every A ∈ X , then F = F. Proof. Let A ∈ X . For every x ∈ A the set F ({x}) is a singleton: F ({x}) = {y} for some y ∈ F (A). Since F ({x}) ⊂ F({x}) = { f (x)}, it follows that F ({x}) = { f (x)}. By the assumption, F (A) ⊂ f (A). It remains to prove that f (A) ⊂ F (A).

(4.1)

Let y ∈ f (A); then y = f (x) for some x ∈ A, whence F (A) ⊃ F ({x}) = { f (x)}. This proves (4.1).

4.2 Symmetrizations of convex sets. The Steiner symmetrization 4.2.1. DEFINITION. A function F : Kn → Kn is a symmetrization if there exists an afﬁne subspace E such that for every A ∈ Kn the set F(A) is symmetric with respect to E, that is, σ E (F(A)) = F(A). This deﬁnition can be extended to all nonempty subsets of Rn . We shall give several examples of symmetrizations. The most commonly used is the following one (compare [30]). 4.2.2. DEFINITION. Let H be a hyperplane in Rn . For every x ∈ H , let L x := A ∈ Kn and x ∈ π H (A), let ax be the symmetry center of A∩ L x , and deﬁne vx := x − ax and A x := A ∩ L x + vx . The set S H (A) is deﬁned by the formula S H (A) := Ax (4.2) H ⊥ (x). For any

x∈π H (A)

(Figure 4.1). The function A → S H (A) is called the Steiner symmetrization (with respect to H ). (Obviously, the set A x is either a segment or a singleton.) It is evident that the values of the function S H are sets symmetric with respect to H . As we shall prove, these sets are compact and convex (Theorem 4.2.7). It is easy to see that S H preserves inclusion: 4.2.3. PROPOSITION. A ⊂ B ⇒ S H (A) ⊂ S H (B). The ﬁxed points of S H are the sets symmetric with respect to H :

42

4. Transformations of the Space Kn of Compact Convex Sets

Figure 4.1.

4.2.4. PROPOSITION. S H (A) = A ⇐⇒ σ H (A) = A. Proof. ⇐: Let π := π H . If H is a hyperplane of symmetry of A, then A x = A ∩ L x for every x ∈ π(A), whence S H (A) = A ∩ L x = A ∩ π −1 π(A) = A, x∈π(A)

because A ⊂ π −1 π(A). ⇒: Let S H (A) = A. Then by (4.2), (A ∩ L x + vx ) = A ∩ Lx; x∈π(A)

x∈π(A)

thus A ∩ L x + vx = A ∩ L x for every x ∈ π(A). Hence vx = 0, i.e., ax ∈ H for every x ∈ π(A). Therefore, H is a hyperplane of symmetry of A. Directly from Deﬁnition 4.2.2 it follows that if two hyperplanes are parallel, then the images of a given set under the corresponding symmetrizations are translates of each other: 4.2.5. PROPOSITION. If v ⊥ H and H = H + v, then for every A ∈ Kn , S H (A) = S H (A) + v.

Let us note the following. 4.2.6. PROPOSITION. (i) If v ⊥ H , then for every A ∈ Kn , S H (A + v) = S H (A). (ii) If vH , then for every A ∈ Kn , S H (A + v) = S H (A) + v. We shall prove

4.2 Symmetrizations of convex sets. The Steiner symmetrization

43

4.2.7. THEOREM. (i) A ∈ Kn ⇒ S H (A) ∈ Kn ; (ii) A ∈ K0n ⇒ S H (A) ∈ K0n . Proof. (i): Let A ∈ Kn . (a) The set A is bounded, whence A ⊂ B(a, α) for some a ∈ H and α > 0. Thus by 4.2.3 and 4.2.4, S H (A) ⊂ S H (B(a, α)) = B(a, α); hence S H (A) is bounded too. (b) Since A is closed in Rn , also S H (A) is closed. Indeed, let yi ∈ S H (A) for i ∈ N, y = lim yi , xi = π H (yi ), x = π H (y), and let z i := yi − vxi for i ∈ N, and z := y − vx . Obviously, z i ∈ A. It is easy to check that lim z i = z (Exercise 4.2); thus z ∈ A because A is closed. Hence y ∈ S H (A). (c) Since A is convex, it follows that so is S H (A): Take two points y1 , y2 ∈ S H (A) and let xi := π H (yi ) for i = 1, 2. Then yi ∈ A xi for i = 1, 2 (compare 4.2.2). Let X = conv(A ∩ L x1 ∪ A ∩ L x2 ), Y = conv(A x1 ∪ A x2 ). Generally, the sets X and Y are trapezoids with their bases perpendicular to H . It is easy to verify that Y = S H (X ). Since A is convex, it follows that X ⊂ A and thus Y ⊂ S H (A) in view of 4.2.3. Therefore, (y1 , y2 ) ⊂ S H (A). (ii): By (i), it sufﬁces to prove intA = ∅ ⇒ intS H (A) = ∅. If B is a ball contained in A, then by 4.2.5 and 4.2.3, the set S H (B) is a ball contained in S H (A). Volume is one of the important invariants of the Steiner symmetrization. 4.2.8. THEOREM. For every A ∈ Kn , Vn (S H (A)) = Vn (A). Proof. Applying Fubini’s theorem twice, we obtain Vn (S H (A)) = V1 (A x )d Vn−1 (x) π(A) = V1 (A ∩ L x )d Vn−1 (x) = Vn (A). π(A)

The next two statements describe relationships between the Steiner symmetrization and the Minkowski operations.

44

4. Transformations of the Space Kn of Compact Convex Sets

4.2.9. THEOREM. If 0 ∈ H , then for every λ > 0, λS H (A) = S H (λA). Proof. Since 0 ∈ H , it follows that λH = H and λL x = L λx for x ∈ H . Hence λ(L x ∩ A) = L λx ∩ (λA), because a homothety with nonzero ratio is a bijection. A homothety maps the midpoint of a segment to the midpoint of its image and preserves parallelism and congruence; hence λA x = (λA)λx (Figure 4.2). By Thales Theorem, π H (λA) = λπ H (A); thus λS H (A) = λA x = (λA)λx = S H (λA). x∈π H (A)

λx∈π H (λA)

Figure 4.2.

4.2.10. THEOREM. If 0 ∈ H , then S H (A + B) ⊃ S H (A) + S H (B). Proof. Let c ∈ S H (A) + S H (B); then c = a + b for some a ∈ S H (A) and b ∈ S H (B). We have to prove that c ∈ S H (A + B). We project a, b, c on the line L 0 := H ⊥ ; let a := π L 0 (a), b = π L 0 (b), c := π L 0 (c).

(4.3)

4.2 Symmetrizations of convex sets. The Steiner symmetrization

45

The map π L 0 is linear, whence c = a + b . Consider the vectors u := a − a, v := b − b, and w := c − c; let A := A + u and B := B + v. Of course, w = u + v, whence A + B = (A + B) + w. We shall ﬁrst show that S H (A + B ) ∩ L 0 ⊃ S H (A ) ∩ L 0 + S H (B ) ∩ L 0 .

(4.4)

Let us note that A ∩ L 0 and B ∩ L 0 are segments (possibly degenerate) on the line L 0 ; further, A ∩ L 0 + B ∩ L 0 ⊂ (A + B ) ∩ L 0 , because L 0 is a linear subspace. Hence V1 (A ∩ L 0 ) + V1 (B ∩ L 0 ) = V1 (A ∩ L 0 + B ∩ L 0 ) ≤ V1 ((A + B ) ∩ L 0 ). Consequently, by Deﬁnition 4.2.2, we obtain V1 (S H (A ) ∩ L 0 ) + V1 (S H (B ) ∩ L 0 ) ≤ V1 (S H (A + B ) ∩ L 0 ).

(4.5)

Observe that all three segments mentioned in (4.5) are contained in L 0 and have the same midpoint; thus (4.5) implies (4.4). In view of 4.2.6.(ii), (S H (A) + u) + (S H (B) + v) = S H (A ) + S H (B ) and

S H (A + B) + w = S H (A + B ),

whence by (4.4), (S H (A) + u) ∩ L 0 + (S H (B) + v) ∩ L 0 ⊂ S H (A + B + w) ∩ L 0 ⊂ S H (A + B + w) = S H (A + B) + w. Thus c + w ∈ S H (A + B) + w, which proves (4.3).

The following corollary is a direct consequence of 4.2.10 combined with 4.2.4. 4.2.11. COROLLARY. For every ε > 0 and A ∈ Kn , (S H (A))ε ⊂ S H ((A)ε ). Let us notice that in Corollary 4.2.11 (and so in Theorem 4.2.10 as well) inclusion cannot be replaced by equality: 4.2.12. EXAMPLE. Let n = 2 and H = {(x1 , x2 ) | x2 = 0}; let A = ((0, 0), (2, 2)) and ε ≤ 12 . For x = (1, 0), the set L x ∩ S H ((A)ε ) is a √ segment with length 2ε 2, while L x ∩ (S H (A))ε is a segment with length 2ε. Hence S H ((A)ε ) = (S H (A))ε (Figure 4.3 a,b). It is easy to generalize this example to arbitrary n (Exercise 4.9). As we already mentioned in Section 4.1, the function S H is continuous on K0n . For the proof we shall need the following simple lemma.

46

4. Transformations of the Space Kn of Compact Convex Sets

Figure 4.3.

4.2.13. LEMMA. Let A ∈ K0n , α, β, δ > 0, and λ > 1. If α Bn ⊂ A ⊂ β Bn , then (i) A + δ B n ⊂ (1 + αδ )A, (ii) λA ⊂ A + (λ − 1)β B n . Proof. (i): α B n ⊂ A ⇒ αδ B n ⊂ δ A ⇒ δ B n ⊂ αδ A. Thus in view of 2.3.6, we obtain (i). (ii): A ⊂ β B n ⇒ λA ⊂ A + (λ − 1)A ⊂ A + (λ − 1)β B n . 4.2.14. THEOREM. For any hyperplane H in Rn , the function S H | K0n is continuous. Proof. Let A, Ak ∈ K0n for k ∈ N and let A = lim Ak .3 We have to show that S H (A) = lim S H (Ak ).

(4.6)

Since translations are continuous with respect to the Hausdorff metric, and by 4.2.5, S H +v (X ) = S H (X ) + v for every X ∈ Kn and v ⊥ H , without loss of generality we may assume that 0 ∈ H . Obviously, 0 ∈ int(A + u) for some unit vector u. Let u = u 1 + u 2 , u 1 H, u 2 ⊥ H. By 4.2.6, S H (A + u) = S H (A + u 1 ) = S H (A) + u 1 and analogously, for every k, S H (Ak + u) = S H (Ak ) + u 1 ; thus we may also assume that 0 ∈ intA. 3 We write here lim instead of lim . H

4.2 Symmetrizations of convex sets. The Steiner symmetrization

47

Since A = lim Ak , it follows that there is a function φ : (0, ∞) → N such that for every δ > 0, ∀k > φ(δ) A ⊂ Ak + δ B n and Ak ⊂ A + δ B n .

(4.7)δ

The set A is bounded and 0 ∈ intA, whence for some α1 , β1 > 0, α1 B n ⊂ A ⊂ β1 B n .

Let k1 := φ

(4.8)

α1 α1 α1 , α := , β := + β1 . 2 2 2

Then by (4.8), α Bn ⊂ A ⊂ β Bn ,

(4.9)

and by (4.8) combined with (4.7)α , for k > k1 , α1 B n ⊂ Ak +

α1 n B ⊂ A + α1 B n ⊂ (α1 + β1 )B n ; 2

hence in view of the cancellation law 3.4.7, ∀k > k1 α B n ⊂ Ak ⊂ β B n .

(4.10)

Now let ε > 0 and k2 = φ(ε · βα ). By condition (4.7)ε· βα , α α A ⊂ Ak + ε · B n and Ak ⊂ A + ε · Bn β β for k > k2 . Hence by (4.9) and (4.10), from Lemma 4.2.13.(i) for δ = ε · βα , it follows that if k > max{k1 , k2 }, then ε ε A ⊂ 1+ Ak and Ak ⊂ 1 + A. β β Thus by 4.2.3 (i) combined with 4.2.9, ε ε S H (A) ⊂ 1 + S H (Ak ) and S H (Ak ) ⊂ 1 + S H (A). β β This together with Lemma 4.2.13.(ii) for λ = 1 +

ε β

yields

S H (A) ⊂ S H (Ak ) + ε B n and S H (Ak ) ⊂ S H (A) + ε B n for sufﬁciently large k. Thus the proof of condition (4.6) is complete.

Theorem 4.2.14 concerns the family of convex bodies K0n . The following example shows that it cannot be generalized on Kn , that is, the function S H : Kn → Kn is not continuous.

48

4. Transformations of the Space Kn of Compact Convex Sets

4.2.15. EXAMPLE. Let H = {(0, t) | t ∈ R} ⊂ R2 . Consider the sequence of segments Ik in R2 convergent to the segment I : 1 Ik = (0, 1), ,0 , k

I = ((0, 1), (0, 0)).

Obviously, 1 S H (Ik ) = (0, 0), ,0 , k

1 1 S H (I ) = 0, − , 0, , 2 2

whence lim S H (Ik ) = S H (I ). It is easy to modify this example for arbitrary n (Exercise 4.10). 4.2.16. THEOREM. The Steiner symmetrization S H : Kn → Kn is not induced by any transformation of Rn . Proof. According to 4.1.5, it sufﬁces to ﬁnd a function F : Kn → Kn that preserves the family of singletons, preserves inclusion, and satisﬁes the following condition: ∀A ∈ Kn F (A) ⊂ S H (A)

and ∃A ∈ Kn F (A) = S H (A).

The simplest example of such a function is (π H )∗ . (Another example is f ∗ S H , where f is an afﬁne map such that H is the set of its ﬁxed points and for every x ∈ H the function f |L x is the homothety of L x with ratio λ < 1.)

4.3 Other symmetrizations The Steiner symmetrization is a particular case of symmetrization with respect to an arbitrary afﬁne subspace (compare [20], p. 58). 4.3.1. DEFINITION. Let E be a k-dimensional afﬁne subspace of Rn for some k ∈ {0, . . . , n − 1}. For any A ∈ Kn and x ∈ π E (A) let A x be the ball in E ⊥ (x) with dimension equal to dim(A ∩ E ⊥ (x)), center x and (n − k)-dimensional volume equal to λn−k (A ∩ E ⊥ (x)). Let S E (A) := Ax . x∈π E (A)

The function A → S E (A) is a symmetrization (with respect to E). Obviously, for k = n − 1 it is the Steiner symmetrization. For k = 1 the symmetrization S E is called the Schwarz symmetrization. As well as the Steiner symmetrization, the Schwarz symmetrization preserves Kn and K0n , preserves volume, and is not induced by any transformation of Rn (Exercises 4.3–4.5).

4.4 Means of rotations

49

We end this section with one more well-known example of symmetrization with respect to a hyperplane. n . For every A ∈ Kn let 4.3.2. DEFINITION. Let H ∈ Gn−1

M H (A) :=

1 (A + σ H (A)). 2

The function M H is called the Minkowski symmetrization or the Blaschke symmetrization (compare [64]).

4.4 Means of rotations We shall now consider the class of transformations of Kn , which will be called means of rotations.4 4.4.1. DEFINITION. A map T : C n → C n is a mean of rotations if there exists m ∈ N and f 1 , . . . , f m ∈ S O(n) such that for every A ∈ C n T (A) =

m 1 f i (A). m i=1

(4.11)

If T is deﬁned by the formula (4.11), we say that it is determined by f 1 , . . . , f m or is the mean of rotations f 1 , . . . , f m . 4.4.2. THEOREM. Let T : C n → C n be a mean of rotations. Then (i) T preserves the classes Kn and K0n ; (ii) T preserves inclusion; (iii) T is linear in the sense of Minkowski: T (α A + β B) = αT (A) + βT (B); (iv) the ball B n is a ﬁxed point of T ; (v) T is continuous (with respect to H ). Proof. (i) follows from 4.4.1 combined with 2.4.2, (ii) from 4.4.1 combined with 2.1.3, and (iii) from 4.4.1. (iv): Let T be determined by f 1 , . . . , f m . Since f i ∈ O(n) for every i, it follows

m that f (B n ) = m1 ( i=1 B n ) = B n , by 2.3.6. (v) is a consequence of (iii), (iv), and (1.4). Means of rotations preserve mean width and do not increase diameter: 4.4.3. THEOREM. Let T be a mean of rotations. For every A ∈ Kn , ¯ (A)) = b(A); ¯ (i) b(T (ii) diamT (A) ≤ diamA. 4 In German Drehmittelungen ([30]). Hadwiger used this notion in a more general sense.

4. Transformations of the Space Kn of Compact Convex Sets

50

Proof. Let T be determined by f 1 , . . . , f m . From Theorem 3.4.4, it follows directly that for every u ∈ S n−1 , h(T (A), u) =

m 1 h( f i (A), u). m i=1

By the deﬁnition of the support function, 3.4.1, h( f i (A), u) = h(A, f i−1 (u)); hence by 3.4.9, for every u ∈ S n−1 b(T (A), u) =

m m 1 1 b( f i (A), u) = b(A, f i−1 (u)). m i=1 m i=1

(4.12)

Thus by 2.2.8, since the spherical measure σ is invariant under linear isometries, we obtain 1 ¯b(T (A)) = b(T (A), u) dσ (u) σ (S n−1 ) S n−1 m 1 ¯ = b(A, v) dσ (v) = b(A). mσ (S n−1 ) i=1 S n−1 From (4.12) and 2.2.7 it follows that diamT (A) = sup b(T (A), v) ≤ diamA. v

Means of rotations do not increase the values of the function r0 (Deﬁnition 2.2.12). 4.4.4. THEOREM. Let T be a mean of rotations. For every A ∈ Kn , r0 (T (A)) ≤ r0 (A). Proof. In view of 4.4.2, A ⊂ α B n ⇒ T (A) ⊂ α B n ; thus {α > 0 | A ⊂ α B n } ⊂ {α > 0 | T (A) ⊂ α B n }. This completes the proof.

The set of means of rotations is closed under superposition (Exercise 4.6): 4.4.5. PROPOSITION. If T1 and T2 are means of rotations, then T1 T2 is also a mean of rotations. Generally, means of rotations are not induced by any transformations of Rn :

4.4 Means of rotations

51

4.4.6. THEOREM. Let T be a mean of rotations f 1 , . . . , f m . Then T = f ∗ for some f : Rn → Rn if and only if m = 1 or f 1 = · · · = f m .

(4.13)

Proof. If condition (4.13) is satisﬁed, then T = ( f 1 )∗ . Assume that it is not satisﬁed and let 1 ∗ T := ( f1 + · · · + fm ) . m Then T preserves the family of singletons, preserves inclusion, and T (A) ⊂ T (A) for every A. We shall show that T = T. Indeed, let A = B n . On the one hand, by 4.4.2 (v), T (A) = A; on the other hand, T (A) is the image of the unit ball under the map m1 ( f 1 + · · · + f m ), which is a linear map but is not an isometry (Exercise 4.7), whence T (A) = A. Hence T is not induced by any transformation of Rn (compare with Theorem 4.1.5).

5 Rounding Theorems

Roughly speaking, rounding theorems1 (5.1.2 and 5.3.2) show how to turn an arbitrary convex body into a ball, iterating Steiner symmetrizations or using means of rotations.

5.1 The ﬁrst rounding theorem 5.1.1. DEFINITION. For any A ∈ K0n , let n S(A) := {S Hk · · · S H1 (A) | Hi ∈ Gn−1 for i = 1, . . . , k, k ∈ N}.

5.1.2. THEOREM. For every A ∈ K0n there exists a sequence in S(A) that is Hausdorff convergent to a ball with center 0 and volume Vn (A). Proof. Let A ∈ K0n . We deﬁne α := inf{r0 (A ) | A ∈ S(A)} (compare Deﬁnition 2.2.12). There exists a sequence (Ai )i∈N in S(A) with α = lim r0 (Ai ). i

For every i, the set Ai is obtained from A by iterating Steiner symmetrizations with respect to hyperplanes passing through 0. Since by 4.2.3 combined with 4.2.4 no Steiner symmetrization increases the value of r0 , it follows that 1 Kugelungstheoreme in [29].

54

5. Rounding Theorems

∀i r0 (Ai ) ≤ r0 (A). Thus the sequence (Ai )i∈N is bounded, whence by 1.2.8 and 1.2.4, it has a convergent subsequence (Aki )i∈N . Without loss of generality we may assume that (Ai )i∈N is convergent; let A0 := lim Ai . (5.1) Since the function r0 is continuous (Theorem 2.2.13), it follows that α = r0 (A0 ).

(5.2)

Let us notice that α > 0. Indeed, by Theorem 4.2.8, Vn (Ai ) = Vn (A); thus in view of 2.2.11, Vn (A0 ) ≥ lim sup Vn (Ai ) = Vn (A) > 0;

(5.3)

hence intA0 = ∅. Let now B0 := α B n . We shall prove that B0 is the “snowball” we are looking for, i.e., A0 = B0 . (5.4) By (5.2), A0 ⊂ B0 ; suppose that A0 = B0 . Let S0 := bdB0 . Then S0 \ A0 is a nonempty open subset of S0 , whence there exists a “spherical ball” (the intersection of S0 with a ball in Rn ) C0 contained in S0 and disjoint from A0 . By compactness of the sphere S0 , its covering by all congruent copies of C0 has a ﬁnite subcovering {C j | j = 0, . . . , m}. For every j ∈ {1, . . . , m} there is a hyperplane H j 0 with σ H j (C j−1 ) = C j . It is easy to show (by induction on j) that S Hm · · · S H1 (A0 ) ∩ S0 = ∅, whence S Hm · · · S H0 (A0 ) ⊂ intB0 . Since the symmetrizations S H j are continuous on K0n (Theorem 4.2.14), by (5.1) it follows that S Hm · · · S H1 (A0 ) = limi S Hm · · · S H1 (Ai ). Hence, for some i 0 , S Hm · · · S H1 (Ai0 ) ⊂ intB0 . Therefore, r0 (S Hm · · · S H1 (Ai0 )) < α, though the set on the left-hand side belongs to S(A). This completes the proof of (5.4). It remains to prove that Vn (B0 ) = Vn (A); in view of (5.3), it sufﬁces to prove the inequality Vn (B0 ) ≤ Vn (A).

(5.5)

5.2 Applications of the ﬁrst rounding theorem

55

We apply 4.2.11 and 4.2.8. Since Ai ∈ S(A) for every i ∈ N, it follows that ∀ε > 0 ∀i Vn ((A)ε ) ≥ Vn ((Ai )ε ). By (5.4), B0 = lim Ai ; thus B0 ⊂ (Ai )ε for almost all i, whence Vn ((A)ε ) ≥ Vn (B0 ) for every ε > 0. Passing to the lower bound with respect to ε, we obtain (5.5).

5.2 Applications of the ﬁrst rounding theorem As an application of Theorem 5.1.2, we shall give a proof of the Brunn–Minkowski inequality, which is the ﬁrst part of the well-known Brunn–Minkowski Theorem (Theorem 5.2.1). The idea of this proof (for n = 3) is partially due to Wilhelm Blaschke (Blaschke applies Steiner symmetrizations, but does not use the rounding theorem; compare [5]). For a proof of the second part, concerning the equality case, see [64]. 5.2.1. THEOREM. Let A0 , A1 ∈ Kn . Then (i) for every t ∈ [0, 1], 1

1

1

(Vn ((1 − t)A0 + t A1 )) n ≥ (1 − t)(Vn (A0 )) n + t (Vn (A1 )) n ; (ii) the equality for some t holds if and only if A0 and A1 either are contained in parallel hyperplanes or are homothetic. Proof of (i). The inequality is evidently true for sets A0 , A1 at least one of which has empty interior. Thus, let us assume that A0 , A1 ∈ K0n . For every t ∈ [0, 1] let At := (1 − t)A0 + t A1

and

1

f (t) := (Vn (At )) n .

Then the inequality in Theorem 5.2.1 is equivalent to the condition f (t) ≥ (1 − t) f (0) + t f (1).

(5.6)

The inequality (5.6) is obviously true for t = 0 or t = 1; let t ∈ (0; 1). By Theorem 5.1.2, in the family S(At ) (Deﬁnition 5.1.1) there exists a sequence (i) (At )i∈N Hausdorff convergent to the ball Bt with center 0 such that Vn (Bt ) = Vn (At ), (i)

(5.7) (i)

and for every i, the set At is obtained from At by means of a ﬁnite iteration Ft of Steiner symmetrizations with respect to hyperplanes passing through 0: (i)

(i)

At = Ft (At ). Let

(i) (i) A˜ 0 := Ft (A0 ),

(i) (i) A˜ 1 := Ft (A1 ).

(5.8)

56

5. Rounding Theorems

By 4.2.3, 4.2.9, and 4.2.10, from (5.8) it follows that (i)

(i)

(i)

At ⊃ (1 − t) A˜ 0 + t A˜ 1 .

(5.9)

Two sequences deﬁned by (5.8) are not necessarily convergent, but there exists an (k ) (k ) increasing sequence of indices (ki )i∈N such that ( A˜ 0 i )i∈N and ( A˜ 1 i )i∈N converge, respectively, to some A˜ 0 , A˜ 1 . From (5.9) it follows that Bt ⊃ (1 − t) A˜ 0 + t A˜ 1 .

(5.10)

(i)

Further, by 5.1.2, there exists a sequence (F0 )i∈N of iterations of Steiner sym(i) metrizations such that lim F0 ( A˜ 0 ) = B0 ; hence (5.10) implies Bt ⊃ (1 − t)B0 + Aˆ 1 , (l )

where Aˆ 1 = limi F0 i ( A˜ 1 ) for some subsequence (li ). Applying now 5.1.2 to Aˆ 1 , we obtain Bt ⊃ (1 − t)B0 + t B1 . (5.11) For every t, let rt be the radius of Bt . From (5.11) we derive the inequality rt ≥ (1 − t)r0 + tr1 , and consequently, by (5.7), the corresponding inequality (5.6) for the nth roots of n-volumes. 5.2.2. COROLLARY. For every A0 , . . . , Am ∈ Kn , m ∈ N, 1

(Vn (A0 + · · · + Am )) n ≥

m

1

(Vn (Ai )) n .

i=0

In view of 5.2.2, means of rotations do not decrease volumes of compact convex sets. 5.2.3. COROLLARY. Let T : C n → C n be a mean of rotations. For every A ∈ Kn , Vn (T (A)) ≥ Vn (A).

5.3 The second rounding theorem 5.3.1. DEFINITION. For every A ∈ K0n , let T(A) := {T (A) | T is a mean of rotations}. 5.3.2. THEOREM. Let A ∈ K0n and 0 ∈ A. There exists a sequence in T(A) Hausdorff convergent to a ball B0 satisfying the following conditions: ¯ ¯ 0 ) = b(A); (i) b(B (ii) Vn (A) ≤ Vn (B0 ) ≤ (diamA)n κn .

5.3 The second rounding theorem

57

Proof. The reasoning is similar to that in the proof of Theorem 5.1.2. Let A ∈ K0n . Deﬁne α := inf{r0 (A ) | A ∈ T(A)}. There is a sequence (A j ) j∈N in T(A), such that α = lim r0 (A j ). j

Since for every j, the set A j is obtained from A by a mean of rotations, from 4.4.2 it follows that ∀ j A j ⊂ r0 (A)B n . Hence (A j ) j∈N has a convergent subsequence, and thus we may assume that it is convergent; let A0 = lim A j . By the continuity of the function r0 , α = r0 (A0 ). Let us notice that α > 0. Indeed, by 5.2.3, ∀ j Vn (A j ) ≥ Vn (A) > 0, whence from Theorem 2.2.11 it follows that Vn (A0 ) ≥ Vn (A) > 0;

(5.12)

thus int(A0 ) = ∅. Let B0 = α B n . We shall prove that B0 is the ball required, i.e., A0 = B0 .

(5.13)

The inclusion A0 ⊂ B0 is evident. Suppose that A0 = B0 . Let S0 = bdB0 . Then S0 \ A0 is an open nonempty subset of the sphere S0 , whence there exists a “spherical ball” C0 contained in S0 \ A0 . Let {C0 , . . . , Cm } be a covering of S0 by congruent copies of C0 . For every i ∈ {1, . . . , m} there exists f i ∈ S O(n) such that f i (C0 ) = Ci . For every X ∈ Kn , let T (X ) :=

1 ( f 1 (X ) + · · · + f m (X )). m

(5.14)

By the linearity of the support function (Theorem 3.4.4) and its geometric properties (Theorem 3.4.5),

58

5. Rounding Theorems

m m 1 1 h( f i (A0 ), v) = h(A0 , f i−1 (v)). m i=1 m i=1 (5.15) In turn, since A0 ⊂ B0 and h(B0 , u) = α for every u ∈ S n−1 , it follows that

∀v ∈ S n−1 h(T (A0 ), v) =

∀v ∈ S n−1 ∀i h(A0 , f i−1 (v)) ≤ α.

(5.16)

Obviously, for every v ∈ S n−1 there exists i 0 ∈ {1, . . . , m} with αv ∈ Ci0 . Thus α f i−1 (v) = f i−1 (αv) ∈ C0 ⊂ S0 \ A0 ; hence 0 0 h(A0 , f i−1 (v)) < α. 0

(5.17)

From conditions (5.15)–(5.17) it follows that for every v ∈ S n−1 , h(T (A0 ), v) <

m 1 α = α; m i=1

therefore, supv h(T (A0 ), v) < α, because this upper bound is attained (the sphere is compact and the support function is continuous with respect to the second variable). However, supv h(T (A0 ), v) = r0 (T (A0 )), whence by 4.4.5 and the deﬁnition of r0 , for every j, r0 (T (A0 )) < α ≤ r0 (T (A j )).

(5.18)

But T is continuous (Theorem 4.4.2 (v)), whence (5.18) contradicts the continuity of r0 . This proves condition (5.13). It remains to prove (i) and (ii). (i) is a consequence of continuity of the function b¯ (see Theorem 2.2.10) and its invariance under means of rotations (Theorem 4.4.3 (i)). (ii) The ﬁrst inequality is a direct consequence of (5.12) and (5.13). Since 0 ∈ A, it follows that α ≤ diamA. Hence Vn (B0 ) = α n κn ≤ (diamA)n κn .

5.4 Applications of the second rounding theorem As an application of Theorem 5.3.2, we shall prove the following Bieberbach theorem: 5.4.1. THEOREM. Among all the compact subsets of Rn with a given positive diameter, a ball has the maximal volume. Proof. We have to prove that if B is a ball and A ∈ C n , then diamA = diamB ⇒ Vn (A) ≤ Vn (B).

(5.19)

5.4 Applications of the second rounding theorem

59

Since Vn (A) ≤ Vn (convA) and diamA = diam(convA) (compare Exercise 3.7), we may assume that A ∈ Kn . The equality of diameters implies that Vn (B) = ( 12 diamA)n κn ; thus it sufﬁces to prove the inequality 1 Vn (A) ≤ n (diamA)n κn . (5.20) 2 By Theorem 5.3.2, there exists a sequence (Ai )i∈N in T(A) convergent to a ball B0 with Vn (B0 ) ≥ Vn (A). (5.21) Since means of rotations do not increase diameter (Theorem 4.4.3 (ii)), it follows that diamAi ≤ diamA for every i, and thus by 2.2.10, diamB0 ≤ diamA. This condition together with (5.21) implies Vn (B0 ) Vn (A) ≥ , n (diamB0 ) (diamA)n which is equivalent to (5.20).

6 Convex Polytopes

The family of convex polytopes in Rn is a rich subfamily of Kn , which forms a separate subject of research ([28], [44]). We shall deal only with selected problems, in particular, with the role of convex polytopes in geometry of compact convex sets (Theorems 6.3.1 and 6.3.2), analogous to the role of arbitrary geometric polyhedra in the topology of compact subsets of Rn .

6.1 Polyhedra and their role in topology A nonempty subset S of Rn is a simplex if S = conv{a0 , . . . , ak } for some afﬁne independent set of points {a0 , . . . , ak }. Such points a0 , . . . , ak are called the vertices of S. A simplex with vertices a0 , . . . , ak will be denoted by (a0 , . . . , ak ).1 Let us note that vertices are the only points of a simplex that do not belong to the relative interior of any segment contained in this simplex. As a consequence, we obtain the following. 6.1.1. PROPOSITION. Every simplex determines uniquely the set of its vertices: 1 Compare 3.1.

62

6. Convex Polytopes

(a0 , . . . , ak ) = (b0 , . . . , bl ) ⇐⇒ k = l and {a0 , . . . , ak } = {b0 , . . . , bl }. In view of 6.1.1, the dimension of a simplex can be deﬁned as follows: for every k ∈ N, dim (a0 , . . . , ak ) := k; hence the dimension of a simplex is equal to the dimension of aff{a0 , . . . , ak }. For example, every segment is a simplex of dimension 1, and every triangle is a simplex of dimension 2. It is convenient to extend the above deﬁnition of a k-dimensional simplex as follows: every singleton is a simplex of dimension 0; the empty set is the simplex of dimension −1. To introduce the notion of a face of a simplex, let S be a simplex of dimension k ≥ 0 and let S (0) be the set of its vertices; then every simplex with vertices in the set S (0) is a face of S; in particular, if k = 0, then S and ∅ are the only faces of S. A face S of S is said to be proper if S = S. Thus for instance, {a0 } and {a1 } are proper faces of the segment (a0 , a1 ); vertices (more precisely, singletons {a0 }, {a1 }, {a2 }) and sides are proper faces of the triangle (a0 , a1 , a2 ). Let S (i) bethe set of i-dimensional faces of a simplex S. We deﬁne the relation ≺ in the set i S (i) : S1 ≺ S2 ⇐⇒ S1 is a face of S2 . 6.1.2. DEFINITION. A set T of simplices is called a simplicial complex if (i) S1 , S2 ∈ T ⇒ S1 ∩ S2 ≺ Si for i = 1, 2, (ii) S ∈ T and S ≺ S ⇒ S ∈ T . A set P ⊂ Rn is a (geometric) polyhedron if there exists a simplicial complex T such that (iii) P = T . We then say that T is a triangulation of the polyhedron P, and P is a geometric realization of the complex T : P = |T | (compare with Exercises 6.1–6.3). The set of simplices in T of dimension i will be denoted by T (i) and called the i-dimensional skeleton of T . The dimension of T is the maximum of the dimensions of its simplices: dim T := max{dim S | S ∈ T }. Let us note that “polyhedron” is often understood as a topological polyhedron, i.e., a set homeomorphic to a geometric polyhedron. However, since we deal only with geometric polyhedra, we omit the adjective “geometric.” Polyhedra play an important role in the topology of Rn ; for instance, every compact subset can be (in some sense) approximated by a sequence of polyhedra (compare [14], Theorem 1.10.18).

6.1 Polyhedra and their role in topology

63

To any polyhedron P in Rn an integer χ (P) is assigned, the Euler–Poincar´e characteristic of P. The function χ is deﬁned as follows. First, the characteristic of a simplicial complex is deﬁned (we denote it by the same symbol χ , since it will not lead to confusion): 6.1.3. DEFINITION. For any simplicial complex T , let ki (T ) be the number of its i-dimensional simplices. Then χ (T ) :=

dim T

(−1)i ki (T ).

i=0

6.1.4. THEOREM. If the geometric realizations of complexes T and T coincide, then χ (T ) = χ (T ) (compare with Exercise 6.14). In view of Theorem 6.1.4, the Euler–Poincar´e characteristic of a polyhedron P can be deﬁned by the formula χ (P) := χ (T ) for any triangulation T of P. 6.1.5. EXAMPLE. Let S be an n-dimensional simplex. Then χ (S) = 1

and

χ (bdS) = 1 − (−1)n .

Indeed, it sufﬁces to verify these two formulae for the simplest triangulations T and T of the polyhedra S and bdS, respectively: T consists of all the nonempty faces of S and T consists all the proper (nonempty) faces of S. of Evidently, ki (T ) = n+1 i+1 for i ∈ {0, . . . , n} , while ki (T ) = ki (T ) for i < n and kn (T ) = 0. Hence n n+1 n+1 n+1 χ (S) = = = 1; (−1)i (−1) j−1 i +1 j i=0 j=1 therefore, χ (bdS) = 1 − (−1)n . An important property of the Euler–Poincar´e characteristic is its topological invariance (see [36], p. 242, for the proof): 6.1.6. THEOREM. If polyhedra P1 and P2 are homeomorphic, then χ (P1 ) = χ (P2 ). In view of 6.1.6, the function χ can be extended over the family of all topological polyhedra.

64

6. Convex Polytopes

6.2 Convex polytopes Convex polytopes may be deﬁned as convex polyhedra; however, we prefer the traditional Deﬁnition 6.2.1, which is much simpler. In view of Theorem 6.2.4, these two deﬁnitions are equivalent.2 6.2.1. DEFINITION. A nonempty subset P of Rn is a convex polytope if there exists a ﬁnite X ⊂ Rn with convX = P. The dimension of a convex polytope P is deﬁned by the formula dim P := dim affP.

(6.1)

Let P n be the family of all convex polytopes in Rn and let P0n := P n ∩ K0n . By 3.2.8, every convex polytope is compact, whence 6.2.2. P n ⊂ Kn . 6.2.3. LEMMA. If P ∈ P n and H is a support hyperplane of P, then H ∩ P ∈ Moreover, for every ﬁnite X ,

Pn.

P = convX ⇒ H ∩ P = conv(H ∩ X ). Proof. Obviously, we may assume that P is not contained in H . By 6.2.1, P = convX for some ﬁnite set X . We shall prove that H ∩ P = conv(H ∩ X ).

(6.2)

Since H ∩ P = H ∩ convX ⊃ conv(H ∩ X ), it remains to prove the inclusion ⊂ in (6.2). Let E + be the support half-space of P with boundary H . Assume that x ∈ H ∩ P; then exist k ∈ N, x1 , . . . , xk ∈ X , and t1 , . . . , tk > 0

k x ∈ H and there k such that i=1 ti = 1 and x = i=1 ti xi . It sufﬁces to prove that xi ∈ H for every i ∈ {1, . . . , k}. If k ≥ 2, then by the Carath´eodory Theorem 3.1.5, there exists a simplex S of dimension at least 1 with vertices in {x1 , . . . , xk } ⊂ P such that x ∈ relintS. Thus S ⊂ E + , whence x ∈ intE + , contrary to the assumption that x ∈ H . Hence k = 1, and thus x1 = x ∈ H . A nonempty subset F of a convex polytope P is called a proper face of P if F = P and F is a support set of P: ∃u ∈ S n−1 F = P ∩ H (P, u). The polytope P is the improper face of itself. 2 In the literature, the word “polytope” has different meanings; see, e.g., [12]

6.2 Convex polytopes

65

Hence, in view of Lemma 6.2.3, every face of a convex polytope is again a convex polytope. Faces of dimension 0 are called vertices, faces of dimension 1 are called edges, and (n − 1)-dimensional faces are called facets. The set of all proper faces of a convex polytope P is denoted by F(P), and the set of faces of dimension i by F (i) (P). We shall now prove 6.2.4. THEOREM. For every subset P of Rn , the following conditions are equivalent: (i) P ∈ P n ; (ii) P is a convex geometric polyhedron. Proof. (i) ⇒ (ii): We have to prove that for every P ∈ P n there exists a triangulation T (P) of P. If dim P = 0, then the complex T (P) consists of one vertex. Let dim P = k ≥ 1 and assume that the assertion is true for the polytopes of dimension k − 1; in particular, we already have T (F) for every proper face F of P, so that F ≺ F implies T (F ) ⊂ T (F). We choose p ∈ relintP and deﬁne T (P) as follows: T (P) := (T (F) ∪ {conv(S ∪ { p}) | S ∈ T (F)}). F∈F k−1 (P)

Thus, T (P) is built of the given triangulations of all facets of P and all the kdimensional simplices with one vertex p and remaining vertices in relbdP. It is clear that T (P) is a triangulation of P. Hence P is a polyhedron. By the assumption, it is convex. (ii) ⇒ (i): Assume now that P is a convex polyhedron. Let T be its triangulation. It sufﬁces to prove that P = conv|T (0) |.

(6.3)

Of course, |T (0) | ⊂ P, whence conv|T (0) | ⊂ P, because the set P is convex. In turn, if x ∈ P, then there exists S ∈ T such that x ∈ S. Since the vertices of the simplex S belong to |T (0) |, it follows that S ⊂ conv|T (0) |. This completes the proof of (6.3). Every convex polytope is the convex hull of the set of its vertices: 6.2.5. THEOREM. Let {a0 , . . . , ak } be the set of vertices of a convex polytope P in Rn . Then P = conv{a0 , . . . , ak }. Proof. The inclusion ⊃ is obvious. ⊂: In view of Deﬁnition 6.2.1, the set P is the convex hull of a ﬁnite set X . Thus it sufﬁces to prove that every vertex belongs to X . Suppose, to the contrary, that there exists a vertex a of P that does not belong to X . Then by the Carath´eodory

66

6. Convex Polytopes

Theorem 3.1.5, there exist an afﬁne independent subset {x0 , . . . , xm } of X and t0 , . . . , tm such that a=

m

ti xi , 0 ≤ ti < 1,

i=0

m

ti = 1.

i=0

Thus P contains a simplex S of positive dimension such that a ∈ relintS,

contrary to the assumption that a is a vertex of P.

It is easy to prove that the class P n is afﬁne invariant (compare Exercise 6.6): 6.2.6. PROPOSITION. For every afﬁne automorphism f of Rn , P ∈ P n ⇒ f (P) ∈ P n . The class P n is closed with respect to Minkowski addition: 6.2.7. PROPOSITION. P, Q ∈ P n ⇒ P + Q ∈ P n . Proof. Let P = conv{a0 , . . . , ak }, Q = conv{b0 , . . . , bl }. By 3.2.6, P + Q = conv{ai + b j | i = 1, . . . , k, j = 1, . . . , l}.

For any P ∈ P0n and any vector v = 0, let P(v) be the support set of P with outer normal vector v (compare with 2.2.1) and let ˆ P(v) := conv({0} ∪ P(v)).

(6.4)

6.2.8. LEMMA. Let P, Q ∈ P0n and v ∈ S n−1 . If at least one of the faces P(v), Q(v) is of dimension n − 1, then dim(P + Q)(v) = n − 1. Proof. By 2.2.3, (P + Q)(v) = P(v) + Q(v). Since P(v) and Q(v) are contained in hyperplanes parallel to v ⊥ , it follows that P(v) + Q(v) is also contained in such hyperplane; hence dim(P + Q)(v) ≤ n − 1. But Minkowski addition does not decrease dimension (Exercise 6.7), whence dim(P + Q)(v) = n − 1.

The following theorem describes the Minkowski sum of convex polytopes in Rn with nonempty interiors.

6.3 Approximation of convex bodies by polytopes

67

6.2.9. THEOREM. Let P, Q ∈ P0n , 0 ∈ intQ, and let {v1 , . . . , vk } ⊂ S n−1 be the set of outer normal vectors to the facets of P or Q. Then P+Q=P∪

k

ˆ i )), (P(vi ) + Q(v

i=1

ˆ i ), and P(v j ) + Q(v ˆ j ) for i = j have pairwise where all the sets P, P(vi ) + Q(v disjoint interiors. Proof. Evidently, the inclusion ⊃ holds, and P+Q=P∪ bd(P + λQ). 0<λ≤1

Since for every λ ∈ (0; 1] and v ∈ S n−1 , ˆ ˆ λ Q(v) ⊂ Q(v) (see (6.4)), to prove the inclusion ⊂ it sufﬁces to show that ∀λ ∈ (0; 1] bd(P + λQ) ⊂

k

ˆ i )). (P(vi ) + λ Q(v

(6.5)

i=1

Let us note that (by Lemma 6.2.8) for any P, Q ∈ P0n , if {v1 , . . . , vk } is the set of all outer normal unit vectors for the facets of P or Q , then bd(P + Q ) =

k i=1

(P + Q )(vi ) =

k

(P(vi )+ Q (vi )) ⊂ P ∪

i=1

k

(P(vi )+ Qˆ (vi )).

i=1

Setting Q := λQ, we obtain (6.5). It remains to prove that the summands on the right-hand side of (6.5) have pairwise disjoint interiors. We leave it to the reader (Exercise 6.8).

6.3 Approximation of convex bodies by polytopes 6.3.1. THEOREM (see [30]). Let A ∈ Kn , ε > 0, and let X be a polytope contained in A. Then there exists P ∈ P n such that X ⊂ P ⊂ A ⊂ (P)ε . Proof. Since A is compact, its covering by all the balls with centers in A has a ﬁnite subcovering {B1 , . . . , Bk } such that Bi = {xi }ε and all the vertices of X belong to {x1 , . . . , xk }. Let P := conv{x1 , . . . , xk }.

68

6. Convex Polytopes

Then X ⊂ P ⊂ A, because A is convex and by 6.2.5, the polytope X is the convex hull of the set of its vertices. It remains to prove that A ⊂ (P)ε . Let x ∈ A. There exists i ∈ {1, . . . , k} such that x ∈ Bi , whence (x, P) ≤ (x, xi ) ≤ ε. Thus x ∈ (P)ε .

Kn

6.3.2. THEOREM (compare [30]). For every A ∈ with 0 ∈ relintA there exists a family of convex polytopes, (P(λ))λ>1 , such that ∀λ > 1 P(λ) ⊂ A ⊂ λ · P(λ) and lim H (P(λ), A) = 0.

λ→1

Proof. Since affA is isometric to Rm for m = dim A, without loss of generality we may assume that A ∈ K0n . Let X be an n-dimensional cube with center 0 contained in A, and let α be the length of its edge. For every λ > 1, let ε(λ) :=

α (λ − 1). 2

By Theorem 6.3.1, for any λ > 1 there is a convex polytope P(λ) that satisﬁes the condition X ⊂ P(λ) ⊂ A ⊂ (P(λ))ε(λ) . Since limλ→1 ε(λ) = 0, it follows that H (A, P(λ)) → 0 if λ → 1. It sufﬁces to prove that for every λ, (P(λ))ε(λ) ⊂ λ · P(λ).

(6.6)

Let x ∈ λ · P(λ) for some λ; then there exists a point y with (0, x) ∩ bd(λ · P(λ)) = {y}; hence there exists a facet F of P(λ) such that y ∈ λF. Let H = affF. Obviously, H is a support hyperplane of P(λ). Let H be the hyperplane parallel to H and passing through x. Then H ∩ λ · P(λ) = ∅ and (x, P(λ)) ≥ (x, H ) = dist(H , H ) > dist(λH, H ) = (0, λH ) − (0, H ) = (λ − 1)(0, H ) ≥ (λ − 1)(0, bdP) ≥ (λ − 1)(0, bdX ) α ≥ (λ − 1) = ε(λ). 2 Thus x ∈ (P(λ))ε(λ) .

6.4 Equivalence by dissection

69

6.4 Equivalence by dissection For a given group G, the binary relation ∼G on the set P0n is deﬁned as follows. 6.4.1. DEFINITION. Let G be a group of transformations of Rn . Polytopes P, Q ∈ P0n are equivalent by dissection with respect to G (in symbols P ∼G Q) if there exist k ∈ N and P1 , . . . , Pk , Q 1 , . . . , Q k ∈ P0n such that P=

k

Pi , Q =

i=1

k

Q i , Pi ≡G Q i for i = 1, . . . , k,

i=1

and int(Pi ∩ P j ) = ∅ = int(Q i ∩ Q j ) for i = j. Here G will be either the group of isometries or the group Tr of translations of Rn . In the ﬁrst case, the relation ∼G will be called simply equivalence by dissection and will be denoted by ∼. 6.4.2. THEOREM. Let P, Q ∈ P02 . Then P ∼ Q ⇐⇒ V2 (P) = V2 (Q). This theorem was proved at the beginning of the nineteenth century (compare [7] and also [40] p. 246). The three-dimensional version of 6.4.2 was the subject of the third Hilbert problem: is it true that every two polytopes P, Q ∈ P03 with equal volumes are equivalent by dissection?3 The solution was given by Dehn a few months after Hilbert announced the problems (compare [57]). The following two three-dimensional simplices S1 , S2 have equal volumes but are not equivalent by dissection: S1 = conv(A ∪ {(0, 0, 1)}), S2 = conv(A ∪ {(0, 1, 1)}), where A = ((0, 0, 0), (1, 0, 0), (0, 1, 0)). To prove that S1 is not equivalent to S2 , Dehn found invariants that distinguish one simplex from another. They are called Dehn’s invariants. Generally, a function f : P0n → R is a Dehn invariant if for every ndimensional polytopes P0 , P1 , P2 , P1 ∼ P2 ⇒ f (P1 ) = f (P2 ) and (P0 = P1 ∪ P2 and dim(P1 ∩ P2 ) < n) ⇒ f (P0 ) = f (P1 ) + f (P2 ). In the example considered, f (S) is the sum of products of lengths of the edges of a three-dimensional simplex S by measures of the dihedral angles corresponding to these edges. 3 See Section 8.6 of [39].

70

6. Convex Polytopes

A complete classiﬁcation of n-dimensional polytopes with respect to the relation ∼Tr is due to Hugo Hadwiger. Invariants that allow one to distinguish nonequivalent polytopes are called Hadwiger’s invariants (see [57], p. 43). As a consequence of Theorem 6.2.9 we obtain Theorem 6.4.4, the so-called theorem on decomposition; we shall need it only for n ≤ 3. To formulate it, we extend in the obvious way the relation ∼Tr over ﬁnite unions of polytopes with disjoint interiors, and introduce the notion of cylindric polytope: 6.4.3. DEFINITION. A convex polytope X ∈ P0n is a cylindric polytope if there exist polytopes X 1 , X 2 of positive dimensions such that X is their direct sum: X = X 1 ⊕ X 2; i.e., X = X 1 + X 2 , Rn = affX 1 + affX 2 , and affX 1 ∩ affX 2 is a singleton. 6.4.4. THEOREM. Let P, Q ∈ P0n , n ≥ 2, and 0 ∈ intQ. There exist polytopes Q 1 , . . . , Q k ∈ P0n and cylindric polytopes X 1 , . . . , X m such that P+Q=P∪

k

Qi ∪

i=1

m

X j,

j=1

k

i=1 Q i ∼Tr Q, and all the sets on the right-hand side of the formula have pairwise disjoint interiors. Sketch of proof (see Figure 6.1). Let w1 , . . . , wk be the outer normal unit vecˆ k ) (see ˆ 1 ), . . . , Q(w tors corresponding to the facets of Q. We dissect Q into Q(w (6.4)): k ˆ i ). Q= Q(w i=1

Further, let v1 , . . . , vl be among the outer unit normal vectors corresponding to the facets of P that do not belong to {w1 , . . . , wk }. Then in view of 6.2.9, P+Q=P∪

k

ˆ i )) ∪ (P(wi ) + Q(w

i=1

l

ˆ j )). (P(v j ) + Q(v

j=1

ˆ i ) is a translate of Q(w ˆ i ); otherwise, it is If dim P(wi ) = 0, then P(wi ) + Q(w ˆ i ). the union of some cylindric polytopes and a translate of Q(w By the assumption on v1 , . . . , vl , all of Q(v1 ), . . . , Q(vl ) have dimensions less ˆ j ) = 1), then P(v j ) + Q(v ˆ j ) has than n − 1. If dim Q(v j ) = 0 (that is, dim Q(v ˆ j) dimension n and is a cylindric polytope. If dim Q(v j ) > 0, then P(v j ) + Q(v is the union of some cylindric polytopes. The details are left to the reader (see Exercise 6.9). The following corollary can be derived from 6.4.2 (Exercise 6.11). 6.4.5. COROLLARY. Every cylindric polytope in R3 is equivalent by dissection to a cube.

6.5 Spherical polytopes

71

Figure 6.1.

6.5 Spherical polytopes We shall now introduce the notion of a spherical polytope, which will be useful in Chapter 7. It is closely related to that of a convex polytope in Rn . 6.5.1. DEFINITION. A nonempty convex subset X of Rn will be called a (linear) convex cone if ∀λ ≥ 0 λX ⊂ X (see [64]). It would be natural to refer to a translate by c of a linear convex cone as a convex cone with vertex c. However, we consider here only linear convex cones; thus we omit the adjective “linear.” 6.5.2. PROPOSITION. For every A ∈ Kn , the set posA is a convex cone. Proof is left to the reader (compare Exercise 6.12). 6.5.3. DEFINITION. A convex cone X in Rn is a polyhedral convex cone if there exist u 1 , . . . , u k ∈ S n−1 such that k X = conv posu i . i=1

Any support set of the cone X is a face; a one-dimensional face is an edge.

72

6. Convex Polytopes

6.5.4. DEFINITION. A subset Y of S n−1 is a spherical polytope if there exists a polyhedral convex cone X with X ∩ S n−1 = Y. 6.5.5. PROPOSITION. For all convex cones X 1 , X 2 in Rn , X 1 ∩ S n−1 = X 2 ∩ S n−1 ⇒ X 1 = X 2 . Proof. It sufﬁces to observe that for a convex cone X , pos(X ∩ S n−1 ) = X.

Hence we can deﬁne faces of a spherical polytope as follows: 6.5.6. DEFINITION. Let k ∈ {0, . . . , n − 1}. A set S is a k-dimensional face of a spherical polytope Y in S n−1 if posS is a (k + 1)-dimensional face of the cone posY . In particular, vertices of Y are the intersection points of edges of posY with S n−1 .

7 Functionals on the Space Kn. The Steiner Theorem

7.1 Functionals on the space Kn In Chapters 2, 3, and 5, some functionals (that is, functions with real values) deﬁned on Kn or C n , have already been considered. Before we return to those examples, let us specify the properties of the functionals in which we are interested. 7.1.1. DEFINITION. Let G be a group of transformations of Rn . A functional : Kn → R is invariant with respect to G if for every A, B ∈ Kn , A ≡G B ⇒ (A) = (B). If is invariant with respect to the group of isometries of Rn , we simply say that it is invariant. 7.1.2. DEFINITION. A functional : Kn → R is a valuation if for every A, B ∈ Kn with A ∪ B ∈ Kn , (A ∪ B) + (A ∩ B) = (A) + (B).

(7.1)

The property described by (7.1) is sometimes called additivity (see [30]); however, to avoid a misunderstanding, we use this term only for Minkowski additivity. There is a connection between these two notions: 7.1.3. PROPOSITION. Every Minkowski additive functional : Kn → R is a valuation. Proof. For every A, B ∈ Kn with convex union A ∪ B, (A ∪ B) + (A ∩ B) = A + B

74

7. Functionals on the Space Kn. The Steiner Theorem

(compare Exercise 2.1). Hence, from the additivity of it follows that (A) + (B) = (A + B) = ((A ∪ B) + (A ∩ B)) = (A ∪ B) + (A ∩ B).

Generally, continuity of will be understood as continuity with respect to the Hausdorff metric. 7.1.4. DEFINITION. A functional : Kn → R is increasing if for every A, B ∈ Kn , A ⊂ B ⇒ (A) ≤ (B). 7.1.5. DEFINITION. Let p ∈ R. A functional : Kn → R is homogeneous of degree p if for every A ∈ Kn and λ > 0, (λA) = λ p (A); is homogeneous if it is homogeneous of degree 1. 7.1.6. EXAMPLES. Minimal width, diameter, and mean width are continuous functionals (Theorem 2.2.10), as is the function r0 (Theorem 2.2.13). The ﬁrst three are invariant with respect to the group of all isometries, while the last one is invariant with respect to O(n). Among these four functionals, only the mean width is a valuation (Exercise 7.1). All of them are increasing and homogeneous. Obviously, the set of all functionals on Kn , with addition and multiplication by scalars inherited from Rn , is a linear space. The following theorem is a direct consequence of the above deﬁnitions. 7.1.7. THEOREM. The set of functionals invariant with respect to a group G, the set of valuations, and the set of continuous functionals are linear subspaces of the space of all functionals on Kn . The set of increasing functionals is closed under addition and multiplication by positive scalars. All these considerations (Deﬁnitions 7.1.1, 7.1.2, 7.1.4, 7.1.5 and Theorem 7.1.7) can be restricted to any subset of Kn , in particular to P n . Since nonempty compact convex sets can be approximated by convex polytopes (Theorem 6.3.2), a natural question is, when are the values of a functional on Kn uniquely determined by its values on P n ? The answer is given by Theorem 7.1.11. 7.1.8. LEMMA. If a functional 0 : P n → R is increasing, then for every A ∈ Kn , (i) the set {0 (P) | P ⊃ A} is bounded from below; (ii) the set {0 (P) | P ⊂ A} is bounded from above. Proof. (i): Let a ∈ A. Then for every P ∈ P n , P ⊃ A ⇒ 0 (P) ≥ 0 ({a}).

7.1 Functionals on the space Kn

75

(ii): Let Q be an n-cube containing A. Then for every P ∈ P n P ⊂ A ⇒ 0 (P) ≤ 0 (Q).

In view of Lemma 7.1.8, we may admit the following deﬁnition: 7.1.9. DEFINITION. For every increasing functional 0 : P n → R, let , : → R be deﬁned by the formulae

Kn

(A) := inf{0 (P) | P ⊃ A, P ∈ P n }, (A) := sup{0 (P) | P ⊂ A, P ∈ P n }. 7.1.10. LEMMA. If 0 : P n → R is increasing, then and are also increasing and |P n = 0 = |P n . (7.2) If 0 is homogeneous of degree p, then and are also homogeneous of degree p. Proof. Let A, B ∈ Kn and let A ⊂ B. Then {0 (P) | P ⊃ B} ⊂ {0 (P) | P ⊃ A}, whence (A) ≤ (B). Thus is increasing. For the proof is analogous. Condition (7.2) is evident. Now let 0 be homogeneous of degree p. Then for every λ > 0, (λA) = sup 0 (P) = sup 0 (λP ) = λ p (A); P⊂λA

P ⊂A

that is, is homogeneous of degree p. For the reasoning is analogous.

7.1.11. THEOREM. (i) If 0 : P n → R is increasing, invariant with respect to Tr, and homogeneous of degree p, then = . (ii) If, moreover, 0 is uniformly continuous, then is continuous, whence it is a unique continuous extension of 0 over Kn . Proof. (i): Let A ∈ Kn . Since 0 is increasing, it follows that for every P, Q ∈ n P , P ⊂ A ⊂ Q ⇒ 0 (P) ≤ 0 (Q); thus, passing ﬁrst to sup P and next to inf Q , in view of Deﬁnition 7.1.9 we obtain the inequality (A) ≤ (A). Hence by Lemma 7.1.10, P ⊂ A ⊂ Q ⇒ 0 (P) ≤ (A) ≤ (A) ≤ 0 (Q).

(7.3)

Since 0 is invariant with respect to Tr, we may assume that 0 ∈ relintA. Thus by Theorem 6.3.2, there exists a subfamily (P(λ))λ>1 of P n such that

76

7. Functionals on the Space Kn. The Steiner Theorem

∀λ > 1 P(λ) ⊂ A ⊂ λP(λ)

(7.4)

lim H (A, P(λ)) = 0.

(7.5)

and λ→1

From (7.3), (7.4), and homogeneity of degree p of 0 it follows that for every λ > 1, 0 (P(λ)) ≤ (A) ≤ (A) ≤ 0 (λP(λ)) = λ p 0 (P(λ)). This condition together with (7.5) implies lim sup 0 (P(λ)) = (A) ≤ (A) ≤ lim sup λ p 0 (P(λ)) = (A). λ

λ

Therefore (A) = (A).

(7.6)

By Lemma 7.1.10, the functional is increasing and homogeneous of degree p. (ii): Assume 0 to be uniformly continuous. Let Ai ∈ Kn for i ∈ N and A = lim H Ai . In view of Theorem 6.3.2, there exist sequences of polytopes (Pk )k∈N and (Pi,k )k∈N for i ∈ N such that A = lim Pk and Ai = lim Pi,k . H

H

By the properties of limit, there exists an increasing sequence of indices (k(i))i∈N such that A = lim Pi,k(i) . H

Hence limi H (Pi,k(i) , Pi ) = 0. Let ε > 0. Since 0 is uniformly continuous, it follows that ε ∃i 0 ∀i > i 0 |0 (Pi,k(i) ) − 0 (Pi )| < . 3 By 7.1.9 combined with (7.6), ∃i 1 ∀i > i 1 |0 (Pi,k(i) ) − (Ai )| < and ∃i 2 ∀i > i 2 |0 (Pi ) − (A)| <

ε 3

ε . 3

Thus ∀i > max{i 0 , i 1 , i 2 } |(A) − (Ai )| < ε. We have proved that is continuous. In view of 6.3.2, the set P n is dense in Kn , whence this continuous extension over Kn is unique.

7.2 Basic functionals. The Steiner theorem

77

The following proposition is sometimes called the “theorem on simultaneous approximation”: 7.1.12. PROPOSITION. If j : P n → R, j = 1, . . . , m, are increasing, then for every ε > 0 and A ∈ Kn there exist P, Q ∈ P n such that P ⊂ A ⊂ Q, | j (P) − j (A)| ≤ ε, | j (Q) − j (A)| ≤ ε for j = 1, . . . , m. Proof. Let A ∈ Kn and ε > 0. By Deﬁnition 7.1.9, for every j ∈ {1, . . . , m} there exist P j , Q j ∈ P n such that P j ⊂ A ⊂ Q j , | j (P j ) − j (A)| ≤ ε, | j (Q j ) − j (A)| ≤ ε. Let P := conv

m j=1

P j , Q :=

m

Q j.

j=1

Then P ⊂ A ⊂ Q, j (P) ≤ j (A) ≤ j (P j ) + ε ≤ j (P) + ε, and j (Q) ≥ j (A) ≥ j (Q j ) − ε ≥ j (Q) − ε. This completes the proof.

7.2 Basic functionals. The Steiner theorem We are now going to deﬁne a ﬁnite sequence of functionals on P n and extend them over Kn . We begin with the notions of an outer normal angle and its measure. 7.2.1. DEFINITION. Let P ∈ P n , k ∈ {0, . . . , n − 1}, and F ∈ F k (P). The set nor(P, F) and the real γ (P, F) are deﬁned as follows: nor(P, F) := {u ∈ S n−1 | P(u) = F}, σn−k−1 (nor(P, F)) γ (P, F) := . ωn−k The set nor(P, F) is the outer normal angle of the polytope P with respect to the face F; the number γ (P, F) is its (normed) measure. 7.2.2. EXAMPLE. (Figure 7.1). Let dim P = n and F ∈ F(P). (a) If F is a facet of P with outer unit normal vector v, then nor(P, F) = {v} and γ (P, F) = 12 . (b) If dim F = n − 2, then F is the intersection of two facets F1 , F2 , and nor(P, F) is an arc of a great circle of the sphere S n−1 ; this arc lies in the twodimensional linear subspace orthogonal to F, and its endpoints are outer normals of F1 and F2 . (c) If F = {a} (that is, F is a vertex of P), and F1 , . . . , Fm are the facets intersecting in a, then nor(P, F) is the spherical polytope whose vertices are outer normals of the facets F1 , . . . , Fm . (Compare Deﬁnitions 6.5.4 and 6.5.6.)

78

7. Functionals on the Space Kn. The Steiner Theorem

Figure 7.1.

7.2.3. EXAMPLE. Let dim P = n − 1, H := affP, and let v ∈ S n−1 be a normal vector of the hyperplane H . If dim F = n − 2, then nor(P, F) is the semicircle with endpoints v and −v in the two-dimensional linear subspace orthogonal to affF (Figure 7.2). Hence γ (P, F) = 12 .

Figure 7.2.

We shall use the following convention: the sum of numbers indexed by the empty set equals 0. 7.2.4. DEFINITION. Let P ∈ P n and k ∈ {0, . . . , n − 1}. λ (F)γ (P, F). Vk (P) := F∈F k (P) k

7.2 Basic functionals. The Steiner theorem

79

The functionals V0 , . . . , Vn are called basic functionals or intrinsic volumes. The term “basic functionals” is justiﬁed by Theorem 8.1.6. The term “intrinsic volumes” was introduced by P. McMullen [45]. In earlier literature, instead of the basic functionals the so-called Quermass integrals W0 , . . . , Wn were considered. The relationship between V0 , . . . , Vn and W0 , . . . , Wn is described by the formula −1 n Wk (P) := κk Vn−k (P) (7.7) k (see [64]). As we shall see in a while, (Theorem 7.2.5), unlike W0 , . . . , Wn , basic functionals are independent of the dimension n of the space, and the symbol Vk for basic functional is compatible with the symbol Vn for n-dimensional volume, i.e., for n-dimensional Lebesgue measure: 7.2.5. THEOREM. Let P ∈ P n . If dim P ≤ k, then Vk (P) = λk (P). Proof. Let k ∈ {0, . . . , n − 1}. By Deﬁnition 7.2.4, if dim P < k, then Vk (P) = 0, because F k (P) = ∅; thus Vk (P) = λk (P). Hence, we may assume that dim P = k. Then P has a unique k-dimensional face: F k (P) = {P}, nor(P, P) is an (n − k − 1)-dimensional subsphere of S n−1 , and γ (P, P) = 1. Thus Vk (P) = λk (P), by Deﬁnition 7.2.4. 7.2.6. EXAMPLE. (a) If P ∈ P0n , i.e., dim P = n, then Vn−1 (P) equals onehalf of the (n − 1)-dimensional “surface area” of P: 1 Vn−1 (P) = λ (F). F∈F n−1 (P) n−1 2 This is a direct consequence of Deﬁnition 7.2.4 combined with Example 7.2.2 (a). (b) From 7.2.4 and 7.2.2 (c), it follows that for every P ∈ P n , V0 (P) = 1. 7.2.7. THEOREM. The functional Vn−1 : P n → R is increasing. Proof. Let P, Q ∈ P0n and P ⊂ Q. Then every facet F of P is the image under orthogonal projection of a subset X F of the boundary of Q; moreover, for different facets F1 , F2 the set X F1 ∩ X F2 has dimension less than n − 1. Since orthogonal projection does not increase λn−1 , by Example 7.2.6 (a) it easily follows that Vn−1 (P) ≤ Vn−1 (Q). We leave to the reader the proof for polytopes P, Q at least one of which has dimension less than n (Exercise 7.4).

80

7. Functionals on the Space Kn. The Steiner Theorem

Directly from the deﬁnitions of generalized ball and outer normal angle (1.1.5 and 7.2.1), we obtain the following theorem on the decomposition of α-hull of a polytope (Figure 7.3): 7.2.8. THEOREM. For every P ∈ P n and α > 0, (P)α = P ∪ (F ⊕ α · conv({0} ∪ nor(P, F))), F∈F

where all the summands of the union on the right-hand side of the equality have pairwise disjoint interiors, and the summands of the direct sum are contained in orthogonal afﬁne subspaces.

Figure 7.3.

We shall now prove the Steiner theorem for polytopes. It says that the volume of the α-hull of a polytope P is a polynomial in α whose coefﬁcients depend only on Vn (P), . . . , V0 (P). 7.2.9. THEOREM. For every P ∈ P n and α > 0, Vn ((P)α ) =

n

α n−k κn−k Vk (P).

k=0

Proof. By Theorem 7.2.8, Vn ((P)α ) = Vn (P) +

n−1 k=0

F∈F k (P)

λk (F) · λn−k (α · conv({0} ∪ nor(P, F))).

Since λn−k is homogeneous of degree n − k and λn−k (conv({0} ∪ nor(P, F))) σn−k−1 (nor(P, F)) = = γ (P, F), κn−k ωn−k from (7.8) and Deﬁnition 7.2.4 it follows that

(7.8)

7.2 Basic functionals. The Steiner theorem

Vn ((P)α ) = Vn (P) +

n−1

α n−k κn−k

k=0

=

n

F∈F k

81

λk (F)γ (P, F)

α n−k κn−k Vk (P).

k=0

We shall now give a partial proof of the following theorem, which makes it possible to extend the basic functionals V0 , . . . , Vn over Kn . 7.2.10. THEOREM. For every k ∈ {0, . . . , n}, the functional Vk : P n → R is homogeneous of degree k, invariant, and increasing. Proof. Homogeneity of degree k and invariance follow directly from Deﬁnition 7.2.4. We prove that Vk is increasing if n ≤ 3. Obviously, Vn is increasing. By Theorem 7.2.7 combined with Example 7.2.6 (b), Vn−1 and V0 are also increasing. Let us show that V1 is increasing: for every P, Q ∈ P n , P ⊂ Q ⇒ V1 (P) ≤ V1 (Q).

(7.9)

Indeed, for every α > 0, P ⊂ Q ⇒ (P)α ⊂ (Q)α ⇒ Vn ((P)α ) ≤ Vn ((Q)α ), and in view of the Steiner theorem for polytopes, 7.2.9, from the last inequality we infer that α n−1 κn−1 V1 (P) +

n

α n−k κn−k Vk (P)

k=2

≤ α n−1 κn−1 V1 (Q) +

n

α n−k κn−k Vk (Q),

k=2

that is, α n−1 κn−1 (V1 (Q) − V1 (P)) +

n

α n−k κn−k (Vk (Q) − Vk (P)) ≥ 0.

k=2

Dividing both sides of the inequality by α n−1 and passing to the limit for α → ∞, we obtain V1 (Q) − V1 (P) ≥ 0, which completes the proof of (7.9). If n ≤ 3, the proof is complete. If n > 3, it remains to show that V2 , . . . , Vn−2 are increasing. We omit this part of proof, because it exceeds the scope of this book (compare with [64], p. 211). The following corollary is a consequence of 7.2.10, 7.1.10, and 7.1.11 (i).

82

7. Functionals on the Space Kn. The Steiner Theorem

7.2.11. COROLLARY. For every k ∈ {0, . . . , n}, the functional Vk : P n → R can be extended over Kn according to Deﬁnition 7.1.9. The extended functional is also homogeneous of degree k and increasing. To simplify the notation, we shall use the same symbols V0 , . . . , Vn for extended basic functionals. We are now ready to prove the Steiner theorem for arbitrary compact convex sets, which is a generalization of 7.2.9. 7.2.12. THEOREM. For every A ∈ Kn and α > 0, Vn ((A)α ) =

n

α n−k κn−k Vk (A).

(7.10)

k=0

Proof. From the theorem on simultaneous approximation, 7.1.12, we deduce the existence of sequences (Pi )i∈N and (Q i )i∈N in P n such that Pi ⊂ A ⊂ Q i for every i and lim Vk (Pi ) = Vk (A) = lim Vk (Q i ) i

i

for k = 0, . . . , n.

For these sequences (Pi )α ⊂ (A)α ⊂ (Q i )α , whence Vn ((Pi )α ) ≤ Vn ((A)α ) ≤ Vn ((Q i )α ). Applying the Steiner theorem for polytopes, 7.2.9, and passing to the limit for i → ∞, we obtain the formula (7.10). 7.2.13. EXAMPLE. Let us calculate Vk (B n ) for k = 1, . . . , n. For any α > 0, n n n−k n n Vn ((B )α ) = (1 + α) κn = α κn . k k=0 By Theorem 7.2.12, comparing the coefﬁcients of α n−k , we obtain n κn n Vk (B ) = . k κn−k Hence by (7.7), Wn−k (B n ) = κn .

7.3 Consequences of the Steiner theorem In view of the Steiner theorem, the n-dimensional volume Vn ((A)α ) of the α-hull of a compact convex subset A of Rn is a polynomial in α whose coefﬁcients are (up to some constant factors) intrinsic volumes of A. Using this theorem we shall prove its analogue for the intrinsic volume Vk ((A)α ) of the α-hull of A, where k ∈ {0, . . . , n − 1}.

7.3 Consequences of the Steiner theorem

83

7.3.1. THEOREM. For every A ∈ Kn , α > 0, and k ∈ {0, . . . , n}, Vk ((A)α ) =

1 κn−k

k n − i k−i α κn−i Vi (A). k −i i=0

(7.11)

Proof. Let us ﬁx an α > 0. For every β > 0, (A)α+β = ((A)α )β ; hence, applying the Steiner theorem 7.2.12 twice, we obtain for every β > 0 the following equality: n

(α + β)n−i κn−i Vi (A) =

i=0

n

(β n−k κn−k Vk ((A)α ).

(7.12)

k=0

Denote the left-hand side of (7.12) by L β . Then n n−i n−i j n−(i+ j) Lβ = α β κn−i Vi (A). j i=0 j=0 For every k ∈ {0, . . . , n}, the coefﬁcients of β n−k on both sides of (7.12) must be equal. In L β , the (n − k)th power of β corresponds to j = k − i; thus k n − i k−i α κn−i Vi (A) = κn−k Vk ((A)α ). k −i i=0 To obtain (7.11), it sufﬁces to divide both sides of the last equality by κn−k .

We shall now make use of the Steiner theorem to examine properties of the basic functionals. For this purpose, we consider the family of functionals α : Kn → R for α > 0: (7.13) α (A) := Vn ((A)α ). Let us ﬁrst prove the following. 7.3.2. THEOREM. (i) For every α > 0, the functional α is an invariant valuation; (ii) For every A0 ∈ Kn and every α0 > 0, the family (α )α∈(0,α0 ] is equicontinuous in A0 . Proof. (i): Let α > 0. Invariance of α (with respect to the isometries) is evident. Let A1 , A2 , A1 ∪ A2 ∈ Kn . Let us note that (A1 ∪ A2 )α = (A1 )α ∪ (A2 )α

(7.14)

(A1 ∩ A2 )α = (A1 )α ∩ (A2 )α .

(7.15)

and

7. Functionals on the Space Kn. The Steiner Theorem

84

Indeed, (A1 ∪ A2 )α =

x∈A1 ∪A2

x +α B n =

x +α B n ∪

x∈A1

x +α B n = (A1 )α ∪(A2 )α .

x∈A2

Moreover, (7.14) holds for arbitrary nonempty closed subsets A1 , A2 of Rn . In (7.15), the inclusion ⊂ is evident (for arbitrary nonempty sets), and the inclusion ⊃ follows directly from the condition (x, A1 ∩ A2 ) ≤ max (x, Ai ), i=1,2

which is true for arbitrary A1 , A2 ∈ Kn with A1 ∪ A2 ∈ Kn (Exercise 2.3). Since Vn is a valuation, from (7.13)–(7.15) it follows that α (A1 ∪ A2 ) + α (A1 ∩ A2 ) = α (A1 ) + α (A2 ); that is, α is a valuation. (ii): Let A0 ∈ Kn and α0 > 0. We have to prove that for every ε > 0 there is a δ > 0 such that ∀α ∈ (0, α0 ] ∀A ∈ Kn H (A, A0 ) < δ ⇒ |α (A) − α (A0 )| < ε.

(7.16)

Let ε > 0. Without loss of generality we may assume that ε < 1. Let us ﬁrst note that if H (A, A0 ) < ε, then there exists a ball B such that (A)α0 ∪ (A0 )α0 ⊂ B. Indeed, A ⊂ (A0 )ε , whence (A)α0 ⊂ (A0 )α0 +ε ⊂ B for some ball B. Now let n−1 ε β := κn−k Vk (B), δ := . β k=0

(7.17)

(7.18)

Then β > 1, because from 7.2.6 (b) it follows that V0 (B) = 1. Hence δ < 1. Let 0 < α ≤ α0 and H (A, A0 ) < δ. Then A0 ⊂ (A)δ and A ⊂ (A0 )δ and thus (A0 )α ⊂ ((A)α )δ and (A)α ⊂ ((A0 )α )δ . By the Steiner Theorem 7.2.12, from (7.19) it follows that α (A0 ) ≤ α (A) +

n−1

δ n−k κn−k Vk ((A)α )

k=0

and α (A) ≤ α (A0 ) +

n−1 k=0

δ n−k κn−k Vk ((A0 )α ).

(7.19)

7.3 Consequences of the Steiner theorem

85

Applying now (7.17) and (7.18), we obtain α (A0 ) < α (A) + δβ = α (A) + ε, and analogously, α (A) < α (A0 ) + ε. Hence |α (A0 ) − α (A)| < ε.

7.3.3. THEOREM. The basic functionals are invariant and continuous valuations. Proof. By 7.3.2 (i), for every α > 0, the function α deﬁned by (7.13) is a valuation. Thus, by the Steiner theorem, for arbitrary A1 , A2 ∈ Kn with A1 ∪ A2 ∈ Kn , n

α n−k κn−k (Vk (A1 ∪ A2 ) + Vk (A1 ∩ A2 )) =

k=0

n

α n−k κn−k (Vk (A1 ) + Vk (A2 )).

k=0

Comparing the coefﬁcients of the corresponding powers of α, we obtain Vk (A1 ∪ A2 ) + Vk (A1 ∩ A2 ) = Vk (A1 ) + Vk (A2 ) for k = 0, . . . , n. Thus V0 , . . . , Vn are valuations. By 7.3.2 (i), the function α is invariant: A ≡ B ⇒ α (A) = α (B). Hence, applying again the Steiner theorem and comparing the coefﬁcients of the corresponding powers of α, we obtain invariance of the basic functionals. It remains to prove continuity. Let lim H Ai = A0 and α0 > 0. From 7.3.2 (ii) it follows that ∀α ∈ (0; α0 ] lim α (Ai ) = α (A0 ).

(7.20)

i

Let us take a sequence of distinct positive numbers α1 , . . . , αn+1 Let M(α1 , . . . , αn+1 ) be the Vandermonde matrix; ⎛ 1 α1 (α1 )2 · · · (α1 )n ⎜1 α2 (α2 )2 · · · (α2 )n ⎜ M(α1 , . . . , αn+1 ) := ⎜ . . .. .. .. ⎝ .. .. . . .

∈ (0; α0 ]. ⎞ ⎟ ⎟ ⎟ ⎠

1 αn+1 (αn+1 )2 · · · (αn+1 )n

and let f : Rn+1 → Rn+1 be the linear transformation with the matrix M(α1 , . . . , αn+1 ) (in the canonical basis).

86

7. Functionals on the Space Kn. The Steiner Theorem

By the Steiner theorem, for i ∈ N ∪ {0}, j = 1, . . . , n + 1, α j (Ai ) =

n

(α j )n−k κn−k Vk (Ai ).

k=0

Therefore, for a ﬁxed i, f (κ0 Vn (Ai ), . . . , κn V0 (Ai )) = (α1 (Ai ), . . . , αn+1 (Ai )). Since M(α1 , . . . , αn+1 ) is nonsingular ([50]), it follows that f −1 (α1 (Ai ), . . . , αn+1 (Ai )) = (κ0 Vn (Ai ), . . . , κn V0 (Ai )). Hence from (7.20) it follows that lim Vk (Ai ) = Vk (A0 ) i

for k = 0, . . . , n. This proves the continuity of the basic functionals.

We are now ready to prove the following. 7.3.4. PROPOSITION. σ (Sn−1 ) = n · κn . Proof. In view of 6.3.2 we can approximate the ball B n by a sequence of convex polytopes (Pk )k∈N contained in B n . Of course, we may assume that the vertices of each Pk belong to S n−1 . Let Tk be a triangulation of bdPk . Then Pk = conv({0} ∪ ), ∈Tk

whence by continuity of Vn (see 7.3.3), κn = Vn (B n ) = lim Vn (Pk ), k

where Vn (Pk ) =

1 ∈Tk n (Vn−1 () · (0, )).

Thus κn = n1 σ (S n−1 ).

Let us observe that Proposition 7.3.4 can also be derived from the following well-known theorem on change of variables (compare with [66], formula (5.2.3) in Theorem 5.2.2). 7.3.5. THEOREM. If f is a nonnegative measurable function on Rn , then ∞ f (x)dλn (x) = f (tu)t n−1 dtdσ (u). Rn

S n−1

0

Let ρ A : S n−1 → R+ be the radial function of a convex body A in Rn with 0 ∈ A: ρ A (u) := sup{α ≥ 0 | αu ∈ A}. (7.21) Setting f = 1 A , the characteristic function of A, as a direct consequence of Theorem 7.3.5 we obtain

7.3 Consequences of the Steiner theorem

87

7.3.6. COROLLARY. For every A ∈ K0n , 1 Vn (A) = ρ A (u)n dσ (u). n S n−1 Of course, Prop. 7.3.4 is a particular case of Corollary 7.3.6, for A = B n . At the end of this section, we mention the traditional notation for n = 3 (compare [29]). Let 4 V := V3 , F := 2V2 , M := π V1 , C := π ; 3 here V is the volume, F is the surface area (double for sets with empty interior), M is called the mean curvature, and C the integral curvature. Let us notice that (in accordance with 7.2.6) V = κ0 V3 , F = κ1 V2 , M = κ2 V1 , C = κ3 V0 . Hence the Steiner formula for n = 3 can be rewritten as follows: V ((A)α ) = V (A) + α F(A) + α 2 M(A) + α 3 C(A).

(7.22)

8 The Hadwiger Theorems

8.1 The ﬁrst Hadwiger theorem From 7.1.7 and 7.3.3 it follows that every linear combination of basic functionals is a continuous invariant valuation. We shall now deal with the ﬁrst Hadwiger theorem, which says that the family of continuous, invariant valuations coincides with the set of such linear combinations (Theorem 8.1.5). Until 1995, the Hadwiger’s original proof was the only one to be found in the literature. For the case n = 3, see [29]. The proof for arbitrary n was presented in [30]. It is very complicated and difﬁcult to follow. In 1995, Daniel Klain gave a new proof of the Hadwiger theorem ([38]); however his methods go beyond the scope of this book. In this situation, we present the proof of Theorem 8.1.5 only for n = 3; the proof for n = 2 is left to the reader (Exercise 8.1). It is worthwhile to note that to generalize the proof for n = 3 to higher dimensions, it seems natural to generalize Lemmas 8.1.3 and 8.1.4. However, Theorem 6.4.5 on cylindric polytopes in R3 , involved in the proof of Lemma 8.1.3 fails for n > 3 (compare with remarks following 6.4.2, and Exercise 6.13): some cylindric polytopes in Rn for n > 3 are not equivalent by dissection to an n-dimensional cube. Lemmas 8.1.1–8.1.3 concern the so-called simple valuations. A valuation : P n → R is simple (compare [38]) if for every P ∈ P n , dim P < n ⇒ (P) = 0.

90

8. The Hadwiger Theorems

8.1.1. LEMMA. Let : P n →R be a simple valuation. Then for every m Pi and pairwise disjoint interiors, P1 , . . . , Pm ∈ P n with convex union i=1

m

Pi

m

=

i=1

(Pi ).

(8.1)

i=1

Proof. Induction on m. For m = 1, equality (8.1) is an identity; for m = 2 it follows directly from the assumption. Let m ≥ 3 and let the equality hold for m − 1. Obviously, we may assume that intPi = ∅ for i = 1, . . . , m, because vanishes for polytopes with empty interiors. m Pi . Since P is convex, there exist i 1 , i 2 such that dim(Pi1 ∩ Let P := i=1 Pi2 ) = n − 1 (Exercise 6.10). Let H = aff(Pi1 ∩ Pi2 ). Then the hyperplane H dissects P into two convex polytopes P , P : P = P ∪ P , P ∩ P = P ∩ H. Assume that Pi1 ⊂ P , Pi2 ⊂ P , and let Pi = Pi ∩ P , Pi = Pi ∩ P . Evidently, Pi1 and Pi2 are not dissected by H , whence Pi1 = Pi1 , Pi2 = Pi2 ; thus

P = Pi1 ∪

i =i 1 ,i 2

Pi ,

P = Pi2 ∪

i =i 1 ,i 2

Pi .

By the inductive assumption (since decompositions of P and P have fewer than m elements), (P ) = (Pi1 ) + (Pi ), (P ) = (Pi2 ) + (Pi ); i =i 1 ,i 2

i =i 1 ,i 2

hence (P) = (P ) + (P ) = (Pi1 ) + (Pi2 ) +

i =i 1 ,i 2

((Pi ) + (Pi )).

(8.2)

But Pi and Pi have disjoint interiors and convex union Pi , whence (Pi ) + (Pi ) = (Pi ) for i = i 1 , i 2 . This condition combined with (8.2) implies (8.1).

8.1 The ﬁrst Hadwiger theorem

91

The following lemma follows directly from 8.1.1. 8.1.2. LEMMA. Let : P n → R be an invariant simple valuation. Then, for arbitrary P ∈ P n and Q 1 , . . . , Q m ∈ P n with pairwise disjoint interiors, P∼

m

Q j ⇒ (P) =

j=1

m

(Q j ).

j=1

8.1.3. LEMMA. Let n ≤ 3 and let : P n → R be an invariant and continuous simple valuation satisfying the condition (i) (∃α > 0 P ∼ α I n ) ⇒ (P) = 0. Then is Minkowski linear. Proof. We have to prove that for every m ∈ N, (

m i=1

ti Pi ) =

m

ti (Pi ).

i=1

Since, by 6.2.6 and 6.2.7, a linear combination of convex polytopes is a convex polytope, induction with respect to m is trivial. Thus it sufﬁces to prove this condition for m = 2. Since is a simple valuation, we may assume that dim Pi = n for i = 1 or i = 2; since is invariant, we may assume that 0 ∈ intP2 . In view of Theorem 6.4.4 on decomposition, P1 + P2 = P1 ∪ Qk ∪ X j, k

j

where k Q k ∼Tr P2 , each of X j is a cylindric polytope, and all the summands on the right-hand side of the equality have pairwise disjoint interiors. Since is a simple valuation, by Lemma 8.1.1 it follows that (P1 + P2 ) = (P1 ) + (Q k ) + (X j ). (8.3) k

j

In turn, since n ≤ 3, in view of 6.4.5 each X j is equivalent by dissection to a cube, whence, by (i), ∀ j (X j ) = 0. Hence by (8.3) combined with Lemma 8.1.2, (P1 + P2 ) = (P1 ) + (P2 );

(8.4)

thus is additive. It remains to prove that is homogeneous: for every P ∈ P n and t ∈ R, (t P) = t(P). (8.5) For a ﬁxed P, let us deﬁne f : R → R by the formula

92

8. The Hadwiger Theorems

f (t) := (t P). Since multiplication by a scalar, t → t P, is continuous (Exercise 2.2) and, by the assumption, is continuous, it follows that f is continuous. In view of (8.4), the function f is additive: f (t1 +t2 ) = ((t1 +t2 )P) = (t1 P +t2 P) = (t1 P)+(t2 P) = f (t1 )+ f (t2 ). As is well known (compare [58] Theorem 10, p. 123), any continuous and additive real function is linear, and thus homogeneous; hence f is homogeneous. Therefore, f (t) = t f (1); that is, (8.5) is satisﬁed. 8.1.4. LEMMA. If : K3 → R is an invariant and continuous simple valuation satisfying condition (i) of Lemma 8.1.3 for every P ∈ P 3 , then = 0. Proof. If P has empty interior, then (P) = 0. Let P ∈ P03 . Since is invariant, we may assume that 0 ∈ intP. By the second rounding theorem, 5.3.2, there exists a sequence (Ai )i∈N in T(P) that is ¯ Hausdorff convergent to a ball B0 with diameter b(P) and center 0. Let i ∈ N. 1 m g (P) for some g , . . . , g ∈ O(n), by Lemma 8.1.3 for Since Ai = m j m 1 j=1 n = 3 it follows that m 1 (Ai ) = (g j (P)), m j=1 where (g j (P)) = (P) because is invariant. Hence (Ai ) = (P) for every i ∈ N, and thus by continuity, (P) = (B0 ).

(8.6)

Condition (8.6) holds for every P ∈ P03 ; thus in particular, it holds for P = I 3 . Therefore, by (8.6), it follows that (P) = 0 for every polytope P. Since is continuous, to complete the proof, it now remains to apply the approximation theorem 6.3.2. 8.1.5. THEOREM. For every : Kn → R the following conditions are equivalent: (i) is an invariant and continuous valuation; (ii) there exist α0 , . . . , αn ∈ R such that =

n

αi Vi .

(8.7)

i=0

Proof. Implication (ii) ⇒ (i) follows from 7.1.7 combined with 7.3.3. (i) ⇒ (ii) for n = 3: Let be an invariant, continuous valuation.

We are looking for α0 , . . . , α3 such that the values of the functionals and 3n=0 αi Vi

8.1 The ﬁrst Hadwiger theorem

93

are the same for every P ∈ P 3 , and so in particular, for P = {0}, I, I 2 , I 3 , where I k is a k-dimensional cube in R3 , with 0 being a vertex (I 0 := {0}). Deﬁne α0 , . . . , α3 : α0 := ({0}), α1 := ( − α0 V0 )(I ), α2 := ( − α0 V0 − α1 V1 )(I 2 ), α3 := ( − α0 V0 − α1 V1 − α2 V2 )(I 3 ). Let : K3 → R be deﬁned by the formula := −

3

αi Vi .

i=0

It sufﬁces to prove that = 0. To this end, let 1 := − α0 V0 , 2 = 1 − α1 V1 , 3 := 2 − α2 V2 ; notice that = 3 − α3 V3 .

(8.8)

Evidently, 1 , 2 , 3 , are invariant continuous valuations, (compare with 7.1.7) and αi = i (I i ) for i = 1, 2, 3. Since is invariant and V0 = 1 by 7.2.6 (b), it follows that 1 ({a}) = 0 for every a ∈ R3 . Evidently, 2 (I ) = 0; we shall show that moreover, 2 (P) = 0 for every segment P.

(8.9)

Since 2 is invariant, it sufﬁces to prove (8.9) for P = t · I , where t is an arbitrary positive number. Let us note that the function f : R → R deﬁned by the formula f (t) := 2 (t · I ) is continuous, because 2 is continuous, and is additive in view of Lemma 8.1.2 for n = 1; thus f is homogeneous. Hence 2 (P) = t2 (I ) = 0, which proves (8.9). Now let P be a convex polygon in R3 ; then 2 (P) depends only on the area, V2 (P). Indeed, by Theorem 6.4.2 and Lemma 8.1.2 for n = 2, V2 (P ) = V2 (P) ⇒ P ∼ P ⇒ 2 (P) = 2 (P ). Obviously, 3 (I 2 ) = 0, whence 3 (P) = 0 for every convex polygon P with area equal to 1. Let us show that moreover,

94

8. The Hadwiger Theorems

3 (P) = 0 for every convex polygon P.

(8.10)

We deﬁne g : R → R by the formula √ g(t) := 3 ( t · I 2 ). Since 3 (P) depends only on the area of a polygon P, it follows that g(t) = 3 (P) for arbitrary polygon P with area V2 (P) = t. By Lemma 8.1.2 for n = 2, the function g is additive. Indeed, any square P with area t1 + t2 can be dissected into rectangles P1 , P2 with areas t1 , t2 , respectively, and any rectangle is equivalent by dissection to a suitable square. Moreover, this function is continuous, whence it is homogeneous ([58], p. 123). Thus 3 (P) = t · g(1) = t3 (I 2 ) = 0, which proves (8.10). From (8.8) and (8.10) it follows that vanishes for polygons, whence it is a simple valuation. Let us show that it satisﬁes condition (i) of Lemma 8.1.3. Of course, (I 3 ) = 0, and by the invariance of its value for any cube P depends only on V3 (P). Let 1 h(t) := (t 3 I 3 ). The function h is additive, because for t = t1 + t2 a cube P with volume t can be dissected into rectangular parallelepipeds P1 , P2 , with volumes t1 , t2 . These parallelepipeds are equivalent by dissection to suitable cubes. Since h is also continuous, it follows that h is homogeneous. Hence for every polytope P equivalent by dissection to a cube, with volume equal to t, (P) = h(t) = th(1) = t(I 3 ) = 0. Thus = 0 (by Lemma 8.1.4), which completes the proof.

In view of the Hadwiger Theorem 8.1.5, the linear space of invariant and continuous valuations on Kn is generated by the set {V0 , . . . , Vn } of the intrinsic volumes. Moreover, this set is a basis: 8.1.6. THEOREM. The sequence (V0 , . . . , Vn ) is a basis of the linear space of invariant continuous valuations on Kn . Proof. In view of 8.1.5, it sufﬁces to prove that the system of basic functionals is linearly independent: n

ti Vi = 0 ⇒ ti = 0 for i = 0, . . . , n.

(8.11)

i=0

Let us consider a sequence (A0 , . . . , An ) in Kn , with dim Ak = k; for instance, let Ak := (a0 , . . . , ak ) for k = 0, . . . , n. By the predecessor of the implication (8.11), for every k ∈ {0, . . . , n}

8.2 The second Hadwiger theorem n

95

ti Vi (Ak ) = 0.

i=0

If k = 0, then Vi (Ak ) = 0 for i ≥ 1; since V0 = 1 (Example 2.6 (b)), it follows that t0 = 0. Assume that k > 0 and ti = 0 for i ≤ k − 1. Then by the predecessor of (8.11), n

ti Vi (Ak ) = 0.

i=k

But in view of 7.2.5, Vk (Ak ) = 0 and Vi (Ak ) = 0 for every i ∈ {k + 1, . . . , n}, whence tk = 0. This proves (8.11).

8.2 The second Hadwiger theorem Since Vk ≥ 0 for k = 0, . . . , n, from 7.1.7 combined with 7.2.10 it follows that every linear combination of the basic functionals with nonnegative coefﬁcients is an invariant increasing valuation. The second Hadwiger theorem, 8.2.2, states that moreover, the class of invariant increasing valuations with nonnegative values coincides with the set of such linear combinations. A proof of this theorem for n = 3 can be obtained by a suitable modiﬁcation of the proof of 8.1.5 presented above (compare with Exercise 8.2). However, there is another possibility: this theorem can be derived from 8.1.5 and the following McMullen result (see Theorem 11.5 in [45]), which we cite without proof (compare with Exercise 8.3): 8.2.1. THEOREM. (P. McMullen) If a valuation : Kn → R is increasing and invariant with respect to the translations, then is continuous. 8.2.2. THEOREM. For every functional : Kn → R+ the following conditions are equivalent: (i) is an invariant and increasing valuation; (ii) there exist α0 , . . . , αn ∈ R+ such that =

n i=0

αi Vi .

9 Applications of the Hadwiger Theorems

9.1 Mean width and mean curvature At the end of Chapter 7 we introduced the notion of mean curvature M for convex bodies in R3 . Generally, for Kn , mean curvature is the functional M deﬁned by the formula1 2π Vn−2 . M := (9.1) n−1 The Hadwiger Theorem 8.1.5 yields a relationship between the mean width and the functional V1 (Theorem 9.1.1), and thus for n = 3, between the mean width and mean curvature (Corollary 9.1.2). 9.1.1. THEOREM. For every A ∈ Kn , 2κn−1 ¯ b(A) = V1 (A). nκn Proof. By 7.1.6, the mean width b¯ is an invariant and continuous valuation. Hence in view of Theorem 8.1.5, there exist α0 , . . . , αn ∈ R such that b¯ =

n

αi Vi .

i=0

Obviously, for the ball B := r · B n with arbitrary radius r , 1 This functional is also called the integral of mean curvature (compare with [64], p. 210).

(9.2)

98

9. Applications of the Hadwiger Theorems

¯ b(B) = 2r ; since by 7.2.10, the functional Vi is homogeneous of degree i, from (9.2) it follows that n ∀r > 0 2r = αi Vi (B n ) · r i . i=0

Thus αi = 0 for i = 1 and α1 V1 (B n ) = 2. Let us substitute α0 , . . . , αn in (9.2); we obtain the formula 2 b¯ = V1 , V1 (B n ) which, combined with 7.2.13, implies the required relationship between b¯ and V1 . Directly from 9.1.1 and (9.1) we deduce 9.1.2. COROLLARY. For every A ∈ K3 , 1 ¯ b(A) = M(A). 2π The experienced reader will certainly appreciate Theorem 9.1.1 and Corollary 9.1.2. While it usually is difﬁcult to calculate the mean width of a convex set A directly from Deﬁnition 2.2.8 (even if this set is simple and regular), it is much easier to ﬁnd V1 (A) (i.e., if n = 3, the mean curvature of A) and apply Theorem 9.1.1 or Corollary 9.1.2.

9.2 The Crofton formulae The Crofton formulae (Theorem 9.2.6) are one of the most important and most interesting applications of the Hadwiger theorems. These integral formulae express, for k ∈ {1, . . . , n}, the value of Vk at any A ∈ Kn by means of the values of Vk−1 at sections of A by afﬁne subspaces of some dimension i ≤ k − 1. Let us observe that our consideration may be restricted to i = n − 1, that is, to sections by hyperplanes. Indeed, every section of A by a subspace of dimension i for some i ≥ 1 can be obtained as the result of n − i operations of cutting a compact convex set contained in a subspace of dimension j by a subspace of dimension j − 1, for j = n, . . . , i + 1. Hence, let us consider the family E n of hyperplanes in Rn . In Section 2.5 we used the parametric representation φ : S n−1 × R+ → E n (see (2.7)) to deﬁne the limit in E n (Deﬁnition 2.5.2). Thus we introduced a topology in E n . A measure µ on the family B(E n ) of Borel sets in E n is deﬁned as follows. 9.2.1. DEFINITION. Let µ0 : B(S n−1 × R) → R be the product measure of σ and λ1 . The function µ : B(E n ) → R is deﬁned by the formula

9.2 The Crofton formulae

99

∀X ∈ B(E n ) µ(X ) := µ0 (φ −1 (X )). 9.2.2. PROPOSITION. The function µ is a measure. (Compare with Exercise 9.2.) 9.2.3. EXAMPLES. (a) Fix a v0 ∈ S 2 and α > 0. Let X := {E ∈ E 3 | dist(0, E) ∈ [0, α], v0 ⊥ E}. The set X is closed in E 3 , so it is a Borel set; since φ −1 (X ) = {v0 } × [0, α] ∪ {(−v0 , 0)}, it follows that µ(X ) = µ0 (φ −1 (X )) = 0 · α = 0. (b) Let A ∈ K0n , 0 ∈ A and let X = {H (A, u) | u ∈ S n−1 }. Then by 3.4.5, φ −1 (X ) = {(u, h A (u)) | u ∈ S n−1 } = graphh A ; thus φ −1 (X ) is measurable as the graph of the continuous function h A |S n−1 , and by Fubini’s theorem, µ(X ) = µ0 (φ −1 (X )) = 0. For any : Kn → R and A ∈ Kn we deﬁne the function A : E n → R by the formula (A ∩ E) if A ∩ E = ∅ A (E) := (9.3) 0 if A ∩ E = ∅. The following statement is a direct consequence of 2.5.6 combined with 9.2.3 (b). 9.2.4. PROPOSITION. If a functional : Kn → R is continuous, then for every A ∈ Kn the function A is continuous µ-almost everywhere, and thus µintegrable. To every continuous functional : Kn → R we now assign the functional ˆ : Kn → R deﬁned by ˆ (A) := A (E)dµ(E). (9.4) En

ˆ preserves the properties considered in Let us show that the assignment → the previous chapter. For every A ∈ Kn , let E A := {E ∈ E n | A ∩ E = ∅}. 9.2.5. THEOREM. For every continuous : Kn → R, ˆ is continuous; (i) ˆ is a valuation; (ii) if is a valuation, then also ˆ (iii) if is invariant, then also is invariant.

(9.5)

100

9. Applications of the Hadwiger Theorems

Proof. By (9.3)–(9.5), for every A ∈ Kn , ˆ (A) = (A ∩ E)dµ(E). EA

Condition (i) follows from Theorem 2.5.7 combined with Example 9.2.3 (b), since a hyperplane disjoint from the interior of A is either disjoint from A or is a support hyperplane of A. Condition (ii) follows from the additivity of integral and deﬁnition of valuation. (iii): Assume to be invariant. Let f : Rn → Rn be an isometry. Then ˆ f (A)) = ( f (A) ∩ E ) dµ(E ) = ( f (A) ∩ f (E)) dµ(E) ( E f (A)

=

EA

f −1 (E f (A) )

( f (A ∩ E)) dµ(E) =

EA

ˆ (A ∩ E) dµ(E) = (A).

9.2.6. THEOREM. Let n ≥ 2. For k ∈ {1, . . . , n}, let βn,k := Then for every A ∈ Kn ,

2 κk−1 . · k κk κn−1

Vk (A) = βn,k

EA

Vk−1 (A ∩ E)dµ(E).

Proof. Let := Vk−1 . Then by (9.3)–(9.5), for every A, ˆ (A) = Vk−1 (A ∩ E)dµ(E). EA

(9.6)

(9.7)

Since, by Theorem 7.3.3, the functional Vk−1 is a continuous, invariant valuation, ˆ By the Hadwiger Theorem 8.1.5, there exist in view of Theorem 9.2.5, so is . α0 , . . . , αn such that n ˆ = αi Vi . (9.8) i=0

ˆ B n ) for arbitrary positive r . To ﬁnd α0 , . . . , αn , let us calculate (r Every hyperplane E is of the form φ(v, t), where v ⊥ E and t = dist(0, E) (compare (2.7) and 2.5.1); thus by deﬁnition of the measure µ (Deﬁnition 9.2.1) combined with (9.7), r ˆ Bn ) = (r Vk−1 (r B n ∩ φ(v, t))dtdσ (v). S n−1

0

Since for t ∈ (0;√ r ) the intersection of r B n and φ(v, t) is an (n − 1)-dimensional ball with radius r 2 − t 2 , and Vk−1 is homogeneous of degree k − 1 (Theorem 7.2.11), it follows that

9.2 The Crofton formulae

ˆ B n ) = σ (S n−1 )Vk−1 (B n−1 ) (r

r

101

( r 2 − t 2 )k−1 dt.

0

By substitution t = r sin s for s ∈ [0, π2 ], we obtain π 2

ˆ B n ) = σ (S n−1 )Vk−1 (B n−1 ) (r

cos s ds r k . k

0

But, by Fubini’s theorem,

1

κk = Vk (B ) = 2 k

(

1 − t 2 )k−1 Vk−1 (B k−1 )dt

= 2κk−1

0

(Figure 9.1), whence

π 2

cosk s ds

0

π 2

cosk s ds =

0

κk . 2κk−1

Figure 9.1.

Thus

ˆ B n ) = σ (S n−1 )Vk−1 (B n−1 ) (r

κk · rk. 2κk−1

By this formula together with (9.8), comparing the coefﬁcients of r i , we obtain αi = 0 for i = k and αk = σ (S n−1 )

Vk−1 (B n−1 ) κk . Vk (B n ) 2κk−1

(9.9)

Since by 7.3.4, σ (S n−1 ) = nκn , from 7.2.13 combined with (9.9) it follows that αk =

kκn−1 κk . 2κk−1

It now remains to insert α0 , . . . , αn into (9.8).

102

9. Applications of the Hadwiger Theorems

Using the traditional notation for n = 3 (see formulae (7.22) at the end of Chapter 7), we can rewrite the Crofton formulae for this particular case as follows. 9.2.7. THEOREM. Let A ∈ K3 . Then 1 V (A) = F(A ∩ E)dµ(E), 4π E A 4 F(A) = 3 M(A ∩ E)dµ(E), π EA 1 M(A) = C(A ∩ E)dµ(E) 4π E A

(Exercise 9.4).

From the Crofton theorem we derive the following simple relationship between the measure of E A and the mean width of A: 9.2.8. COROLLARY. For every A ∈ Kn , µ(E A ) = κn−1 V1 (A) =

1 ¯ nκn · b(A). 2

Proof. The ﬁrst equality follows from (9.6) for k = 1; applying Theorem 9.1.1 we obtain the second one.

9.3 The Cauchy formulae The so-called Cauchy formulae (Theorem 9.3.2) are another important application of the Hadwiger theorems. These integral formulae express, for k ∈ {1, . . . , n}, the value of Vk at A ∈ Kn by means of the values of Vk for projections of A on linear subspaces of different dimensions. As with the Crofton formulae, we can restrict our considerations to linear subspaces of dimension n − 1. n Let G n := Gn−1 be the Grassmannian, that is, the family of linear hyperplanes n in R . Evidently, G n is a subset of E n :

G n = {E ∈ E n | ∃v ∈ S n−1 E = φ(v, 0)}; hence in G n we have the subspace topology induced by the limit (Deﬁnition 2.5.2). We deﬁne the measure ν : B(G n ) →R: for every X ∈ B(G n ), ν(X ) := σ {v ∈ S n−1 | φ(v, 0) ∈ X }.

(9.10)

˜ : Kn → R deﬁned by Now, with any functional : Kn →R we associate the formula

9.3 The Cauchy formulae

˜ (A) :=

103

Gn

(π E (A))dν(E).

(9.11)

9.3.1. THEOREM. For every : Kn →R, ˜ is also continuous; (i) if is continuous, then ˜ is also a valuation; (ii) if is a valuation, then ˜ (iii) if is invariant, then is also invariant. Proof. Since the orthogonal projection π E on a linear subspace E is a weak contraction, it follows that for every A ∈ Kn and α > 0, π E ((A)α ) ⊂ (π E (A))α . Thus in view of (1.4), A = lim Ak ⇒ π E (A) = lim π E (Ak ), H

H

and thus condition (i) holds. By additivity of the integral, to verify (ii) it sufﬁces to prove that for every A1 , A2 ∈ Kn with convex union A1 ∪ A2 , π E (A1 ∪ A2 ) = π E (A1 ) ∪ π E (A2 )

(9.12)

π E (A1 ∩ A2 ) = π E (A1 ) ∩ π E (A2 ).

(9.13)

and Condition (9.12) holds for arbitrary A1 , A2 (this is a particular case of the formula for the image of the union under any function). Similarly, the inclusion ⊂ in (9.13) is satisﬁed by arbitrary A1 , A2 . For the inclusion ⊃, the convexity of the union is essential (Exercise 2.11). ⊃: Obviously, we may assume that A1 ∩ A2 = Ai for i = 1, 2.

(9.14)

Let y ∈ π E (A1 ) ∩ π E (A2 ). Then there exist x1 , x2 such that π E (x1 ) = y = π E (x2 ). If x1 = x2 , then y ∈ π E (A1 ∩ A2 ). If x1 = x2 , then (x1 , x2 ) ⊂ π E−1 (y) ∩ (A1 ∪ A2 ), because A1 ∪ A2 and π E−1 (y) are convex (the second set is a line). By (9.14), the set A1 ∩ A2 disconnects A1 ∪ A2 ; hence (x1 , x2 ) ∩ (A1 ∩ A2 ) = ∅. Let x ∈ (x1 , x2 ) ∩ (A1 ∩ A2 ); then x ∈ π E−1 (y), whence π E (x) = y ∈ π E (A1 ∩ A2 ). This completes the proof of (9.13). Condition (iii) is obvious.

104

9. Applications of the Hadwiger Theorems

9.3.2. THEOREM. Let n ≥ 2. For k = 0, . . . , n − 1, let γn,k :=

κn−k−1 1 · . n − k κn−1 κn−k

Then for every A ∈ Kn , Vk (A) = γn,k

Gn

Vk (π E (A))dν(E).

Proof. Let := Vk . Then according to (9.11), ˜ (A) = Vk (π E (A))dν(E). Gn

(9.15)

˜ is a continuous invariant valuaBy 7.3.3 combined with 9.3.1, the functional tion, whence in view of the Hadwiger Theorem 8.1.5, there exist α0 , . . . , αn such that n ˜ = αi Vi . (9.16) i=0

˜ B n ) for arbitrary positive r . We shall determine α0 , . . . , αn , calculating (r By (9.15) and (9.10), n ˜ (r B ) = Vk (πφ(v,0) (r B n ))dσ (v). S n−1

Since the orthogonal projection of r B n on any hyperplane is a ball of dimension n − 1 with the same radius, by the homogeneity of degree k of Vk , from 7.2.13 combined with 7.3.4 it follows that n − 1 κn−1 k ˜ B n ) = nκn (r r . k κn−k−1 On the other hand, by (9.16) and 7.2.13, ˜ Bn ) = (r

n κn i αi r . i κn−i i=0

n

Comparing coefﬁcients of corresponding powers of the variable r , we obtain ⎧ ⎨0 if i = k, κn−1 κn−k αi = if i = k. ⎩(n − k) κn−k−1 To complete the proof, it now remains to insert α0 , . . . , αn into (9.16).

9.3 The Cauchy formulae

105

In particular, for n = 3, we obtain the classical Cauchy formulae. They can be written in terms of the functionals M and F for K3 deﬁned by (7.22) and the functionals f and l for K2 , where f (A) is the area of A and l(A) is the perimeter of A (compare with Exercise 9.5). 9.3.3. THEOREM. Let A ∈ K3 . Then 1 M(A) = l(π E (A))dν(E), 2π G 3 1 F(A) = f (π E (A))dν(E). π G3

Part II

10 Curvature and Surface Area Measures

Every set A ∈ Kn determines two ﬁnite sequences of measures: curvature measures i (A, ·) : B(Rn ) → R for i = 0, . . . , n and surface area measures i (A, ·) : B(S n−1 ) → R for i = 0, . . . , n. For each i, both i and i are “localizations” of intrinsic volume Vi (see Theorem 10.1.7 (b) and 10.2.6): i (A, Rn ) = Vi (A) = i (A, S n−1 ). The notion of surface area measures was introduced independently by A.D. Alexandrov in 1937 and by W. Fenchel and B. Jessen in 1938 ([1] and [19]). The notion of curvature measures was introduced by H. Federer in 1959 for a larger class of sets than Kn ([18]); this class will be the subject of Section 11.1. Twenty years later, R. Schneider in [62] developed the theory of curvature measures for convex compact sets. Also, his survey in [63] deserves particular attention.

10.1 Curvature measures For any A ∈ Kn and ε > 0, the set colε A := (A)ε \ A will be called the ε-collar of A, For every ε > 0 we deﬁne the function Uε : Kn × B(Rn ) → R : Uε (A, X ) := λn ((colε A) ∩ ξ A−1 (X ∩ A)).

(10.1)

110

10. Curvature and Surface Area Measures

Thus, Uε (A, X ) is the volume (that is, n-dimensional Lebesgue measure) of the intersection of the ε-collar of A and the inverse image of X ∩ A by the metric projection on A (compare with 3.3.2). Examples are shown in Figure 10.1.1 Evidently, Uε (A, X ) = Uε (A, X ∩ A)

(10.2)

Uε (A, Rn ) = Vn ((A)ε ) − Vn (A).

(10.3)

and

Figure 10.1.

Let us note the following. 10.1.1. PROPOSITION. For every A ∈ Kn and ε > 0, the function Uε (A, ·) : B(Rn ) → R is a ﬁnite measure on Rn . Proof. Obviously, this function is nonnegative and bounded (by (10.2)). Directly from deﬁnition (10.1) it follows that for every sequence (X i )i∈N of pairwise disjoint Borel sets in Rn , Uε A, Xi = Uε (A, X i ). i

i

We present without proof the following theorem on weak convergence of measures (see (3.3) in [62]): 1 In [62], U (A, X ) is the volume of intersection of (A) and ξ −1 (X ∩ A). ε ε A

10.1 Curvature measures

111

10.1.2. PROPOSITION. If A = lim H Ak , then for every ε > 0, w

Uε (Ak , ·) → Uε (A, ·). As we did for intrinsic volumes (Deﬁnition 7.2.4), we ﬁrst deﬁne curvature measures for the class P n : 10.1.3. DEFINITION. For every P ∈ P n and X ∈ B(Rn ), i F∈F i (P) H (F ∩ X )γ (P, F) for i = 0, . . . , n − 1, i (P, X ) := λn (P ∩ X ) for i = n. (compare with Deﬁnition 7.2.1.) Evidently, i (P, Rn ) = Vi (P).

(10.4)

By (10.3) and (10.4), the following theorem is a generalization of the Steiner theorem for polytopes, 7.2.9. Its proof is analogous to that of 7.2.9. 10.1.4. THEOREM. For every P ∈ P n , X ∈ B(Rn ) and ε > 0, Uε (P, X ) =

n−1

ε n−i κn−i i (P, X ).

(10.5)

i=0

According to Deﬁnition 10.1.3, the functions 0 , . . . , n are deﬁned on the Cartesian product P n × B(Rn ). We shall now extend them over Kn × B(Rn ). To this end, let us substitute ε = 1, . . . , n in formula (10.5). It is then evident that (U1 (P, X ), . . . , Un (P, X )) is the image of the vector (0 (P, X ), . . . , n−1 (P, X )) under the linear transformation with matrix M(n, . . . , 1)nj=1 κ j , which is nonsingular as the product of the Vandermonde matrix M(n, . . . , 1) and a number different from 0. Hence, for every i ∈ {0, . . . , n − 1} there exists (a unique) sequence (αi,1 , . . . , αi,n ), independent of P and X , such that i (P, X ) =

n

αi, j U j (P, X ).

j=1

Now let for every A ∈ Kn and X ∈ B(Rn ), n j=1 αi, j U j (P, X ) for i = 0, . . . , n − 1, i (P, X ) := λn (A ∩ X ) for i = n. As a direct consequence of 10.1.2, we obtain the following. 10.1.5. PROPOSITION. If A = lim H Ak , then for every i ∈ {0, . . . , n}, w

i (Ak , ·) → i (A, ·). Let us prove

(10.6)

112

10. Curvature and Surface Area Measures

10.1.6. THEOREM. Let A ∈ Kn . (i) For i ∈ {0, . . . , n} the function i (A, ·) : B(Rn ) → R is a measure. (ii) For every ε > 0 and X ∈ B(Rn ), Uε (A, X ) =

n−1

ε n−i κn−i i (A, X ).

i=0

Proof. (i) follows from 10.1.1 combined with (10.6). (ii): We approximate A by a sequence of polytopes and apply, in turn, 10.1.2, 10.1.4, and 10.1.5. The function i (A, ·) is the i-dimensional curvature measure (or curvature measure of order i) of A. 10.1.7. PROPOSITION. (a) For every A ∈ Kn , X ∈ B(Rn ), and i ∈ {0, . . . , n − 1}, i (A, X ) = i (A, X ∩ bdA). (b) For every A ∈ Kn and i, i (A, Rn ) = Vi (A). Proof. Condition (a) follows from (10.6) combined with (10.2); condition (b) follows from (10.6) combined with (7.10) (the Steiner formula for Kn ). As we shall see below (Theorem 10.1.11), curvature measures of orders 0 and n − 1 have a simple geometric interpretation ([62], (3.20) and (3.21)). Let A(u) be the support set of A in the direction of a vector u (compare Section 2.2): A(u) := A ∩ H (A, u). 10.1.8. DEFINITION. Let A ∈ Kn and X ∈ B(Rn ). The spherical image of X with respect to A, in symbols σ (A, X ), is deﬁned by the formula σ (A, X ) := {u ∈ S n−1 | A(u) ∩ X = ∅}. It is easy to check that this deﬁnition is consistent with that given by Schneider in [64] (compare with Exercise 10.2). Let us observe (Exercise 10.3) the following simple relationship between the notion of spherical image and that of outer normal angle (see Deﬁnition 7.2.1): 10.1.9. PROPOSITION. If P ∈ P n and F is a proper face of P, then σ (P, relintF) = nor(P, F). In particular, for dim F = 0 the function γ (see Deﬁnition 7.2.1) can be extended in an obvious way over Kn × B(Rn ) (compare Exercise 10.4):

10.1 Curvature measures

113

10.1.10. DEFINITION. For A ∈ Kn and X ∈ B(Rn ), let γ0 (A, X ) :=

σn−1 (σ (A, X )) . ωn

10.1.11. THEOREM. For every A ∈ Kn and X ∈ B(Rn ), 0 (A, X ) = γ0 (A, X ), 1 n−1 H (bdA ∩ X ) if int A = ∅ n−1 (A, X ) = 2 Hn−1 (A ∩ X ) if int A = ∅.

(10.7) (10.8)

The following theorem describes basic properties of curvature measures. 10.1.12. THEOREM. For every i ∈ {0, . . . , n}, (i) i : Kn × B(Rn ) → R is invariant under isometries: for any isometry f : Rn → Rn , i ( f (A), f (X )) = i (A, X ); (ii) i is homogeneous of degree i: for every t > 0, i (t A, t X ) = t i i (A, X ); (iii) for every X ∈ B(Rn ), the function i (·, X ) : Kn → R is a valuation. The proof of (i) and (ii) is based only on the deﬁnition of curvature measures (Exercise 10.5). For the proof of (iii) the reader is referred to [62]. The curvature measures are deﬁned locally. It means that they have the property described by the following theorem. 10.1.13. THEOREM. Let A1 , A2 ∈ Kn and let X be an open subset of Rn . If X ∩ A1 = X ∩ A2 , then ∀X ∈ B(Rn ) X ⊂ X ⇒ i (A1 , X ) = i (A2 , X ) for i = 0, . . . , n. Let us observe that in Theorem 10.1.13 the assumption that X is open is essential: 10.1.14. EXAMPLE. Let a be a common vertex of polytopes P1 and P2 and let P1 \ {a} ⊂ intP2 . Then by (10.7), 0 (P1 , {a}) = γ0 (P1 , {a}) = γ0 (P2 , {a}) = 0 (P2 , {a}). The Hadwiger Theorem 8.1.5 characterizes linear combinations of basic functionals as the continuous invariant valuations. In Theorem 10.1.15, an analogue of 8.1.5 for curvature measures, valuations with values in R are replaced by valuations whose values are Borel measures.

114

10. Curvature and Surface Area Measures

A function φ from Kn in the set of Borel measures on Rn is • •

a valuation if and only if for every X ∈ B(Rn ) the function A → φ(A)(X ) is a valuation, weakly continuous if and only if w

A = lim Ak ⇒ φ(Ak ) → φ(A), H

• invariant (under isometries) if and only if φ( f (A))( f (X )) = φ(A)(X ) •

for every isometry f : Rn → Rn , deﬁned locally if and only if for every A ∈ Kn the measure φ(A) is deﬁned locally.

10.1.15. THEOREM. For any function φ from Kn into the set of Borel measures on Rn , the following conditions are equivalent: (i) φ is a weakly continuous, locally deﬁned, invariant valuation; (ii) there exist α0 , . . . , αn ≥ 0 such that for every A ∈ Kn and X ∈ B(Rn ), φ(A)(X ) =

n

αi i (A, X ).

i=0

It is easy to prove that the set of weakly continuous, locally deﬁned, invariant valuations is closed under addition and multiplication by nonnegative scalars (Exercise 10.6). Thus the implication (ii) ⇒ (i) in Theorem 10.1.15 follows from 10.1.5, 10.1.12, and 10.1.13. We omit the (difﬁcult) proof of the converse implication. Also, the Crofton formulae (Theorem 9.2.6) have their counterpart for curvature measures. As in Section 9.2, we restrict our consideration to sections by hyperplanes. We keep in mind the notation introduced there. 10.1.16. THEOREM. Let n ≥ 2. For i ∈ {1, . . . , n}, let βn,i :=

2 κi−1 . · i κi κn−1

Then for every A ∈ Kn and X ∈ B(Rn ), i (A, X ) = βn,i i−1 (A ∩ E, X )dµ(E). EA

Applying (10.8), it is easy to prove that (n − 1)-dimensional curvature measure n−1 (A, ·) determines uniquely the set A (Exercise 10.7). The natural question arises whether the isometry group is the maximal group of transformations of Rn preserving curvature measures.

10.1 Curvature measures

115

10.1.17. PROBLEM. Prove or disprove the following: if f is a homeomorphism of Rn satisfying the condition ∀X ∈ B(Rn ) ∀i ∈ {0, . . . , n − 1} i (( f (A), f (X )) = i (A, X )

(10.9)

for some A ∈ K0n , then f |bdA is an isometry of bdA onto bd f (A). The following Schneider theorem ([62], (9.1)) gives a partial solution to Problem 10.1.17 (see Exercise 10.8). 10.1.18. THEOREM. Let A1 , A2 ∈ Kn , 0 ∈ int(A1 ∩ A2 ), and let f : bdA1 → bdA2 be the central projection from 0. If there exists an i ∈ {1, . . . , n − 1} such that i (A1 , X ) = i (A2 , f (X )) for every Borel set X ⊂ bdA1 , then A1 = A2 . The proof of Theorem 10.1.18 requires a more general version of Theorem 10.1.16, namely the version concerning sections by afﬁne subspaces of arbitrary dimension k < n. We conﬁne ourselves to the proof of a weaker theorem, 10.1.18 . Let us ﬁrst notice that in Theorem 10.1.16 the integral over the set E A of hyperplanes intersecting A can be replaced by the integral over the whole set E n if we allow A ∩ E to be empty and deﬁne i (∅, X ) := 0 for every i. 10.1.18 . THEOREM. Let A1 , A2 ∈ Kn , 0 ∈ int(A1 ∩ A2 ), and let f : bdA1 → bdA2 be the central projection from 0. If for every Borel set X ⊂ bdA1 , 1 (A1 , X ) = 1 (A2 , f (X )), then A1 = A2 . Proof. Suppose A1 = A2 . Then bdA1 \ A2 = ∅ or bdA2 \ A1 = ∅, because otherwise bdA1 ⊂ A2 and bdA2 ⊂ A1 , whence A1 = A2 , a contradiction. (Of course, we make use of the assumption that A1 and A2 are convex.) Thus, we may assume that X := bdA1 \ A2 = ∅. Of course, X ∈ B(Rn ). We shall ﬁrst prove that for every E ∈ E n , 0 (A1 ∩ E, X ) ≥ 0 (A2 ∩ E, f (X )).

(10.10)

116

10. Curvature and Surface Area Measures

Notice that f (X ) ⊂ intA1 ; hence A1 ∩ E = ∅ ⇒ f (X ) ∩ E = ∅ ⇒ 0 (A2 ∩ E, f (X )) = 0 (compare with 10.1.7 (a)). Therefore, if A1 ∩ E = ∅, then (10.10) is satisﬁed. Let A1 ∩ E = ∅ = A2 ∩ E. Then for every (n − 2)-dimensional hyperplane H (A2 ∩ E, u) in E supporting A2 ∩ E at some point of f (X ), there exists the parallel (n − 2)-dimensional hyperplane H (A1 ∩ E, u) in E supporting A1 ∩ E at some point of X . Hence in this case condition (10.10) is satisﬁed as well. Finally, if A1 ∩ E = ∅ = A2 ∩ E, (10.11) then 0 (A1 ∩ E, X ) = 0 = 0 (A2 ∩ E, f (X )), whence in (10.10) the inequality is sharp. Since the set of hyperplanes satisfying (10.11) is of positive measure µ, in view of Theorem 10.1.16 it follows that 1 (A1 , X ) > 1 (A2 , f (X )), contrary to the assumption. Let us complete this section with some examples.

10.1.19. EXAMPLE. Let n = 2. Let A := (a, b, c) and X 1 , X 2 , X 3 be as in Figure 10.1. We calculate i (A, X j ) for i = 0, 1 and j = 1, 2, 3. By 10.1.3 combined with 7.2.1, i (A, X j ) = Hi (F ∩ X j )(ω2−i )−1 σ1−i (nor(A, F)), F∈F i (A) where nor(A, F) = {u ∈ S n−1 | A(u) = F}. Thus it is easy to see that 0 (A, X 1 ) = 0, 1 0 (A, X 2 ) = (π − (a, b, c)), 2π 1 0 (A, X 3 ) = (π − (a, c, b)), 2π and 1 (A, X 1 ) =

1 1 a − b , 1 (A, X 2 ) = 0, 1 (A, X 3 ) = c − c . 2 2

10.1.20. EXAMPLE. Let n = 3. Consider the cylinder A = {(x1 , x2 , x3 ) ∈ R3 | x12 + x22 ≤ 1, 0 ≤ x3 ≤ 1} and let X 1 = A ∩ aff(e1 , e2 ), X 2 = relbdX 1 , and X 3 = ((1, 0, 0), (1, 0, 1)). To calculate the curvature measures i (X j ), for i = 0, 1, 2 and j = 1, 2, 3, we apply 10.1.6(ii). Take ε ∈ (0; 12 ). The counterimage of X j under the metric

10.2 Surface area measures

117

projection onto A is the half-space H = {(x1 , x2 , x3 ) | x3 ≤ 0} for j = 1, the set H \ {(x1 , x2 , x3 ) | x12 + x22 ≤ 1, −ε ≤ x3 ≤ 0} for j = 2, and the half-plane {(x1 , 0, x3 ) | x1 ≥ 1} for j = 3. To calculate the volumes of intersections of colε (A) with those three counterimages, we use the Guldin formula for volume of the set obtained by rotating a plane ﬁgure D about the axis disjoint from D (here the axis lin(e3 )). For j = 1, we obtain Uε (A, X 1 ) = πε +

π 2 ε2 2 + π ε3 ; 2 3

hence by 10.1.6(ii), πε +

π 2 ε2 2 4 + πε 3 = ε3 π0 (A, X 1 ) + ε 2 π 1 (A, X 1 ) + 2ε2 (A, X 1 ). 2 3 3

Comparing the coefﬁcients of corresponding powers of ε on both sides of the equation above, we obtain 0 (A, X 1 ) = 12 and i (A, X 1 ) = π2 for i = 1, 2. For j = 2, ξ A−1 (X 2 ) = H \ {(x1 , x2 , x3 ) | (x1 , x2 , 0) ∈ relintX 1 }, whence

1 2 2 2 3 π ε + πε . 2 3 Thus, comparing again the coefﬁcients, we obtain Uε (A, X 2 ) =

0 (A, X 2 ) =

1 π , 1 (A, X 2 ) = , 2 (A, X 2 ) = 0. 2 2

For j = 3, since the left side of the formula in 10.6 (ii) is 0, we obtain i (A, X 3 ) = 0 for i = 0, 1, 2. We leave to the reader the details of the above calculations (see Exercise 10.11).

10.2 Surface area measures Unlike the curvature measures, which are Borel measures on Rn , the surface area measures are Borel measures on S n−1 . For every A ∈ Kn , the function u A : Rn \ A → S n−1 is deﬁned by u A (x) := (compare [64], p. 9). From 1.1.2 and 3.3.4 it follows that

x − ξ A (x) (x, A)

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10. Curvature and Surface Area Measures

10.2.1. For every A ∈ Kn the function u A is continuous. By 10.2.1, inverse images (with respect to u A ) of Borel sets are Borel sets: n Y ∈ B(S n−1 ) ⇒ u −1 A (Y ) ∈ B(R ).

Thus for every ε > 0 the function Vε : Kn × B(S n−1 ) → R may be deﬁned by the formula Vε (A, Y ) := λn ((colε (A)) ∩ u −1 (10.12) A (Y )) (Figure 10.2).

Figure 10.2.

Evidently, directly from deﬁnition (10.12) we obtain an analogue of (10.3): Vε (A, S n−1 ) = Vn (Aε ) − Vn (A).

(10.13)

The following statement is a counterpart of Theorem 10.1.1 concerning Uε (A, ·). 10.2.2. For every A ∈ Kn and ε > 0, the function Vε (A, ·) is a ﬁnite measure on S n−1 . As with the curvature measures (Deﬁnition 10.1.3), the surface area measures are deﬁned ﬁrst for convex polytopes: 10.2.3. DEFINITION. For every P ∈ P n , Y ∈ B(S n−1 ), and i ∈ {0, . . . , n − 1} i (P, Y ) :=

F∈F i (P)

σn−1−i (σ (P, F) ∩ Y ) i H (F). ωn−i

In view of (10.13), the following theorem is a generalization of the Steiner theorem for polytopes (Theorem 10.1.4).

10.2 Surface area measures

119

10.2.4. THEOREM. For every P ∈ P n , Y ∈ B(S n−1 ), and ε > 0, Vε (P, Y ) =

n−1

ε n−i κn−i i (P, Y ).

i=0

For every i ∈ {0, . . . , n − 1}, the function i : P n × B(S n−1 ) → R can be extended over Kn × B(S n−1 ) (compare Section 10.1). As before, we use the same symbol for the extension as for the original function. The following statement is a counterpart of 10.1.6. 10.2.5. THEOREM. For every A ∈ Kn and i ∈ {0, . . . , n − 1}, the function i (A, ·) is a ﬁnite measure on S n−1 and for every ε > 0, Vε (A, ·)) =

n−1

ε n−i κn−i i (A, ·).

i=0

The function i (A, ·) is called the i-dimensional surface area measure of A. The following statement is a counterpart of 10.1.7 (b). 10.2.6. PROPOSITION. For every A ∈ Kn , i (A, S n−1 ) = Vi (A). 10.2.7. DEFINITION. Let A ∈ Kn and Y ∈ B(S n−1 ). The reverse spherical image of Y with respect to A is the set τ (A, Y ) deﬁned by τ (A, Y ) := A(u). u∈Y

The surface area measures are deﬁned locally but in a more general sense than the curvature measures (compare 10.1.13): 10.2.8. THEOREM. Let A1 , A2 ∈ Kn and Y ∈ B(S n−1 ). If τ (A1 , Y ) = τ (A2 , Y ), then for every i, i (A1 , Y ) = i (A2 , Y ). Further, let us note an analogue of 10.1.12 (ii): 10.2.9. PROPOSITION. For every Y ∈ B(S n−1 ), the function i (·, Y ) is homogeneous of degree i: ∀t > 0 i (t A, Y ) = t i i (A, Y ). A classical theorem of Alexandrov, Fenchel, and Jessen (1937, 1938) concerns determination (up to a translation) of a convex body by the surface area measures ([63]). Like the previous statements in this section, we mention it without proof.

120

10. Curvature and Surface Area Measures

10.2.10. THEOREM. Let j ∈ {1, . . . , n − 1}. Let A1 , A2 ∈ Kn , and let dim Ai > j for i = 1, 2. If j (A1 , ·) = j (A2 , ·), then ∃v ∈ Rn A2 = A1 + v. (Compare Exercise 10.7 and Theorem 10.1.18 on curvature measures.)

10.3 Curvature and surface area measures for smooth, strictly convex bodies Let A ∈ K0n . A point a ∈ bdA is a singular point of A if A has more than one support hyperplane at a, or equivalently, dim σ (A, {a}) > 0. Otherwise, a is a regular point of A. Thus, for any regular point a there is a unique support hyperplane of A at a; that is, the spherical image σ (A, {a}) is a singleton. Similarly, the direction of u ∈ S n−1 is a singular direction for A if the support set A(u) has dimension at least one. Otherwise, it is a regular direction for A. Thus if the direction of u is regular, then A(u) is a singleton. Points and directions can be classiﬁed more precisely by means of the dimension of the spherical image or the dimension of the support set, respectively (see [64]). From the point of view of those classiﬁcations, the family P0n has “extreme” properties: for every n-dimensional convex polytope P in Rn and every k ∈ {0, . . . , n − 1}, • •

there exist points with k-dimensional spherical image; they are the points of the (n − 1 − k)-dimensional faces of P, there exist vectors with k-dimensional support sets; they are the outer normal vectors of the k-dimensional faces of P.

We shall now consider the “opposite extreme”: the class of convex bodies with all points and all directions regular. 10.3.1. PROPOSITION. (i) If all the points of a convex body A are regular, then the formula σ A (x) := σ (A, {x}) deﬁnes a function σ A : bdA → S n−1 . (ii) If all directions are regular for A, then the formula τ A (u) := τ (A, {u}) deﬁnes a function τ A : S n−1 → bdA.2 2 Compare 10.2.7.

10.3 Curvature and surface area measures for smooth, strictly convex bodies

121

In case (i) we say that A is smooth; the function σ A is then referred to as the spherical map of A; in case (ii) we say that A is strictly convex; the function τ A is then referred to as the reverse spherical map. Let us note the following. 10.3.2. PROPOSITION. If A is a smooth, strictly convex body, then σ A is a homeomorphism and σ A−1 = τ A (Exercise 10.9). For smooth convex bodies as well as for strictly convex bodies the curvature measures are closely related to the surface area measures (Theorem 4.2.4 in [64]): 10.3.3. THEOREM. Let A ∈ K0n , i ∈ {0, . . . , n − 1}. If A is smooth, then for every Y ∈ B(S n−1 ), i (A, Y ) = i (A, σ A−1 (Y )); if A is strictly convex, then for every X ∈ B(Rn ), i (A, X ) = i (A, τ A−1 (X )). 10.3.4. COROLLARY. If A is a smooth, strictly convex body in Rn , then for i ∈ {0, . . . , n − 1}, i (A, X ) = i (A, σ A (X ∩ bdA)). Smooth convex bodies are a subject of interest of classical differential geometry. Hence let us express their curvature measures in terms of differential geometry. Let A be a smooth convex body in Rn and let a ∈ bdA. There exists a neighborhood S of a in bdA that is a smooth sheet, which means that there is a domain U in Rn−1 and a homeomorphism p : U → S of the class C 1 with linearly ∂p independent partial derivatives pi := ∂u for i = 1, . . . , n − 1. Let i pn :=

×( p1 , . . . , pn−1 ) , × ( p1 , . . . , pn−1 )

where × is the vector product in Rn (see [50]), and let u 0 := p −1 (a). Then of course, pn (u 0 ) is a unit normal vector of S at a. If the choice of the parametric representation p is suitable, then this vector is the inner normal vector of A, i.e., − pn is the outer normal vector. Assume now that p is of the class C 2 . Let L be the normal line at a: L := a + lin( pn ). For every plane N containing L, the intersection N ∩ S is an arc (if S is small enough); it is the so-called normal section. This arc is the image p(L) of some

122

10. Curvature and Surface Area Measures

arc L contained in U with parametrization u : (α; β) → U of the class C 1 ; consequently, S ∩ N = { pu(s) | s ∈ (α; β)} and u 0 = u(s0 ) for some s0 ∈ (α; β). Let r := pu. We may assume that r is normalized, i.e., its derivative r has norm equal to 1. Let r1 := r . Then r1 (s) =

n−1

pi (u(s))u i (s),

(10.14)

i=1

whence pn ⊥ r1 ; thus (a; r1 (s0 ), pn (u 0 )) is an (orthogonal) afﬁne basis of the plane N . In the plane N oriented by the choice of this basis, the oriented normal vector r 2 coincides with pn ; hence by the Frenet formulae for an oriented curve, the oriented curvature κ of the normal section S ∩ N has the form κ = (r1 ) ◦ pn . Let pi j :=

(10.15)

∂ pi ∂u j

and bi j := pi j ◦ pn . From (10.14) and (10.15) it follows that κ=

n−1

bi j u i u j .

(10.16)

i, j=1

Hence we have expressed the oriented curvature of the normal section S ∩ N as

the value of the second fundamental form i j bi j xi x j for xi := u i . Eigenvalues of the matrix (bi j (u 0 ))i, j=1,...,n−1 are principal curvatures of S at u0. In the case n = 3, principal curvatures are extremal values of the curvature κ =: κ N treated as a function of N . Let κ 1 , . . . , κ n−1 be the principal curvatures. In the situation considered, either κ i ≥ 0 for all i or κ i ≤ 0 for all i; since, by the assumption, pn is the inner normal unit vector, it follows that they are nonnegative. The functions H j : U → R for j = 1, . . . , n − 1 are deﬁned by H j :=

−1

n−1 j

1≤i 1 ,...,i j ≤n−1 i κ
κ i1 · · · κ i j .

The ﬁrst one, H1 , is the mean curvature, and the last, Hn−1 , the Gaussian curvature of S:

10.3 Curvature and surface area measures for smooth, strictly convex bodies

123

1 (κ + · · · + κ n−1 ), n−1 1 = κ 1 · · · κ n−1 .

H1 = Hn−1

Since p is a homeomorphism, the H j can be treated as functions of a point x ∈ S and can be integrated over X ∩ A with respect to the (n − 1)-dimensional Hausdorff measure Hn−1 . Covering bdA by a ﬁnite family of closures of pairwise disjoint open subsets, we can treat each H j as a function over bdA. The proof of the following theorem can be found in [64], Section 2.5. The 2 used there denotes the class of convex bodies whose boundaries can symbol C+ be parametrized (locally) by functions of the class C 2 , and the Gaussian curvature is positive. 2 and every X ∈ B(Rn ), 10.3.5. THEOREM. For every A ∈ Kn of class C+ n−1 n 1 Uε (A, X ) = ε n−i Hn−1−i dHn−1 . (10.17) i n i=0 X ∩bdA

Comparing (10.17) with the Steiner formula for curvature measures (Theorem 10.1.6 (ii)), we obtain a theorem on curvature measures of smooth convex bodies (Theorem 10.3.6). To formulate this theorem, let us replace i by the following function Ci , also referred to as the curvature measure: −1 n Ci := n κn−i i . (10.18) i The function Ci is a localization of the functional Wn−i (see (7.7)), in the similar sense similar as i is a localization of Vi (compare with 10.1.7 (b)): Ci (A, Rn ) = nWn−i (A).

(10.19)

Unlike i , the function Ci depends on the dimension of the ambient space. 2 in Rn , for every 10.3.6. THEOREM. For every convex body A of class C+ n X ∈ B(R ), and i ∈ {0, . . . , n − 1}, Ci (A, X ) = Hn−1−i dHn−1 . X ∩bdA

Theorem 10.3.6 combined with (10.18) and (10.19) yields the following corollary concerning the functionals W1 , . . . , Wn (and thus concerning also the intrin2 (see [64] (4.2.28)). sic volumes) for convex bodies of class C+ 2, 10.3.7. COROLLARY. For every A ∈ Kn of the class C+ 1 Wi (A) = Hi−1 dHn−1 . n bdA

Hence in particular, for convex bodies of this class, the functional W2 is the integral of the mean curvature H1 (compare footnote to (9.1)). In [64] the reader can also ﬁnd analogues of Theorem 10.3.6 and Corollary 10.3.7 for surface area measures.

11 Sets with positive reach. Convexity ring

11.1 Sets with positive reach As was already mentioned at the beginning of Chapter 10, H. Federer deﬁned curvature measures for a larger class of sets than K0n : for the so-called sets with positive reach. 11.1.1. DEFINITION. For any nonempty closed subset A of Rn , let D(A) := {x ∈ Rn | ∃1 a ∈ A (x, a) = (x, A)}; reachA := sup{r ≥ 0 | {x |(x, A) < r } ⊂ D(A)}; reachA := ∞ if this upper bound does not exist. Thus reachA is the upper bound of radii of the open generalized balls around A, every point of which has only one nearest point in A. 11.1.2. EXAMPLES. (a) Let A be a broken line in R2 : A = (a, b) ∪ (b, c) for noncollinear points a, b, c with a − b = b − c. Of course, reachA = 0, because every point of the bisectrix of the pair of halflines b + pos(a − b), b + pos(c − b), excluding their common point, has two nearest points in A. (b) Let A be the sphere in Rn , with center 0 and radius r > 0. Every point x ∈ Rn \ {0} has exactly one nearest point in A, while for the point 0 every point of A is the nearest point. Hence D(A) = Rn \ {0} and reachA = r. (c) In view of Theorem 3.3.1, if A is a nonempty closed convex subset of Rn , then reachA = ∞.

126

11. Sets with Positive Reach. Convexity Ring

11.1.3. DEFINITION. A subset A of Rn is a set with positive reach if 0 < reachA ≤ ∞. Thus, Theorem 3.3.1 can be reformulated as follows: 11.1.4. THEOREM. A subset A of Rn is nonempty, closed, and convex if and only if reachA = ∞. Let us note that Federer’s deﬁnition of reach is slightly different; namely, he ﬁrst deﬁnes the local reach, reach(A, a), for a ∈ A: 11.1.5. DEFINITION. Let A ⊂ Rn and a ∈ A. Then reach(A, a) := sup{r > 0 | (x, a) < r ⇒ x ∈ D(A)}. Further, he deﬁnes the reach of A as the lower bound of reach(A, a) for a ∈ A. For compact sets his notion of reach coincides with that deﬁned by 11.1.1 above: 11.1.6. If A is compact, then reachA = inf{reach(A, a) | a ∈ A}. Proof. For every a ∈ A, the set {r > 0 | (x, A) < r ⇒ x ∈ D(A)} is contained in {r > 0 | (x, a) < r ⇒ x ∈ D(A)}, whence reachA ≤ reach(A, a) for every a ∈ A. Thus reachA ≤ inf{reach(A, a) | a ∈ A}. Let r = inf{reach(A, a) | a ∈ A}. Since A is compact, it follows that there exists a ∈ A with r = reach(A, a); hence {x | (x, a) < r } ⊂ D(A). This implies {x | (x, A) < r } ⊂ D(A), which means that r ≤ reachA.

11.1.7. THEOREM. ([18] Remark 4.2, p. 432) (i) The function A a → reach(A, a) is continuous. (ii) for every a ∈ bdA, 0 ≤ reach(bdA, a) ≤ reach(A, a) ≤ ∞. Federer deﬁnes the set Tan(A, a) of vectors tangent to A ⊂ Rn at a ∈ A: 11.1.8. DEFINITION. ([18], 4.3) u ∈ Tan(A, a) :⇐⇒ u = 0 or ∀ε > 0 ∃x ∈ A 0 < x − a < ε and |

x −a u − | < ε. x − a u

11.1.9. THEOREM. (compare [18] Remark 4.20, p. 450). Let A be a set with positive reach in Rn . The following conditions are equivalent: (i) A is a k-dimensional manifold, (ii) for every a ∈ A the set Tan(A, a) is a k-dimensional linear space.

11.2 Convexity ring

127

The same remark 4.20 in [18] contains some conditions sufﬁcient for a smooth k-dimensional manifold to have positive reach. Already in 1939, H. Weyl in [67] established an analogue of the Steiner theorem for compact k-dimensional submanifolds of Rn , of class C 2 , for 1 ≤ k ≤ n − 1. We shall present here without proof the following Federer theorem (compare with 5.6, p. 455, in [18]): 11.1.10. THEOREM. For every A ⊂ Rn with positive reach there exists a unique sequence of Borel measures (0 (A, ·), . . . , n (A, ·)) such that for every A and every X ∈ B(Rn ), if 0 ≤ r < reachA, then λn ((A)r ∩ ξ A−1 (X )) =

n

r n−i κn−i i (A, X ).

i=0

Evidently, by Theorem 10.1.6 combined with Example 11.1.2 (c), for any A ∈ Kn the function i (A, ·) is the ith curvature measure of A. Hence 11.1.10 is a generalization of 10.1.6 (local version of the Steiner theorem) on the class of sets with positive reach.

11.2 Convexity ring The class Kn ∪{∅} is closed under intersection (compare with 3.2.1), but of course, it is not closed under union. This class can be extended to the family U n deﬁned by the formula A ∈ U n :⇐⇒ ∃A1 , . . . , Am ∈ Kn ∪ {∅} A =

m

Ai .

(11.1)

i=1

11.2.1. The family U n is a ring of sets (i.e., a ring with respect to ∪ and ∩) generated by Kn ∪ {∅}. Proof. Since the power set of any nonempty set, thus of Rn in particular, is a ring with respect to ∪ and ∩, it sufﬁces to prove that U n is closed under these two operations. Directly from Deﬁnition 11.2.1, it follows that U n is closed under ∪. It is closed under ∩ as well: m Let A = i=1 Ai and B = kj=1 B j for some A1 , . . . , Am , B1 , . . . , Bk ∈ Kn . Since Ai ∩ B j ∈ Kn ∪ {0} for every i ∈ {1, . . . , m}, j ∈ {1, . . . , k}, and A ∩ B = n i, j Ai ∩ B j , it follows that A ∩ B ∈ U . The family U n will be called the convexity ring.1 1 Hadwiger in [30] and Schneider in [64] use the name Konvexring and convex ring, respectively; it

seems, however, that their terminology is confusing.

128

11. Sets with Positive Reach. Convexity Ring

The study of the convexity ring is motivated by the possibility of generalizing on U n some theorems concerning Kn , for instance the Crofton Theorem 9.2.6. Of course, to do this, we have to extend basic functionals over U n . We need Lemma 11.2.3, whose proof is due to R. Schneider ([64] p. 174). This lemma concerns some special valuations; following Schneider, we call them fully additive: 11.2.2. DEFINITION. Let F be a nonempty family of subsets of Rn , closed under intersection, and let U(F) be the family of ﬁnite unions of members of F. A function : F → R is fully additive on F if for every A1 , . . . , Am ∈ F m with i=1 Ai ∈ F, m m Ai = (−1)r −1 (Ai1 ∩ · · · ∩ Air ). (11.2) i=1

i 1 <···
r =1

Obviously, every fully additive function is a valuation (that is, it satisﬁes (11.2) for m = 2). We admit the following notation. For every m ∈ N, let S(m) := {ν ⊂ {1, . . . , m} | ν = ∅}

and |ν| := cardν for ν ∈ S(m).

For any ν = {i 1 , . . . , ir } ∈ S(m) and A1 , . . . , Am ∈ F, let Aν := Ai1 ∩ · · · ∩ Air . Then (11.2) can be written as m Ai = (−1)|ν|−1 (Aν ).

(11.3)

ν∈S(m)

i=1

11.2.3. LEMMA. If : F → mR is fully additive on F, then for any A1 , . . . , Am , B1 , . . . , Bk ∈ F, with i=1 Ai = kj=1 B j , (−1)|ν|−1 (Aν ) = (−1)|λ|−1 (Bλ ). ν∈S(m) λ∈S(k) Proof. Let α := (−1)|ν|−1 (Aν ), ν∈S(m) and

m

β :=

Ai = X =

i=1

k

λ∈S(k)

(−1)|λ|−1 (Bλ ),

Bj.

j=1

Then for every ν ∈ S(m) and λ ∈ S(k), Aν = X ∩ Aν =

k j=1

Aν ∩ B j

(11.4)

11.2 Convexity ring

129

and m

Bλ = X ∩ Bλ =

Ai ∩ Bλ .

(11.5)

i=1

By (11.4) and (11.3) (full additivity of ), α= =

ν∈S(m)

ν∈S(m)

(−1)

|ν|−1

(−1)

|ν|−1

k (Aν ∩ B j )

j=1 λ∈S(k)

(−1)|λ|−1 (Aν ∩ Bλ ).

By (11.5) and (11.3), β= =

m |λ|−1 (−1) (A ∩ B ) i λ λ∈S(k)

i=1

λ∈S(k)

(−1)

|λ|−1

ν∈S(m)

(−1)|ν|−1 (Aν ∩ Bλ ).

Hence α = β.

In view of Lemma 11.2.3, every fully additive function : F → R can be extended over U(F) to the function deﬁned by (X ) := for any A1 , . . . , Am ∈ F with

ν∈S(m)

m i=1

(−1)|ν|−1 (Aν )

(11.6)

Ai = X .

Let us note that Lemma 11.2.3 is indispensable, because it guarantees that the value of for X ∈ U(F) is independent of the representation of X as a union of elements of F. The following is well known (compare with Exercise 11.3). 11.2.4. PROPOSITION. The extension of a fully additive valuation : F → R, deﬁned by (11.6), is a valuation on U(F). (Proposition 11.2.4 is a part of Theorem 3.4.11 in [64]). We shall prove that such an extension is fully additive: 11.2.5. PROPOSITION. The function deﬁned by (11.6) is fully additive on U(F). Proof. In view of 11.2.4, the assertion is true for m = 2. Let m > 2; we admit the inductive assumption for m − 1. Then

130

11. Sets with Positive Reach. Convexity Ring

m m−1 m−1 m−1 Ai = Ai ∪ A m = Ai + (Am ) − (Ai ∩ Am ) i=1

i=1

=

i=1

m−1

r =1

i 1 <···
(−1)r −1

−

m−1

m

(Ai1 ∩ · · · ∩ Air ) + (Am )

(−1)r −1

(Ai1 ∩ · · · ∩ Air ∩ Am )

i 1 <···
r =1

=

i=1

(−1)r −1

(Ai1 ∩ · · · ∩ Air ).

i 1 <···
r =1

In Section 6.1, the Euler–Poincar´e characteristic χ was introduced for geometric polyhedra. According to 6.1.5–6.1.7, since every compact convex subset of Rn is homeomorphic to a simplex, we deﬁne χ (A) := 1

for

A ∈ Kn

and χ (∅) = 0.

(11.7)

Of course, every geometric polyhedron in Rn belongs to the convexity ring Our next task is to extend the function χ deﬁned on the family of geometric polyhedra and compact convex sets in Rn to a valuation χ : U n → R. We shall need the following lemma. The proof is due to Groemer ([24]). Un.

11.2.6. LEMMA. The function χ |Kn ∪ {∅} deﬁned by (11.7) is fully additive on ∪ {∅}. m Ai ∈ Kn for some m > 1. Proof. Let A1 , . . . , Am ∈ Kn and A = i=1 If Aν = ∅ for every ν ∈ S(m), i.e., all intersections of elements of the set {A1 , . . . , Am } are nonempty, then the formula (11.2) can be written as Kn

1=

m

(−1)r −1

r =1

which is equivalent to

m , r

m = 0; (−1) r r =0

m

r

but the last equality follows from the Newton formula. Hence, for m = 2 the assertion is true, since A1 ∩ A2 = ∅ as A1 ∪ A2 is convex. Let m > 2; assume that the assertion is true for m − 1. (a) If among A1 , . . . , Am there is a pair of disjoint sets, then these sets can be separated by some closed half-spaces E , E (compare with 3.3.8). We may assume these two sets to be Am−1 , Am ; thus Am−1 ⊂ E ,

Am ⊂ E ,

E ∪ E = Rn ,

and E ∩ E is a hyperplane disjoint from Am−1 ∪ Am .

11.2 Convexity ring

Let

Ai = Ai ∩ E ,

and

A :=

131

Ai = Ai ∩ E ,

Ai ,

A :=

Ai .

By the inductive assumption, χ (A ) =

m−1

r =1

i 1 <···
(−1)r −1

and so for A and Ai . Let E 0 := E ∩ E , Since A0 =

m

A0 := A ∩ A = A ∩ E 0 .

Ai

χ (Ai1 ∪ · · · ∪ Air )

∩ E0 =

i=1

m−2

Ai ∩ E 0 ∈ Kn ,

i=1

from the inductive assumption it follows that χ (A0 ) =

m−2

r =1

i 1 <···
(−1)r −1

χ (Ai1 ∩ · · · ∩ Air ∩ E 0 ),

whence χ (A) = χ (A ∪ A ) = χ (A ) + χ (A ) − χ (A0 ) m = (−1)r −1 χ (Ai1 ∩ · · · ∩ Air ) + χ (Ai1 ∩ · · · ∩ Air ) i 1 <···
r =1

=

m r =1

(−1)r −1

χ (Ai1 ∩ · · · ∩ Air ),

i 1 <···
because Ai1 ∩ · · · ∩ Air = (Ai1 ∩ · · · ∩ Air ) ∪ (Ai1 ∩ · · · ∩ Air ). (b) Assume that every two elements of the set {A1 , . . . , Am } have a nonempty intersection: Ai ∩ A j = ∅ for i, j ∈ {1, . . . , m}. (11.8) Let k0 := max{k ∈ {1, . . . , m} | Ai1 ∩ · · · ∩ Aik = ∅ for every {i 1 , . . . , i k }}. Of course, k0 ≥ 2. We have proved that the assertion is true for k0 = m. Now let k0 < m and let, for example,

132

11. Sets with Positive Reach. Convexity Ring

A1 ∩ · · · ∩ Ak0 +1 = ∅.

(11.9)

We deﬁne B1 , . . . , Bm : Bi =: A1 ∩ · · · ∩ Ak0 −1 ∩ Ai . Evidently, Bi ∈ Kn for i = 1, . . . , m and Bk0 ∩ Bk0 +1 = ∅ by (11.9). Thus we can use (a) for B1 , . . . , Bm . But m

Bi = A ∩ A1 ∩ · · · ∩ Ak0 −1 = A1 ∩ · · · ∩ Ak0 −1

i=1

and Bi1 ∩ · · · ∩ Bir = (Ai1 ∩ · · · ∩ Air ) ∩ (A1 ∩ · · · ∩ Ak0 −1 ), whence (by (a) for Bi ) we obtain χ Bi = (−1)|ν|−1 χ ((Bi )ν ), ν

and this condition in terms of the sets A1 , . . . , Am has the form 1 = χ (A1 ∩ · · · ∩ Ak0 −1 ) m = (−1)r −1 χ ((Ai1 ∩ · · · ∩ Air ) ∩ (A1 ∩ · · · ∩ Ak0 −1 )) i 1 <···
r =1

=

m

(−1)r −1

(χ (Ai1 ∩ · · · ∩ Air ) + χ (A1 ∩ · · · ∩ Ak0 −1 )

r =1

− χ((Ai1 ∩ · · · ∩ Air ) ∪ (A1 ∩ · · · ∩ Ak0 −1 )). Since the last two terms can be reduced (each of them equals 1), and χ (A) = 1 (because A ∈ Kn ), it follows that χ (A) =

m

(−1)r −1

χ (Ai1 ∩ · · · ∩ Air ).

i 1 <···
r =1

In view of 11.2.4 and 11.2.6, the following formula deﬁnes the valuation χ : → R, which is an extension of the characteristic χ : Kn → R: for any m n A ∈ U , if A = i=1 Ai , then

Un

χ (A) :=

ν∈S(m)

(−1)|ν|−1 χ (Aν ).

(11.10)

We shall now prove that χ is also an extension of the Euler–Poincar´e characteristic χ deﬁned earlier for geometric polyhedra in Rn .

11.2 Convexity ring

133

11.2.7. THEOREM. For every geometric polyhedron P in Rn , χ (P) = χ (P). Proof. Let T be a triangulation of the polyhedron P. We select the subset A of T consisting of the simplices that are not proper faces of another. Let A = {A1 , . . . , Am }. Since evidently P = A, and (by 11.2.3) the value of χ is independent of the decomposition of a given set into a union of elements of Kn , it sufﬁces to prove that (−1)|ν|−1 χ (Aν ) = χ (P). ν∈S(m)

For m = 1 the set P is a simplex, whence this condition is satisﬁed. Let m > 1; assume that itis satisﬁed for m − 1. m−1 Let P := i=1 Ai . Then P = P ∪ Am and χ (P) = χ (P ) + χ (Am ) − χ (P ∩ Am ). An analogous formula holds for χ , because it is a valuation on U n : χ (P) = χ (P ) + χ (Am ) − χ (P ∩ Am ). Since by the inductive assumption, the right-hand sides of these two formulas are equal, it follows that χ (P) = χ (P). 11.2.8. REMARK. To any ﬁnite set {A1 , . . . , Am } in Kn a simplicial complex N (A1 , . . . , Am ) can be assigned. i ) ∈ Rm , and let T be the Let S := (e1 , . . . , em ), where ei = (δ1i , . . . , δm natural triangulation of the simplex S: T = {(ei1 , . . . , eir ) | i 1 , . . . , ir ∈ {1, . . . , m}, r = 1, . . . , m}. The complex N (A1 , . . . , Am ) is a subcomplex of T deﬁned as follows: N (A1 , . . . , Am )(0) := S (0) = {e1 , . . . , em },

(11.11)

and for every r ∈ {2, . . . , m}, (ei1 , . . . , eir ) ∈ N (A1 , . . . , Am )(r ) :⇐⇒ Ai1 ∩ · · · ∩ Air = ∅.

(11.12)

By analogy with the well-known topological notion of the nerve of a covering, the complex N (A1 , . . . , Am ) is called the nerve of {A1 , . . . , Am }. Directly from 6.1.3 combined with (11.11) and (11.12) it follows that χ (A1 ∪ · · · ∪ Am ) = χ (N (A1 , . . . , Am )).

For the function χ we use the same symbol χ and we call it the Euler–Poincar´e characteristic . Evidently, χ |Kn = V0 , whence χ is an extension on U n of the basic functional V0 .

134

11. Sets with Positive Reach. Convexity Ring

Let V0 (A)

:=

0 χ (A)

if A = ∅ if ∅ = A ∈ U n .

We shall now extend the remaining basic functionals, V1 , . . . , Vn . We preserve the notation introduced in Chapter 9 (see 9.2.1). 11.2.9. DEFINITION. Let n ≥ 2. For i ∈ {1, . . . , n}, let βn,i := and for A ∈ U n , let Vi (A)

2 κi−1 · , i κi κn−1

:= βn,i

EA

Vi−1 (A ∩ E)dµ(E).

11.2.10. THEOREM. The functionals V0 , . . . , Vn are valuations on U n and = Vi for i = 0, . . . , n. Proof. We already know that theorem holds for i = 0. Let i ≥ 1. By the Crofton Theorem 9.2.6, Vi (A) = Vi (A) for A ∈ Kn . Vi |Kn

Evidently, for every i,

Vi (∅) = 0.

(11.13)

We prove by induction that Vi is a valuation: for i = 0 it is a corollary of 11.2.4 combined with 11.2.6. Let i ≥ 1; we admit the inductive assumption for i − 1. By Deﬁnition 11.2.9, condition (11.13), and the inductive assumption, Vi (A1 ∪ A2 ) + Vi (A1 ∩ A2 ) = βn,i (Vi−1 (A1 ∩ E) + Vi−1 (A2 ∩ E)) dµ(E) En

=

Vi (A1 ) +

Vi (A2 ).

Directly from Deﬁnition 11.2.9 we infer that for the extended basic functionals the Crofton formulae hold.

12 Selectors for Convex Bodies

12.1 Symmetry centers Centrally symmetric sets play a particular role in the geometry of Rn . 12.1.1. DEFINITION. Let A be a nonempty subset of Rn . A point p is a symmetry center of A if and only if σ p (A) = A, i.e., x ∈ A ⇒ σ p (x) ∈ A. Of course, some sets have many symmetry centers (compare Exercise 12.1). Let C0 (A) be the set of all symmetry centers of A. 12.1.2. THEOREM. For every nonempty convex subset A of Rn , the set C0 (A) is a convex subset of A. Proof. We may assume that C0 (A) = ∅. Let p ∈ C0 (A). According to 12.1.1, if x ∈ A, then σ p (x) ∈ A; thus p = 1 2 (x + σ p (x)) ∈ A, because A is convex. Hence C0 (A) ⊂ A. Let p0 , p1 ∈ C0 (A) and pt := (1 − t) p0 + t p1 for t ∈ [0, 1]. For every x ∈ A, σ pt (x) = 2 pt − x = (1 − t)(2 p0 − x) + t (2 p1 − x) and 2 pi − x ∈ A for i = 0, 1; since A is convex, it follows that σ pt (x) ∈ A and thus pt ∈ C0 (A). Hence C0 (A) is convex.

136

12. Selectors for Convex Bodies

12.1.3. THEOREM. Every bounded subset of Rn has at most one symmetry center. Proof. We may assume A = ∅. Suppose that p, q ∈ C0 (A) and α := p−q > 0. From 12.1.2 it follows that p, q ∈ A. Let p1 := p and p2i := σq ( p2i−1 ),

p2i+1 := σ p ( p2i )

(12.1)

for i ≥ 1. Evidently, p j ∈ A for every j ∈ N. All the points of the sequence ( p j ) j∈N belong to aff( p, q) in the following natural order (it means the order that corresponds to the order ≤ or ≥ in R by an isometry R → aff( p, q)): . . . , p2i+3 , p2i+1 , . . . , p, q, p2 , . . . , p2i+2 , p2i+4 , . . . . Hence by (12.1), p2i − p = 2iα

(12.2)

(compare Exercise 12.2), and thus limi→∞ p2i − p = ∞, contrary to the assumption that A is bounded. Let K1n be the family of sets in Kn that have a symmetry center; for every A ∈ K1n , the unique (by 12.1.3) symmetry center will be denoted by c0 (A). 12.1.4. THEOREM. For every A ∈ K1n and every afﬁne automorphism f of Rn , c0 ( f (A)) = f (c0 (A)). Proof. For any x, y ∈ Rn , f

x + y f (x) + f (y) = . 2 2

(12.3)

Indeed, let f¯ be the linear part of f : f¯ := f (x) − f (0); then f

x + y x + y f¯(x) + f¯(y) f (x) + f (y) = f (0) + f¯ = f (0) + = . 2 2 2 2

Since p = c0 (A) ⇐⇒ ∀x ∈ A ∃y ∈ A from (12.3) the assertion follows.

x+y = p, 2

12.2 Selectors and multiselectors

137

12.2 Selectors and multiselectors 12.2.1. DEFINITION. Let F be a nonempty family of convex subsets of Rn . A function φ : F → Rn is said to be a selector for F if ∀A ∈ F φ(A) ∈ A; n

a function : F → 2R is called a multiselector for F if ∀A ∈ F ∅ = (A) ⊂ A. Thus a selector chooses a point from each member of a given family, while a multiselector chooses a subset from each member. 12.2.2. EXAMPLE. Let F1 be the family of all convex subsets of Rn with at least one symmetry center, and let F2 be the subfamily of F1 consisting of n bounded sets. In view of 12.1.2, the function C0 : F1 → 2R described in Section 12.1 is a multiselector for F1 , while in view of 12.1.3, the function c0 : F2 → Rn is a selector for F2 . 12.2.3. DEFINITION. A selector φ : F → Rn is equivariant under a map f : Rn → Rn if ∀A ∈ F φ( f (A)) = f (φ(A)). 12.2.4. EXAMPLE. The selector c0 : K1n → Rn is equivariant under afﬁne automorphisms (compare 12.1.4). As we shall see, for compact, convex, centrally symmetric sets, c0 is the only selector equivariant under all the isometries: 12.2.5. THEOREM. If φ : Kn → Rn is equivariant under the isometries, then φ|K1n = c0 . Proof. Let A ∈ K1n and p = c0 (A). Then by assumption, σ p (φ(A)) = φ(σ p (A)) = φ(A), whence φ(A) is the ﬁxed point of the central symmetry σ p . Thus φ(A) = p.

Theorem 12.2.5 is a particular case of the following one (compare Exercise 12.4): 12.2.6. THEOREM. If a selector φ : Kn → Rn is equivariant under the isometries, then for every A ∈ Kn and every afﬁne subspace E σ E (A) = A ⇒ φ(A) ∈ E.

138

12. Selectors for Convex Bodies

12.3 Centers of gravity The notion of a center of gravity has its roots in physics. For a system of “material points,” i.e., points with weights, one looks for the point for which the system is in the equilibrium state. In terms of the geometry of Rn this can be described as follows. DEFINITION. Let k ∈ N. Let p1 , . . . , pk ∈ Rn , α1 , . . . , αk ≥ 0, and

12.3.1. k i=1 αi = 0. A point x 0 is the center of gravity of ( p1 , . . . , pk ) with the system of weights (α1 , . . . , αk ) if k

αi ( pi − x0 ) = 0.

(12.4)

i=1

12.3.2. THEOREM. Let p1 , . . . , pk ∈ Rn . For every sequence (α1 , . . . , αk ) of nonnegative numbers, at least one different from 0, there exists a unique center of gravity x0 of ( p1 , . . . , pk ) with the system of weights (α1 , . . . , αk ): k

x0 :=

αi pi

i=1 k

.

(12.5)

αi

i=1

Proof. Let us ﬁrst show that the point x0 deﬁned by (12.5) is a center of gravity of ( p1 , . . . , pk ) with the system of weights (α1 , . . . , αk ). Indeed, k

α j ( p j − x0 ) =

j=1

k j=1

αj pj −

1 k

· αi

k j=1

αj

k

αi pi = 0,

i=1

i=1

whence condition (12.4) is satisﬁed. Suppose now that there is another center of gravity y0 of ( p1 , . . . , pk ) with the same system of weights. Then k j=1

thus

k

j=1 α j (x 0

α j ( p j − x0 ) = 0 =

k

α j ( p j − y0 );

j=1

− y0 ) = 0. Hence x0 = y0 , because

k

j=1 α j

= 0.

This elementary notion of a center of gravity has a counterpart for an arbitrary measure space (X, A, µ), in particular for (Rn , B(Rn ), µ) with µ being an arbitrary Borel measure: 12.3.3. DEFINITION. A point x0 is the center of gravity of a set A ∈ B(Rn ) with respect to µ if

12.3 Centers of gravity

139

(x − x0 ) dµ(x) = 0. A

12.3.4. EXAMPLE. If A is ﬁnite, A = { p1 , . . . , pk }, µ is a nonzero measure, and µ({ pi }) = αi for i = 1, . . . , k, then the center of gravity of the set A with respect to µ is the center of gravity of the sequence ( p1 , . . . , pk ) with the system of weights (α1 , . . . , αk ). The following theorem is a generalization of 12.3.2. 12.3.5. THEOREM. Let µ be a Borel measure. For every A ∈ B(Rn ) with 0 < µ(A) < ∞, there is a unique center of gravity x0 of A with respect to µ: " x dµ(x) x0 := A . (12.6) µ(A) Proof. The point x0 deﬁned by (12.6) is a center of gravity of A with respect to µ. Indeed, 1 (x − x0 ) dµ(x) = x dµ(x) dµ(y)) · (µ(A) x dµ(x) − µ(A) A A A A 1 = x dµ(x) µ(A) − dµ(y) = 0. µ(A) A A Suppose there is another center of gravity y0 . Then (x − x0 ) dµ(x) = 0 = (x − y0 ) dµ(x), whence

"

A A (x 0

A

− y0 ) dµ(x) = 0. Thus x0 = y0 , because 0 < µ(A) < ∞.

B(Rn )

12.3.6. DEFINITION. For any set A ∈ and any Borel measure µ with 0 < µ(A) < ∞, let cµ (A) be the center of gravity of A with respect to µ. 12.3.7. LEMMA. If a Borel measure µ on Rn is invariant under an isometry f : Rn → Rn , then for every A ∈ B(Rn ) with 0 < µ(A) < ∞, cµ ( f (A)) = f (cµ (A)). The proof is left to the reader (Exercise 12.6). 12.3.8. THEOREM. If a Borel measure µ is invariant under isometries of Rn , then cµ |K0n is a selector for K0n . If µ = λn , this selector is equivariant under the afﬁne automorphisms. Proof. We are going to show that cµ (A) ∈ A for every A ∈ K0n . Let a = cµ (A). Suppose that a ∈ A. Then there exists a hyperplane H that separates a from A. Let α := 12 (a, A). In view of 12.3.7, we may assume that H = {(x1 , . . . , xn ) ∈ Rn | xn = α}, a = 0,

140

12. Selectors for Convex Bodies

and A ⊂ {(x1 , . . . , xn ) ∈ Rn | xn > α}. Then

xn dµ(x) > A

αdµ(x) = αµ(A) > 0. A

Since

x dµ(x) = (y1 , . . . , yn ), A

where yi =

xi dµ(x), A

by (12.6) it follows that a · µ(A) = 0, contrary to the assumption a = 0. Hence cµ |K0n is a selector. It remains to prove that for µ = λn , the Lebesgue measure, this selector is equivariant under any afﬁne automorphism f of Rn . For translations it is obvious; thus we may assume f to be a linear automorphism. Then Vn ( f (A)) = | det f |Vn (A), and

f (A)

y dλn (y) = | det f | f

x dλn (x) . A

Thus 1 cλn ( f (A)) = y dλn (y) | det f |Vn (A) f (A) 1 = x dλn (x) = f cλn (A). f Vn (A) A

12.3.9. EXAMPLE. Let A = (a0 , . . . , an ). Then 1 1 cλn (A) = c a0 , . . . , an ; , . . . , ; n n thus cλn (A) is the center of gravity of the sequence of vertices of the simplex A, with equal weights (compare Theorem 12.3.2 and Exercise 12.5). For a regular simplex this follows from 12.2.6, because this point is the point of intersection of all the hyperplanes of symmetry of such a simplex. Since every n-dimensional simplex is the image of a regular one under an afﬁne automorphism, for an arbitrary simplex it sufﬁces to apply 12.3.8. The center of gravity cλn is also called the centroid. The next example concerns another center of gravity.

12.3 Centers of gravity

141

12.3.10. EXAMPLE. (See [64], p. 305.) For every A ∈ K0n and i ∈ {0, . . . , n}, let pi (A) be the center of gravity of A with respect to the (n − i)-dimensional curvature measure of A: pi (A) := cn−i (A,·) (A). In particular, p0 (A) is the centroid of A, i.e., p0 (A) = cλn (A) (compare (10.6)). Since by 10.1.12, the curvature measures are invariant under isometries, from 12.3.8 it follows that for each i the map pi is a selector. The following theorem, which describes the position of the centroid of A, has been well known for many years; an idea of a proof was suggested in [10]. The proof given below can be found in [53]. 12.3.11. THEOREM. Let u ∈ S n−1 . (i) For every A ∈ K0n , b(A, u) ≤ (n + 1)(cλn (A), H (A, u)). (ii) If in particular, A is a cone with base contained in H (A, u), then the inequality is an equality. Proof. Let A ∈ K0n and x0 := cλn (A). For every t ∈ R, let Ht = {x ∈ Rn | x ◦ u = t}. If u = en and 0 ∈ A, then u ◦ x0 = =

1 Vn (A) 1 Vn (A)

h(A,u)

−h(A,−u) h(A,u) −h(A,−u)

t dλn−1 (x1 , . . . , xn−1 ) dt A∩Ht

t Vn−1 (A ∩ Ht ) dt.

(ii): Let A be a cone with base B and vertex 0. Then H0 = H (A, −u) and Hb(A,u) = H (A, u). Using two elementary properties of cones, Vn−1 (A ∩ Ht ) = Vn−1 (B) and Vn (A) =

t b(A, u)

n−1

1 Vn−1 (B)b(A, u), n

142

12. Selectors for Convex Bodies

we obtain n b(A, u)n

(x0 , H (A, −u)) = u ◦ x0 =

b(A,u)

t n dt =

0

n b(A, u). n+1

Thus (x0 , H (A, u)) =

1 b(A, u), n+1

which proves (ii). (i): Now let A be an arbitrary convex body. As above, we may assume that u = en and x0 = 0. Let D be a cone with base B ⊂ H (A, u), with vertex in A ∩ H (A, −u), and with D ∩ H0 = A ∩ H0 . Let y0 := cλn (D). In view of (ii), it sufﬁces to prove that (x0 , H (A, u)) ≥ (y0 , H (A, u)), or equivalently, u ◦ y0 ≥ 0. Indeed, it is easy to see that t Vn−1 (D ∩ Ht ) ≥ t Vn−1 (A ∩ Ht ) for every t ∈ [−h(A, −u), h(A, u)]; thus u ◦ y0 ≥

Vn (A) u ◦ x0 = 0. Vn (D)

We complete this section with a useful formula for the centroid of a convex body. 12.3.12. THEOREM. For every A ∈ K0n , cλn (A) =

1 n+1

S n−1

uρ A (u)n+1 dσ (u).

Proof. Since the centroid map is equivariant under translations, we may assume that A ⊂ {(x1 , . . . , xn ) ∈ Rn | xi ≥ 0 for i = 1, . . . , n}. Thus the desired formula follows from Theorem 7.3.5 by simple calculation.

12.4 The Steiner point An interesting example of a selector for Kn is the so-called Steiner point. The history of this notion can be found in [60].

12.4 The Steiner point

143

12.4.1. DEFINITION. For every A ∈ K0n , let 1 s(A) := uh A (u)dσ (u). κn S n−1 The point s(A) is the Steiner point of A. We mention without proof the following result (compare [64] (5.4.12) and [60]). 12.4.2. THEOREM. Let A ∈ K0n . The Steiner point of A is the center of gravity of A with respect to the curvature measure 0 (A, ·): s(A) = c0 (A,·) (A). From 12.2.4 combined with deﬁnition of the function pi (see 12.3.10), it follows that s = pn . (12.7) Thus by 12.3.8, the map s is a selector for K0n . The notion of the Steiner point has a simple geometric interpretation for polytopes: 12.4.3. THEOREM. Let P ∈ P0n and let {a1 , . . . , ar } be the set of vertices of P. Then r s(P) = γ0 (P, {ai }) · ai i=1

(compare Deﬁnition 10.1.10). Proof. Since the measure 0 (P, ·) is concentrated on the set of vertices of P, it sufﬁces to apply 10.1.11 and 12.4.2. n n The following theorem describes important properties of s : K → R ([60]). 12.4.4. THEOREM. The selector s is Minkowski additive, equivariant under similarities, and continuous. Proof. Additivity follows from Theorem 3.4.4 concerning support functions; the proof of continuity and equivariance under similarities is an easy exercise (Exercise 12.8). The following Schneider theorem ([60] Theorem 1) gives a solution of a modiﬁcation of a problem posed by Gr¨unbaum in [27]. We mention it without proof. 12.4.5. THEOREM. If a function φ : Kn → Rn is Minkowski additive, equivariant under the proper isometries, and continuous, then φ = s. Let us note that in view of 12.4.4, the conditions in Theorem 12.4.5 sufﬁcient for a function φ to coincide with s are also necessary. Thus they characterize the selector s. Moreover, as a consequence of these two theorems we obtain the following corollary.

144

12. Selectors for Convex Bodies

12.4.6. COROLLARY. For every φ : Kn → Rn , the following conditions are equivalent: (i) φ is additive, equivariant under similarities, and continuous, (ii) φ is additive, equivariant under proper isometries, and continuous, (iii) φ = s. The selector s is not only continuous but is Lipschitz continuous (compare [64] (1.8.3) and Note 15 p. 61): 12.4.7. THEOREM. (see [56]) For every A, B ∈ Kn , s(A) − s(B) ≤

2κn−1 H (A, B). κn

This estimate is optimal.

12.5 Center of the minimal ring The notion of the minimal ring containing the boundary of a convex body in Rn appeared in the literature in 1924 for n = 2 ([8]). Its short history can be found in the paper by Imre B´ar´any (see [2]). For every A ∈ K0n and x ∈ A, let R A (x) := inf{α > 0 | B(x, α) ⊃ A}

(12.8)

r A (x) := sup{α > 0 | B(x, α) ⊂ A}.

(12.9)

and 12.5.1. LEMMA. (i) The function R A : A → R is convex. (ii) If for distinct x0 , x1 ∈ A, RA

x + x 1 0 1 = (R A (x0 ) + R A (x1 )), 2 2

then there is a unique point p ∈ bdA ∩ aff(x0 , x1 ) such that # x + x x0 + x1 # 0 1 # # and p − xi = R A (xi ) for i = 0, 1. #p − # = RA 2 2 (iii) The function r A : A → R is concave. Proof. (i): Let x0 , x1 ∈ A and xt = (1 − t)x0 + t x1 for t ∈ [0; 1]. We have to prove that R A (xt ) ≤ (1 − t)R A (x0 ) + t R A (x1 ). (12.10) By deﬁnition of R A (see (12.8)), there exists p ∈ bdA such that R A (xt ) = xt − p.

(12.11)

12.5 Center of the minimal ring

145

Since A ⊂ B(xi , R A (xi )) for i = 0, 1, it follows that p ∈ B(xi , R A (xi )) and thus p − xi ≤ R A (xi ) for i = 0, 1. Hence by (12.11), R A (xt ) = (1 − t)(x0 − p) + t (x1 − p) ≤ (1 − t)x0 − p + tx1 − p ≤ (1 − t)R A (x0 ) + t R A (x1 ), i.e., (12.10) holds. (ii): Evidently, the equality in (12.10) occurs if and only if there are only equalities in the last formula, i.e., (1 − t)(x0 − p) + t (x1 − p) = (1 − t)x0 − p + tx1 − p = (1 − t)R A (x0 ) + t R A (x1 ). Thus p ∈ aff(x0 , x1 ) and p − xi = R A (xi ) for i = 0, 1. This implies the uniqueness of p. (iii): We have to prove that r A (xt ) ≥ (1 − t)r A (x0 ) + tr A (x1 )

(12.12)

for every x0 , x1 ∈ A.

Figure 12.1.

Obviously, B(x0 , r A (x0 )) ∪ B(x1 , r A (x1 )) ⊂ A. Let us observe that (Figure 12.1) B(xt , (1 − t)r A (x0 ) + tr A (x1 )) ⊂ conv(B(x0 , r A (x0 )) ∪ B(x1 , r A (x1 ))), because for any convex X, Y ,

146

12. Selectors for Convex Bodies

conv(X ∪ Y ) =

(1 − t)X + tY

t∈[0;1]

(compare Exercise 3.9). Thus B(xt , (1 − t)r A (x0 ) + tr A (x1 )) ⊂ A,

which yields (12.12). K0n

12.5.2. THEOREM (I. B´ar´any [2]). For every A ∈ there is a unique minimizer x0 of the function R A − r A : A → R. Proof. By Lemma 12.5.1, the function R A is convex and r A is concave; hence R A −r A : A → R is convex. The functions R A and r A can be extended over Rn to a convex function R¯ A and concave r¯ A (see [2] and [46]). Since R¯ A − r¯ A : Rn → R is convex, it follows that it is continuous (compare 1.5.1 in [64]). Thus R A −r A , as a continuous function on a compact set A, attains its lower bound at some point x0 of A. It remains to prove the uniqueness. Suppose that there exist two minimizers, x0 and x1 , and let α := R A (xi ) − r A (xi ) We shall show that RA

for i = 0, 1.

x + x x + x 0 1 0 1 − rA < α. 2 2

(12.13)

Suppose, to the contrary, that (12.13) does not hold; then x + x x + x 0 1 0 1 RA − rA = α, 2 2 because R A − r A is convex by 12.5.1 (i). Then by 12.5.1 (i), (iii), x0 , x1 satisfy the assumption of 12.5.1 (ii), whence there is a unique p ∈ bdA ∩ aff(x0 , x1 ) such that # x + x (x0 + x1 ) # 0 1 # # and p − xi = R A (xi ) for i = 0, 1. #p − # = RA 2 2 We may assume that x1 is between x0 and p (Figure 12.2); since r A (x0 ) − r A (x1 ) = R A (x0 ) − R A (x1 ), it follows that B(x1 , r A (x1 )) ⊂ B(x0 , r A (x0 )), and thus x0 − x1 = r A (x0 ) − r A (x1 ). Hence there is exactly one point q ∈ bdA ∩ (x1 , p) with q − x1 = r A (x1 ). But conv(B(x0 , r A (x0 ))∪{ p}) is a subset of A and q belongs to the interior of this convex hull; thus q ∈ intA, a contradiction. So, we have proved the inequality (12.13). From this inequality we infer that inf (R A (x) − r A (x)) < α,

x∈A

contrary to the assumption.

12.5 Center of the minimal ring

147

Figure 12.2. The ﬁgure illustrates an impossible situation; evidently p − x1 = R A (x1 ).

Let c(A) be the unique point x0 whose existence is ensured by Theorem 12.5.2, and let R(A) := R A (c(A)) and r (A) = r A (c(A)). (12.14) Let us consider the set ringA obtained from the ball B(c(A), R(A)) circumscribed about A by deleting the interior of the ball B(c(A), r (A)) inscribed in A: ringA := B(c(A), R(A)) \ intB(c(A), r (A)).

(12.15)

The set ring A is called the minimal ring of A, and the point c(A) is called the center of the minimal ring. Of course, the function c : K0n → Rn is a selector for K0n . Directly from the deﬁnition of center of the minimal ring we obtain (compare Exercise 2.4) 12.5.3. PROPOSITION. The selector c is equivariant under similarities. The following theorem is a consequence of 12.5.3 combined with 12.2.6. 12.5.4. THEOREM. If a convex body A is symmetric with respect to an afﬁne subspace E, then c(A) ∈ E. In particular, we obtain the following. 12.5.5. COROLLARY. If a convex body A is centrally symmetric, then c(A) is the center of symmetry. I. B´ar´any in [2] gave a criterion that allows one to check whether a point of A is the center of the minimal ring. We mention it without proof. 12.5.6. THEOREM ([2] Theorem 2). Let A ∈ K0n and x0 ∈ Rn . A point x0 is the center of the minimal ring of A if and only if there exist p1 , . . . , pk ∈ bdA ∩ bdB(x0 , R A (x0 )) and q1 , . . . , ql ∈ bdA ∩ bdB(x0 , r A (x0 )) such that p −x q − x j 0 i 0 conv | i = 1, . . . , k ∩ conv | j = 1, . . . , l = ∅. (12.16) R A (x0 ) r A (x0 )

148

12. Selectors for Convex Bodies

Applying 12.5.6, B. Zdrodowski found the center of the minimal ring for an arbitrary triangle in R2 : 12.5.7. THEOREM ([68]). For any triangle T in R2 , the point c(T ) is the intersection point of the bisectrix of the longest side and the bisectrix of the smallest angle of T . Proof. Let T = (a, b, c). Assume that a − b ≥ max{a − c, b − c} and that the measure of the angle at the vertex a is not greater than the measures of the two remaining angles. Let x0 be the point of intersection of the bisectrix of (a, b) and the bisectrix of the angle at a. We deﬁne p1 , p2 , q1 , q2 : p1 := a, p2 := b, q1 := ξaff(a,b) (x0 ), q2 := ξaff(a,c) (x0 ). It sufﬁces to show that these points satisfy condition (12.16). To this end let us observe that RT (x0 ) = pi − x0 and r T (x0 ) = q j − x0 . Without any loss of generality we may assume that RT (x0 ) = 1 and x0 = 0 (we use Exercise 12.7). Then condition (12.16) reads as q q2 1 (a, b) ∩ , = ∅, r A (0) r A (0) so evidently it is satisﬁed (Figure 12.3). The selector c is continuous but is not additive ([46]):

12.5.8. THEOREM. Let A, Ak ∈ K0n for k ∈ N. Then lim Ak = A ⇒ lim c(Ak ) = c(A). H

12.5.9. THEOREM. The function c : K0n → Rn is not Minkowski additive. Proof. Suppose the selector c is additive. Since by 12.5.3 it is equivariant under isometries, and by 12.5.8 is continuous, in view of Theorem 12.4.5 it coincides with the Steiner point map: for every A ∈ K0n ,

c(A) = s(A). We shall prove the opposite. Let n = 2 and let A := (a, b, c), where (acb) = π2 , a − c = b − c, and c(A) = 0. By 12.5.7 the point c(A) is the center of the circle inscribed in A; it is easy to check that √ 2 c = a + b. (12.17) 2

12.6 Pseudocenter. G-pseudocenters

149

Figure 12.3.

On the other hand, by Theorem 12.4.3, c =

3π 4 (a

+ b) + π2 c = 0, whence

3 a + b, 2

contrary to (12.17).

It was recently proved by I. Herburt that the center c of the minimal ring is not Lipschitz continuous (see [32]). Finally, let us mention the following application of minimal rings (see [6]). 12.5.10. THEOREM. For every A ∈ K0n , there is a unique ball nearest to A with respect to H . It has center c(A) and radius 12 (R(A) + r (A)). The particular case of this result for n = 2 was already proved by H. Lebesgue in 1921 (see [9], p.52).

12.6 Pseudocenter. G-pseudocenters In 1950 I. F´ary and L. R´edei proved (see [17]) that for every convex body A in Rn there is a unique centrally symmetric convex body contained in A with maximal volume: 12.6.1. THEOREM. For every A ∈ K0n there is a unique C ∈ K1n such that C⊂A

and

Vn (C) ≥ Vn (X ) for every X ∈ K1n with X ⊂ A.

150

12. Selectors for Convex Bodies

Proof. For any convex body A and point x ∈ A, let A x be the intersection of A with its image under the symmetry σx : A x := A ∩ σx (A).

(12.18)

The set A x is symmetric with respect to x and contains every subset of A symmetric with respect to x. Thus it sufﬁces to prove that there exists a unique x ∈ A at which the function ψ : A → R deﬁned by ψ(x) := Vn (A x )

(12.19)

attains its lower bound. The existence of such a point follows from the compactness of A and continuity of ψ (the function Vn is continuous by Theorem 7.3.3; it is easy to prove that the function A x → A x ∈ Kn is continuous (Exercise 12.10)). Hence, it remains to prove the uniqueness. Suppose x1 = x2 and ψ(x1 ) = ψ(x2 ) ≥ ψ(x) for every x ∈ A. Let x0 =

1 2 (x 1

(12.20)

+ x2 ). Then A x0 ⊃

1 (A x1 + A x2 ) 2

(12.21)

(Exercise 12.11). Thus by 5.2.2 (the Brunn–Minkowski inequality) combined with (12.19) and (12.20), ψ(x0 ) ≥

1 1 1 (ψ(x1 ) n + ψ(x2 ) n )n ≥ ψ(x0 ). 2n

Hence in the last formula, both inequalities are equalities, which corresponds to the equality in the Brunn–Minkowski inequality for A x1 and A x2 . By Theorem 6.1.1 in [64], the set A x2 is the image of A x1 under a homothety: A x2 = λA x1 + v. But λ = 1, because the volumes of these sets are equal, and v = 0, because x1 = x2 (Exercise 12.12). Let B := conv(A x1 ∪ A x2 ). Obviously, B ∈ K1n , B ⊂ A, and Vn (Ac0 (B) ) ≥ Vn (B) > Vn (A xi ) for i = 1, 2, contrary to (12.20). The unique convex body C ∈ K1n whose existence is ensured by Theorem 12.6.1 will be called the symmetric kernel of A; the symmetry center of C will be called the pseudocenter of A.1 In view of 12.6.1 and 12.1.3, every convex body A in Rn has exactly one pseudocenter; we denote it by p(A). By 12.1.2, p(A) ∈ A. Thus we have the following. 1 Our terminology differs from that in [17].

12.6 Pseudocenter. G-pseudocenters

151

12.6.2. PROPOSITION. The function p : K0n → Rn is a selector for K0n . The following proposition is easy to prove (see Exercise 12.14): 12.6.3. PROPOSITION. The selector p is equivariant under afﬁne automorphisms of Rn . 12.6.4. EXAMPLE. The pseudocenter of any simplex A coincides with its centroid; thus according to 12.3.10, p(A) = p0 (A). Indeed, by the afﬁne equivariance of the centroid and of the pseudocenter, we may assume A to be regular. Then by 12.2.6, the assertion follows from the equivariance of both selectors under isometries. Let us observe that the problem of existence of a centrally symmetric convex body with the maximal volume contained in A ∈ K0n is a particular case of the following one, which was solved in [52], Theorem 3.8 (see Theorem 12.6.7 below). 12.6.5. PROBLEM. Let G be a subgroup of O(n). Is it true that for every A ∈ K0n there is a unique convex body C contained in A with maximal volume, invariant under τ Gτ −1 for some τ ∈ Tr? Indeed, Theorem 12.6.2 gives a positive answer to this question for G = {idRn , σ0 }, because C is symmetric with respect to x if and only if σx (C) = C (recall that σx = τx σ0 τx−1 , where τx is a translation by x). If a set is invariant under τ Gτ −1 for some τ ∈ Tr, we shall say that it is invariant under G up to a translation, or simply (if it does not lead to confusion) G-invariant. For any group G ⊂ O(n), any convex body A, and a point x ∈ A, let A x,G := τx gτx−1 (A). (12.22) g∈G

(Compare (12.18).) 12.6.6. PROPOSITION. For every group G ⊂ O(n) and any convex bodies A, C in Rn , the following conditions are equivalent: (i) C is a maximal (with respect to the volume) G-invariant convex body contained in A; (ii) there exists p ∈ A such that C = A p,G , and the function ψG : A → R deﬁned by ψG (x) := Vn (A x,G ) (12.23) attains its upper bound at p. (See Exercise 12.15.)

152

12. Selectors for Convex Bodies

Every convex body C satisfying condition (i) (and so also (ii)) of 12.6.6 will be called a G-kernel of A, and every point p for which A p,G is a G-kernel of A will be called a G-pseudocenter of A. The set of G-pseudocenters of A will be denoted by PG (A). 12.6.7. THEOREM. For any nontrivial subgroup G of O(n), the following conditions are equivalent: (i) every A ∈ K0n has a unique G-kernel; (ii) G = {idRn , σ0 }. For the implication (ii) ⇒ (i) see Theorem 12.6.1. Proof of the converse implication is omitted.2 The following problem is open. 12.6.8. PROBLEM. Characterize the class of convex bodies that have a unique G-kernel for every subgroup G of O(n). The following theorem gives a partial solution of this problem (see [52] Theorem 3.9). 12.6.9. THEOREM. If A is strictly convex, then for every nontrivial subgroup G of O(n) there is a unique G-kernel of A. Proof. Suppose A has at least two G-kernels, C0 and C1 . By 12.6.6, they are of the form Ci = A pi ,G for some p0 , p1 ∈ PG (A). Let pt := (1 − t) p0 + t p1 for t ∈ [0; 1] and Ct := (1 − t)C0 + tC1 . Then A pt ,G ⊃ Ct

(12.24)

(see [52]; compare Exercise 12.17). From (12.24) and the Brunn–Minkowski inequality (Theorem 5.2.2) it follows that 1 1 n ψG ( pt ) ≥ (1 − t)ψG ( p0 ) n + tψG ( p1 ) n . Since ψG ( p0 ) = ψG ( p1 ), we obtain ψG ( pt ) ≥ ψG ( pi )

(12.25)

for i = 1, 2. But the function ψG attains its upper bound at pi , whence in (12.25) we have equality. This corresponds to the equality in the Brunn–Minkowski inequality for C0 and C1 . Since these two sets have equal volumes, it follows that C1 = C0 + v for some v = 0. Since A is strictly convex, for every c ∈ C0 the relative interior of (c, c + v) is contained in intA, and thus C 1 ⊂ intA. 2

Let 2 In the proof of Theorem 3.8 in [52], in the last formula and in (3.4), the sign “=” should be replaced

by “≤”.

12.6 Pseudocenter. G-pseudocenters

153

ε := dist C 1 , bdA and C := C 1 + ε B n . 2 2 Then C is a G-invariant subset of A and Vn (C) > Vn C 1 = Vn (Ci ) (as ε > 0), 2 contrary to the assumption. In the case of the group G = σ0 ! (i.e., the group generated by the central symmetry σ0 ), for the (unique) G-kernel of A there exists a unique G-pseudocenter, p(A). For an arbitrary G the situation may be much more complicated: 12.6.10. EXAMPLE. Let G = σ L ! for a line L being an axis of symmetry of a convex body A. Then A is a unique G-kernel of itself, while every point of L ∩ A is a G-pseudocenter of A. Hence, the following problem arises: 12.6.11. PROBLEM. When is PG (A) a singleton? A partial answer to the question 12.6.11 is given by Theorem 12.6.13, which follows from Theorem 12.6.9 combined with the following lemma (compare 3.10 in [52]): 12.6.12. LEMMA. Let A ∈ K0n and let G be an arbitrary subgroup of O(n). If 0 is the unique ﬁxed point of the group G, then for every x, y ∈ PG (A), A x,G = A y,G ⇒ x = y. Proof. Suppose x = y. The set PG (A) is convex (Exercise 12.16), whence its intersection with A x,G is convex as well, and thus (x, y) is contained in this intersection. Therefore, we may assume that x = −y, because otherwise, x, y can be replaced by points of the segment (x, y) that satisfy this condition. By the assumption on G, there exists g ∈ G with g(x + y) = x + y.

(12.26)

Since for every p ∈ A the set A p,G is invariant under the map g p := τ p gτ p−1 , it follows that A x,G (which coincides with A y,G ) is invariant under both gx and g y ; thus it is also invariant under f := gx g −1 y . However, let us observe that for every z ∈ Rn , f (z) = z + (x + y) − g(x + y), i.e., f is a translation by the vector v = (x + y)−g(x + y). Since A x,G is compact, it follows that v = 0. Hence g(x + y) = x + y, contrary to (12.26). 12.6.13. THEOREM. Let G be a subgroup of O(n). If 0 is the only ﬁxed point of the group G, then every strictly convex body has a unique G-pseudocenter. If PG (A) is a singleton, then the unique G-pseudocenter of A will be denoted by pG (A). Hence we arrive at the following.

154

12. Selectors for Convex Bodies

12.6.14. COROLLARY. Let G be a subgroup of O(n). If 0 is the only ﬁxed point of the group G, then the function pG restricted to the family of strictly convex bodies in Rn is a selector for this family. Let us mention that the G-pseudocenter is always equivariant under translations (compare Proposition 2.1 in [53]), but generally, it is not equivariant with respect to an arbitrary isometry of Rn (see Example 2.2 in [53]). For this reason, Rolf Schneider suggested that one consider a modiﬁed notion, which was studied in [53]. We end this section with the following theorem, which says that if G is a subgroup of O(n) with only one ﬁxed point 0, then for most convex bodies in Rn there is a unique G-pseudocenter. To be more precise, let us recall that a subset of a topological space is meager if it is a countable union of nowhere dense sets, while a residual set is the complement of a meager set (see [64], p. 119). 12.6.15. THEOREM. Let X be a subclass of K0n deﬁned as follows: A ∈ X if and only if cardPG (A) = 1 for every subgroup G of O(n) with the unique ﬁxed point {0}. Then X is residual in Kn . Proof. By Theorem 12.6.13, all strictly convex bodies in Rn belong to X . Since, in view of Theorem 2.6.1 in [64], the class of strictly convex bodies is residual in Kn , it follows that X is residual.

12.7 G-quasi-centers. Chebyshev point In [17], F´ary and R´edei asked a question that can be formulated as follows: Does the statement 12.6.1 remain true if the centrally symmetric convex body C ⊂ A with maximal volume is replaced by a centrally symmetric convex body containing A with minimal volume? They noticed that the answer is negative (Example 12.7.2). Any centrally symmetric convex body C containing A with minimal volume will be called a centrally symmetric hull of A, and the symmetry center of C will be called a quasi-center of A. Let Q(A) be the set of quasi-centers of A. 12.7.1. PROPOSITION. (i) For every A ∈ Kn and x ∈ A, the set conv(A ∪ σx (A)) is the smallest convex body symmetric at x, containing A. (ii) The function x → Vn (conv(A ∪ σx (A))) is continuous. Proof. (i) is obvious; (ii) follows from 3.2.11 combined with 7.3.3. 12.7.2. EXAMPLE. Let A be a triangle, A = (a0 , a1 , a2 ) ⊂ R2 . There exists inﬁnitely many centrally symmetric convex bodies with minimal area containing A. The set Q(A) is the triangle with vertices bk := 12 (ai +a j ) for distinct i, j, k ∈ {0, 1, 2}.

12.7 G-quasi-centers. Chebyshev point

155

Indeed, let X := (b0 , b1 , b2 ) and x ∈ A (Figure 12.4).

Figure 12.4.

If x ∈ intX , then conv(A ∪ σx (A)) is the hexagon with vertices a0 , σx (a0 ), a1 , σx (a1 ), a2 , σx (a2 ). Its area equals 2V2 (A). By 12.7.1 (ii), the same holds for x ∈ bdX . If x ∈ X , then x ∈ (ai , b j , bk ) for distinct i, j, k, and the set conv(A∪σx (A)) is a parallelogram with area ai − ak · 2(x, (a j , ak )) > 2V2 (A). Hence X = Q(A). One may suspect that triangles (or generally, simplices) are the only convex bodies with more than one quasi-center. However, it turns out that the situation is different: ˙ 12.7.3. EXAMPLE. (T. Zukowski) For every odd n ≥ 3 there exists a convex 2 n-gon in R with an inﬁnite set of quasi-centers; for instance, any regular n-gon has this property. The proof is left to the reader (Exercise 12.19). Let us consider the following more general problem, in some sense dual to 12.6.5. 12.7.4. PROBLEM. Let G be a subgroup of O(n). Is it true that for every A ∈ K0n there exists a unique G-invariant (up to a translation) convex body with minimal volume, containing A? Examples 12.7.2 and 12.7.3 show that for G = σ0 ! the answer to this question is negative. The following theorem gives the positive answer for G = O(n). 12.7.5. THEOREM. For every A ∈ Kn there is a unique ball with minimal volume containing A. Its center belongs to A. Proof. Suppose that there are two such minimal balls, B1 and B2 , Bi = B(ai , r ) for i = 1, 2.

156

12. Selectors for Convex Bodies

Let C := B1 ∩ B2 and let H be the bisectrix hyperplane of the segment (a1 , a2 ). The set H ∩ C is an (n − 1)-dimensional ball; its center c is the symmetry center of C, and its radius r0 is smaller than r (Figure 12.5). The n-dimensional ball B0 := B(c, r0 ) contains C, and so it contains A (Exercise 12.18). Since r0 < r , it follows that Vn (B0 ) < Vn (Bi ) for i = 1, 2, contrary to the assumption.

Figure 12.5.

Further, for any x ∈ Rn , let R A (x) be the radius of the minimal ball with center x, containing A (compare (12.8)). Observe that if x ∈ / A, then there is a y ∈ Rn such that R A (x) > R A (y).

(12.27)

Indeed, for some α, a support hyperplane H of B(x, α) separates x from A; if y ∈ H ∩ B(x, α), then for every z ∈ A, x − z > y − z, which implies (12.27). It is easy to check that by the continuity of R A ([46]), this completes the proof. The center of the ball with minimal radius (i.e., of the minimal volume) containing a compact convex set A is called the Chebyshev point of A. We denote it by c(A). ˇ As a direct consequence of 12.7.5, we obtain 12.7.6. COROLLARY. The map cˇ : Kn → Rn is a selector for Kn . It is easy to prove that the map cˇ is equivariant under similarities (Exercise 12.20). Notice that {c(A)} ˇ is the unique singleton nearest to A in the sense of the Hausdorff metric (compare Exercise 12.21).3 Generally, the following notion of G-quasi-center is a natural counterpart of that of G-pseudocenter: 3 For related results see [6].

12.7 G-quasi-centers. Chebyshev point

157

12.7.7. DEFINITION. Let G be a subgroup of O(n). A point x ∈ A is a Gquasi-center of a convex body A if there exists a convex body with minimal volume containing A that is invariant under the group τx Gτx−1 . Let Q G (A) be the set of G-quasi-centers of A. If Q G (A) is a singleton, let qG (A) be the unique G-quasi-center. Thus in particular, cˇ = q O(n) . The following statement is a generalization of 12.7.1. 12.7.8. PROPOSITION. Let G be a subgroupof O(n). (i) For every A ∈ Kn and x ∈ A, the set conv g∈G g(A−x)+x is the smallest −1 convex body invariant with respect A. to τx Gτx containing (ii) The function x → Vn conv g∈G g(A − x) + x is continuous. Proof is analogous to that of 12.7.1. As a consequence of Proposition 12.7.8, we obtain 12.7.9. THEOREM. For every subgroup G of O(n) and every A ∈ K0n , Q G (A) = ∅. The following problem is open: 12.7.10. PROBLEM. Is it true that there is no proper subgroup G of O(n) such that Q G (A) is a singleton for every A ∈ K0n ? We shall return to selectors for convex bodies in Chapters 13 and 16. Section 13.5 deals with the Santal´o point map, which is deﬁned in terms of polarity. In Section 16.1 we consider radial center maps, which are restrictions to K0n of some multiselectors for star bodies.

13 Polarity

Among transformations of Kn in Kn (compare Chapter 4) a particular role is played by polarity. For this reason we devote a separate chapter to this operation. Let us ﬁrst brieﬂy recall the notion of a polar hyperplane. In fact, this notion belongs to the theory of algebraic sets in n-dimensional projective space. In that theory the polar hyperplane of a point with respect to an algebraic manifold of degree 2 is deﬁned ([11], [50]). However, the only manifolds we need are spheres, hence we shall deal with Rn , which can be treated as the set of proper points in projective space.

13.1 Polar hyperplane of a point with respect to the unit sphere We begin with the equation of the hyperplane tangent to a sphere S at a, i.e., the support hyperplane of the ball B with bdB = S at that point. 13.1.1. PROPOSITION. Let S be the sphere in Rn with center x0 and radius r , S = r · S n−1 + x0 , and let a ∈ S. Then the hyperplane tangent to S at a is described by the equation (a − x0 ) ◦ (x − x0 ) = r 2 .

(13.1)

This well-known fact can be proved by means of algebraic methods (compare [50]) or methods of differential geometry, but it can also be proved as follows.

160

13. Polarity

Proof. Since S is symmetric with respect to every line passing through x0 , it easily follows that the hyperplane tangent at a (as the union of lines that have only the point a in common with S) is orthogonal to the vector a − x0 . Thus it is described by the equation (a − x0 ) ◦ (x − a) = 0, which is equivalent to (a − x0 ) ◦ (x − x0 ) = r 2 because a − x0 = r . 13.1.2. DEFINITION. Let S := bdB(x0 , r ) for some r > 0 and x0 ∈ Rn . For every a ∈ Rn \ {x0 }, B S (a) := {x ∈ Rn | (a − x0 ) ◦ (x − x0 ) = r 2 }. The hyperplane B S (a) is called the polar hyperplane of a with respect to S. Directly from Deﬁnition 13.1.2 it follows that b ∈ B S (a) ⇒ a ∈ B S (b).

(13.2)

This simple property can be used to construct the polar hyperplane of any point a ∈ Rn \ {x0 } with respect to S (Figure 13.1).

Figure 13.1.

Let C S (a) be the cone with vertex a tangent to S, i.e., the union of all lines tangent to S and passing through a. 13.1.3. THEOREM. Let S = bdB(x0 , r ). (i) If a ∈ S, then B S (a) is the hyperplane tangent to S at a. (ii) If a ∈ Rn \ B(x0 , r ), then B S (a) = aff(C S (a) ∩ S). (iii) If a ∈ int B(x0 , r ) \ {x0 }, hyperplanes H1 , . . . , Hn intersect at a, and Hi ∩ S = C S (bi ) ∩ S for i = 1, . . . , n, then B S (a) = aff(b1 , . . . , bn ). Proof. (i) follows directly from 13.1.1 combined with 13.1.2. (ii): Let us observe that

13.2 Polarity for arbitrary subsets of Rn

C S (a) ∩ S = B S (a) ∩ S.

161

(13.3)

Indeed, x ∈ C S (a) ∩ S ⇐⇒ (x − x0 ) ◦ (a − x) = 0 and (x − x0 ) ◦ (x − x0 ) = r 2 ⇐⇒ (x − x0 ) ◦ (a − x0 ) = r 2 and (x − x0 ) ◦ (x − x0 ) = r 2 ⇐⇒ x ∈ B S (a) ∩ S. By (13.3), aff(C S (a) ∩ S) ⊂ B S (a).

(13.4)

r2 a−x0

Since dist(x0 , B S (a)) = < r , it follows that the set B S (a) ∩ S (and thus also C S (a) ∩ S) is a sphere of dimension n − 2, whence it generates a hyperplane. Thus in (13.4) equality holds. (iii): Condition (13.3) is satisﬁed for every point outside the ball B(x0 , r ), in particular for bi : C S (bi ) ∩ S = B S (bi ) ∩ S; hence Hi = B S (bi ) for i = 1, . . . , n. Therefore, a ∈ B S (bi ), and so by (13.2), bi ∈ B S (a) for i = 1, . . . , n. Thus aff(b1 , . . . , bn ) ⊂ B S (a). It remains to show that dim aff(b1 , . . . , bn ) = n − 1, which is equivalent to det(b1 − x0 , . . . , bn − x0 ) = 0.

(13.5)

Since the system of equations (x − x0 ) ◦ (bi − x0 ) = 0 for i = 1, . . . , n, n T T which describes i=1 Hi , has the matrix n ((b1 − x0 ) , . . . , (bn − x0 ) ), condition (13.5) follows from the assumption i=1 Hi = {a}.

13.2 Polarity for arbitrary subsets of Rn For arbitrary nonempty subsets of Rn , polarity is deﬁned as follows. 13.2.1. DEFINITION. Let A ⊂ Rn , A = ∅. The set A∗ deﬁned by the formula A∗ := {x ∈ Rn | ∀a ∈ A x ◦ a ≤ 1} is polar to A; the function A → A∗ is the polarity. The relationship between the notion of polarity and that of polar hyperplane is evident (see 13.2.2). For every a ∈ Rn , let B(a) := B S n−1 (a) and E(a) := {x ∈ Rn | x ◦ a ≤ 1}.

(13.6)

and for a = 0, the set E(a) is the closed half-space with Then E(0) = bdE(a) = B(a) and 0 ∈ E(a). Rn ,

162

13. Polarity

13.2.2. PROPOSITION. For every nonempty subset A of Rn , A∗ = {E(a) | a ∈ A} = {E(a) | a ∈ A \ {0}}. 13.2.3. EXAMPLES. (a) {a}∗ = E(a); (b) (B n )∗ = B n = (S n−1 )∗ ; (c) (Rn )∗ = {0}. (Compare Exercise 13.2.) The following three theorems describe basic properties of polarity. 13.2.4. THEOREM. For all nonempty subsets A1 , A2 , and A of Rn , (i) A1 ⊂ A2 ⇒ A∗1 ⊃ A∗2 ; (ii) (A1 ∪ A2 )∗ ⊂ A∗1 ∩ A∗2 ; (iii) A ⊂ A∗∗ . Proof. (i) follows from 13.2.2; since Ai ⊂ A1 ∪ A2 for i = 1, 2, (i) implies (ii). (iii): y ∈ A ⇒ ∀x ∈ A∗ x ◦ y ≤ 1 ⇒ y ∈ A∗∗ . 13.2.5. THEOREM. Let ∅ = A ⊂ (i) For every t > 0

Rn .

1 ∗ A . t (ii) For every linear isometry f : Rn → Rn , (t A)∗ =

f (A∗ ) = ( f (A))∗ . Proof. (i): Since (t x) ◦ a = t (x ◦ a) = x ◦ (ta), it follows that (t A)∗ = {x ∈ Rn | ∀a ∈ A (t x) ◦ a ≤ 1} 1 1 = {t x ∈ Rn | ∀a ∈ A (t x) ◦ a ≤ 1} = A∗ . t t (ii): In turn, for every f ∈ O(n), f (x) ◦ f (a) = x ◦ a; thus ( f (A))∗ = { f (x) ∈ Rn | ∀a ∈ A f (x) ◦ f (a) ≤ 1} = f {x ∈ Rn | ∀a ∈ A x ◦ a ≤ 1} = f (A∗ ). 13.2.6. THEOREM. Let A be a nonempty subset of Rn . (i) If A is bounded, then 0 ∈ intA∗ . (ii) If 0 ∈ intA, then A∗ is bounded.

13.3 Polarity for convex bodies

163

Proof. (i): Let A ⊂ r B n for some r > 0. Then by 13.2.4 (i), 13.2.5.(i), and 13.2.3 (b), 1 1 A∗ ⊃ (B n )∗ = B n , r r whence 0 ∈ intA∗ . (ii): Let 0 ∈ intA. Then r B n ⊂ A for some r > 0; thus by 13.2.4 (i), 13.2.3 (b), and 13.2.5 (i), 1 A∗ ⊂ (r B n )∗ = B n , r whence A∗ is bounded.

13.3 Polarity for convex bodies n of convex bodies with We shall now restrict our considerations to the family K00 0 in the interior n K00 := {A ∈ Kn | 0 ∈ intA}.

From 13.2.2 it follows that for any nonempty A ⊂ Rn the set A∗ is closed and convex. Hence, 13.2.6 yields the following. n ⇒ A∗ ∈ Kn . 13.3.1. THEOREM. A ∈ K00 00

We shall now prove that polarity is idempotent: 13.3.2. THEOREM. If A ∈ K0n and 0 ∈ A, then A∗∗ = A. Proof. In view of 13.2.4 (iii), it sufﬁces to prove that A∗∗ ⊂ A.

(13.7)

Let x ∈ Rn \ A. Since A is compact, there exists a hyperplane H such that A and x are in different open half-spaces determined by H . Since 0 ∈ A and H = {y ∈ Rn | y ◦ v = 1} for some v = 0, it follows that A ⊂ {y ∈ Rn | y ◦ v < 1}

(13.8)

x ◦ v > 1.

(13.9)

and By (13.8) we infer that v ∈ A∗ , whence by (13.9), x ∈ A∗∗ . This completes the proof of (13.7).

164

13. Polarity

n is continuous with respect to As we shall see, the operation ∗ restricted to K00 the Hausdorff metric.1 We need the following lemma. n and u ∈ S n−1 , 13.3.3. LEMMA. For every A ∈ K00

h(A∗ , u) = sup

v∈S n−1

u◦v . h(A, v)

Proof. Since h(A, x) = supa∈A a ◦ x, it follows that x ∈ A∗ ⇐⇒ ∀a ∈ A x ◦ a ≤ 1 ⇐⇒ h(A, x) ≤ 1. If v =

x x ,

then h(A, v) =

1 x h(A, x).

Hence for every u ∈ S n−1 , x ◦u v◦u = sup . h(A, x) h(A, v) n−1 x =0 v∈S

h(A∗ , u) = sup{x ◦ u | h(A, x) ≤ 1} = sup

n is continuous with respect to . 13.3.4. THEOREM. Polarity restricted to K00 H n Proof. Let A1 , A2 ∈ K00 . There exists r > 0 such that

r B n ⊂ A1 ∩ A2 . We are going to prove that H ((A1 )∗ , (A2 )∗ ) ≤

1 H (A1 , A2 ). r2

(13.10)

By 3.4.10, there exists u ∈ S n−1 satisfying the condition H ((A1 )∗ , (A2 )∗ ) = |h((A1 )∗ , u) − h((A2 )∗ , u)|; without any loss of generality we may assume that H ((A1 )∗ , (A2 )∗ ) = h((A1 )∗ , u) − h((A2 )∗ , u). In view of Lemma 13.3.3, there exists v ∈ S n−1 such that h((A1 )∗ , u) =

u◦v . h(A1 , v)

Thus H ((A1 )∗ , (A2 )∗ ) =

u◦v u◦v u◦w u◦v − sup ≤ − h(A1 , v) w∈S n−1 h(A2 , w) h(A1 , v) h(A2 , v)

= (u ◦ v) which proves (13.10). 1 The proof is due to K. Przesławski.

h(A2 , v) − h(A1 , v) ≤ r −2 H (A1 , A2 ), h(A1 , v)h(A2 , v)

13.4 Combinatorial duality induced by polarity

165

The support function of the convex body polar to A is closely related to the radial function of A (see (7.21)): n , 13.3.5. THEOREM. For every A ∈ K00

ρA =

1 h A∗

.

Figure 13.2.

Proof (Figure 13.2). Let u ∈ S n−1 and a ∈ (bdA) ∩ posu. Then u = by 13.2.2, H (A∗ , u) = B(a) = {x ∈ Rn | x ◦ a = 1}.

a a ,

and

Hence by 3.4.5, h A∗ (u) = dist(0, H (A∗ , u)) =

1 1 = . a ρ A (u)

13.4 Combinatorial duality induced by polarity For convex polytopes, the term “polarity” is often treated as a synonym of “duality” (see Deﬁnition 13.4.1). The connection between these two notions is described precisely by Theorem 13.4.3. 13.4.1. DEFINITION. Let P, P ∈ P n . (a) A bijection ψ : F(P) → F(P ) is a combinatorial duality for P and P if ψ reverses inclusion, i.e., ∀F0 , F1 ∈ F(P) F0 ⊂ F1 ⇒ ψ(F0 ) ⊃ ψ(F1 ). (b) The polytopes P and P are combinatorially dual if there exists a combinatorial duality for them.

166

13. Polarity

13.4.2. EXAMPLES. Every two m-gons in R2 are combinatorially dual, and so are every two n-dimensional simplices in Rn . Any cube and any octahedron in R3 are combinatorially dual (Exercise 13.4). Let n n P00 := K00 ∩ Pn. n , the polytopes P are P ∗ are combina13.4.3. THEOREM. For every P ∈ P00 torially dual. n we deﬁne ψ : F(P) → F(P ∗ ) by the formula Proof. For any P ∈ P00 P

ψ P (F) := {y ∈ P ∗ | x ◦ y = 1 for every x ∈ F}.

(13.11)

Then by 13.1.2, ψ P (F) = P ∗ ∩

x∈F

B(x) =

P ∗ ∩ B(x).

x∈F

But in view of 13.2.2, the hyperplane B(x) is the supporting hyperplane for P ∗ with outer normal vector x, whence P ∗ ∩ B(x) is the support set for P ∗ , and ψ P (F) is the intersection of support sets of P ∗ for all x ∈ F. Hence ψ P (F) ∈ F(P ∗ ). Let us observe that ψ P is a bijection, because by (13.11) and (13.2), ψ P ∗ ψ P = idF (P)

(13.12)

(Exercise 13.5). Directly from (13.11) it follows that ψ P reverses inclusion. Thus ψ P is a combinatorial duality. By analogy with the notion of face of a convex polytope, a face of an arbitrary convex body A in Rn can be deﬁned: 13.4.4. DEFINITION. A subset F of a convex body A is called a face of A (proper face of A) if F is a support set of the body A. The set of all the faces of A will be denoted by F(A). Of course, for polytopes the notion of face in the sense of Deﬁnition 13.4.4 coincides with that introduced in Chapter 6. The notion of combinatorial duality can also be deﬁned generally, for arbitrary convex bodies. We have restricted our consideration to polytopes; generalization is left to the reader (Exercise 13.3). Some remarks can be found in the introduction to [51], where duality is deﬁned in terms of category theory.

13.5 Santal´o point We shall now deal with one more selector for K0n , which is closely related to polarity.

13.5 Santal´o point

167

Polarity is deﬁned by means of the unit sphere S n−1 , and as can be easily seen, it does not commute with translations: generally, (A − x)∗ = A∗ − x (Exercise 13.6). Moreover, generally, Vn ((A − x)∗ ) = Vn (A∗ ) for x = 0

(13.13)

(Exercise 13.7). Thus, the question arises whether for every A ∈ K0n there exists x ∈ intA such that the volume of (A − x)∗ is minimal. The following theorem gives an answer to this question (see [64], p. 419).2 13.5.1. THEOREM. For every A ∈ K0n there is a unique minimizer x0 of the function A : intA → R deﬁned by A (x) := Vn ((A − x)∗ ). Idea of proof: the function A is strictly convex, whence there is at most one minimizer; further, A (x) → ∞ when x approaches bdA, which yields the exis tence of x0 . The unique minimizer x0 of A is called the Santal´o point of A. We denote it by s0 (A). Let us mention the following interesting relationship between the selectors s0 and cλn ([64], p. 420; [49], Theorem 3.3): 13.5.2. THEOREM. For every A ∈ K0n , s0 (A) = x0 ⇐⇒ cλn ((A − x0 )∗ ) = 0. As a direct consequence of 13.5.2, one obtains n , 13.5.3. COROLLARY. For every A ∈ K00

s0 (A) = 0 ⇐⇒ cλn (A∗ ) = 0.

(13.14)

Finally, let us mention without proof one more theorem on the Santal´o point. 13.5.4. THEOREM. For every A ∈ K0n , Vn (A) · Vn ((A − s0 (A))∗ ) ≤ (κn )2 .

(13.15)

Inequality (13.15) is well known as the Blaschke–Santal´o inequality ([64] (7.4.24)). 2 A more general version of this theorem can be found in [49].

168

13. Polarity

13.6 Self-duality of the center of the minimal ring The relationship between the Santal´o point and the centroid, described by (13.14) (compare also Exercise 13.8), can be treated as a kind of duality. More examples of pairs of selectors dual in this sense are given in [49]. It is a natural question whether there exists a self-dual selector. We shall prove that the answer is positive (Theorem 13.6.2): the center of the minimal ring (see Section 12.5) is self-dual in the above sense. Let us keep the notation used in 12.5. We need the following lemma (3.4 in [51]). n . If A ⊂ B n or B n ⊂ A, then 13.6.1. LEMMA. Let A ∈ K00

S n−1 ∩ bdA = S n−1 ∩ bd(A∗ ).

(13.16)

Proof. Case 1: A ⊂ B n . Then B n ⊂ A∗ by 13.2.3 (b) combined with 13.2.4 (i), whence S n−1 ∩ bdA ⊂ A∗ . Since S n−1 ∩ bdA is disjoint with int(A∗ ), it follows that S n−1 ∩ bdA ⊂ S n−1 ∩ bd(A∗ ). To obtain the converse inclusion, it sufﬁces to prove S n−1 ∩ bd(A∗ ) ⊂ A.

(13.17)

Let a ∈ S n−1 ∩bd(A∗ ). Then bdE(a) is a support hyperplane of A∗ at a (Exercise 13.9). Since 0 ∈ E(a)∩ A∗ , it follows that A∗ ⊂ E(a). Hence a ∈ E( p) for every p ∈ A∗ , i.e., a ∈ A∗∗ . This proves condition (13.17), because A∗∗ = A in view of 13.3.2. Case 2: B n ⊂ A. Since in Case 1 the set A was an arbitrary subset of the unit ball, in view of 13.2.4 (i) we may use Case 1 for A replaced by A∗ . 13.6.2. THEOREM. For every A ∈ K0n , c(A) = 0 ⇒ c(A∗ ) = 0. Proof. Assume that c(A) = 0. Let α := R A (0) and β := r A (0). By deﬁnition of r A and R A (formulae (12.8) and (12.9)) combined with Theorems 13.2.4 (i) and 13.2.5 (i), we infer that R A∗ (0) =

1 β

and r A∗ (0) =

1 . α

(13.18)

By the B´ar´any Theorem 12.5.6, there exist p1 , . . . , pk ∈ (bdA) ∩ αS n−1 and q1 , . . . , ql ∈ (bdA) ∩ β S n−1 such that

13.7 Metric polarity

q p j i | i = 1, . . . , k ∩ conv | j = 1, . . . , = ∅. conv α β

169

(13.19)

Since β B n ⊂ A ⊂ α B n , it follows that 1 1 A ⊂ B n ⊂ A. α β Thus by Lemma 13.6.1 and Theorem 13.2.5 (i), S n−1 ∩

1 1 bdA = S n−1 ∩ αbd(A∗ ) and S n−1 ∩ bdA = S n−1 ∩ βbd(A∗ ). α β

Therefore,

pi 1 n−1 ∩ bd(A∗ ) for i = 1, . . . , k S ∈ α α2

and

q j 1 n−1 ∩ bd(A∗ ) for j = 1, . . . , l. S ∈ β β2

Now let pi :=

pi α2

and q j :=

to S n−1 , and analogously,

qj β

qj ; β2

then

pi α

is the central projection of pi and pi

is the central projection of q j and q j :

p pi = 1i α α

and

q j qj = 1 . β β

Consequently, condition (13.19) can be written as p q j conv 1i | i = 1, . . . , k ∩ conv 1 | j = 1, . . . , l = ∅. α

β

In view of (13.18), by the B´ar´any theorem we infer that c(A∗ ) = 0.

Theorem 13.6.2 is a particular case of Theorem 3.6 in [51], which will be proved in the next section (see Theorem 13.7.5).

13.7 Metric polarity Polarity ∗ depends on the sphere S n−1 , whence it commutes neither with translations nor with homotheties. We are now going to deﬁne a function : K0n → K0n , a modiﬁed polarity that commutes with all the similarities (Theorem 13.7.4) and thus is called the metric polarity (or metric duality (see [51]). The idea is to assign to any convex body A some sphere S(A) related to A.

170

13. Polarity

Figure 13.3.

13.7.1. DEFINITION. For every A ∈ K0n , √ (i) ρ(A) := r (A) · R(A), S(A) := bdB(c(A), ρ(A)); n (ii) A := {x ∈ R | ∀a ∈ A (x − c(A)) ◦ (a − c(A)) ≤ (ρ(A))2 } (see Figure 13.3). There is a simple relationship between the operations ∗ and . 13.7.2. PROPOSITION. Let f : Rn → Rn be the similarity deﬁned by f (x) = ρ(A) · x + c(A). n , Then for every A ∈ K00

(13.20)

f (A∗ ) = ( f (A)).

Proof. f (A∗ ) = { f (x) ∈ Rn | ∀a ∈ A x ◦ a ≤ 1} = {y ∈ Rn | ∀b ∈ f (A) (ρ(A))−2 (y − c(A)) ◦ (b − c(A)) ≤ 1} = ( f (A)).

Metric polarity is an involution: 13.7.3. THEOREM. For every A ∈ Kn , A = A. n . Proof. Let f be deﬁned by (13.20) and let B = f −1 (A). Then B ∈ K00 Applying twice 13.7.2, we obtain

A = ( f (B)) = f (B ∗ ) and

A = ( f (B ∗ )) = f (B ∗∗ ).

Hence by 13.3.2, the assertion follows.

13.7 Metric polarity

171

Metric polarity commutes with similarities: 13.7.4. THEOREM. For every similarity g : Rn → Rn and every A ∈ K0n , g(A) = (g(A)). Proof. For any sphere S in Rn and a point a, let E S (a) be the half-space with boundary B S (a) such that the center of S belongs to E S (A). Then in particular, E(a) = E S n−1 (a) (compare (13.6)). By 13.7.1 (ii), {E S(A) (a) | a ∈ A}.

A =

(13.21)

Let g be a similarity of Rn . Then for every a and a sphere S, g(E S (a)) = E g(S) (g(a))

(13.22)

and by 13.7.1 (i) and the equivariance of the selector c under similarities (see 12.5.3), g(S(A)) = S(g(A)). Hence by (13.21) and (13.22), g(A) = (g(A)).

The following theorem on metric polarity is a generalization of Theorem 13.6.2. 13.7.5. THEOREM. For every A ∈ K0n , c(A) = c(A). Proof. Let f be the similarity deﬁned by (13.20). Let us observe that the assertion of Theorem 13.7.2 can be reformulated as follows: for every A ∈ K0n , f −1 (A) = ( f −1 (A))∗ .

(13.23)

Since by 12.5.3, c( f −1 (A)) = f −1 (c(A)) = 0, from Theorem 13.6.2 and formula (13.23) it follows that c( f −1 (A)) = 0. Hence again applying 12.5.3, we obtain f −1 (c(A)) = 0, whence c(A) = f (0) = c(A).

Part III

14 Star Sets. Star Bodies

The notion of star body is a natural extension of the notion of convex body. Both classes of sets play an important role in geometric tomography (Gardner’s book Geometric Tomography [20] was already mentioned in the Introduction). The main task of geometric tomography is, roughly speaking, to examine properties of subsets of Rn by means of properties of their projections and sections. In the case of projections, it is reasonable to consider only convex bodies because generally, there is no chance to determine a nonconvex set if we know only its projections on hyperplanes: for any subset A of Rn , the projection of A on a hyperplane coincides with the projection of convA. In the case of intersections, the class of star bodies (or more generally, star sets) is suitable.

14.1 Star sets. Radial function In the literature, different approaches to the notion of star set can be found. Many authors deal with the notion of a set star-shaped at some point that belongs to this set: 14.1.1. DEFINITION. Let A ⊂ Rn and a ∈ A. The set A is star-shaped at a if ∀x ∈ A \ {a} (a, x) ⊂ A.

(14.1)

Gardner and Volˇciˇc in [21] introduced the notion of set star-shaped at any point of Rn :

176

14. Star Sets. Star Bodies

14.1.2. DEFINITION. Let A ⊂ Rn and a ∈ Rn . The set A is star-shaped at a if for every line L passing through a the set L ∩ A is connected. We shall follow Deﬁnition 14.1.1. (Gardner in [20] deals with 14.1.2.) 14.1.3. DEFINITION. For every A ⊂ Rn , the kernel of A is the subset ker A of all points of A at which A is star-shaped. The set A is called a star set if ker A = ∅. Obviously, 14.1.4. If A is convex, then ker A = A. Hence 14.1.5. Every nonempty convex set is a star set. The notion of radial function, which was already deﬁned in Chapter 7 for any convex body A with 0 ∈ A (see (7.21)), can be extended to the class of all bounded sets star-shaped at 0. 14.1.6. DEFINITION. Let A be a bounded subset of Rn and let 0 ∈ ker A. The radial function ρ A : Rn \ {0} → R is deﬁned by the formula ρ A (x) := sup{λ ≥ 0 | λx ∈ A}. The restriction of ρ A to S n−1 is denoted by the same symbol. Sometimes we write ρ(A, x) instead of ρ A (x). Of course, in Deﬁnition 14.1.6 the assumption that the set is bounded can be omitted if we allow inﬁnity to be a value of the radial function. The role of the radial function for star sets is similar to the role of the support function for convex sets. However, there is no simple connection between the radial function of a set and the radial function of its translate. We shall return to this matter in Chapter 16. Proof of the following simple statement is left to the reader (Exercise 14.4). k 14.1.7. PROPOSITION. If sets A1 , . . . , Ak are star-shaped at 0, A = i=1 Ai , k and A0 = i=1 Ai , then A and A0 are also star-shaped at 0 and for every u ∈ S n−1 , ρ A (u) = max ρ Ai (u), ρ A0 (u) = min ρ Ai (u). 1≤i≤k

1≤i≤k

As a direct consequence of Deﬁnition 14.1.6, we obtain 14.1.8. PROPOSITION. Let f ∈ G L(n). For every subset A of Rn star-shaped at 0, ∀x = 0 ρ f (A) ( f (x)) = ρ A (x). For sets that are star-shaped at 0 the notion of radial sum is deﬁned as follows:

14.2 Star bodies

177

14.1.9. DEFINITION. (i) For every x1 , x2 ∈ Rn , x + x2 if 0, x1 , x2 are collinear; ˜ 2 := 1 x1 +x 0 otherwise. ˜ 2 := {x1 +x ˜ 2 | xi ∈ Ai for i = 1, 2}. (ii) A1 +A Evidently, 14.1.10. ρ A1 +A ˜ 2 = ρ A1 + ρ A2 . It is easy to see (Exercise 14.5) that ˜ 2 ⊂ A1 + A2 . 14.1.11. A1 +A

14.2 Star bodies 14.2.1. DEFINITION. A nonempty set A ⊂ Rn is a body if A is compact and cl intA = A. A body A is called a star body if ker A = ∅. 14.2.2. Every convex body is a star body. (Compare Exercise 14.6) A characterization of convex bodies in terms of radial function is given in [20], Lemma 5.1.4: 14.2.3. THEOREM. A star body A in Rn with 0 ∈ ker A is convex if and only if for every u, v ∈ S n−1 with u = −v and ρ A (u), ρ A (v), ρ A (u + v) = 0, ρ A (u + v)−1 ≤ ρ A (u)−1 + ρ A (v)−1 . Proof. Let us observe that A is convex if and only if for every u, v ∈ S n−1 , u = −v, x ∈ (posu) ∩ bdA, y ∈ (posv) ∩ bdA, z ∈ pos(u + v) ∩ (x, y) ⇒ z ∈ A. (14.2) Indeed, condition (14.2) is evidently necessary; the reader can easily prove that it is also sufﬁcient (compare Exercise 14.12). Hence it sufﬁces to prove that (14.2) is equivalent to the inequality required. Condition (14.2) can be reformulated as follows: z := (1 − t)uρ A (u) + tvρ A (v) = z

u+v ⇒ z ≤ u + vρ A (u + v). u + v

Let u = v (for u = v there is nothing to prove). By the linear independence of u, v, z (1 − t)ρ A (u) = = tρ A (v), u + v

178

14. Star Sets. Star Bodies

whence t=

ρ A (u) ρ A (u)ρ A (v) and z = u + v . ρ A (u) + ρ A (v) ρ A (u) + ρ A (v)

Thus (14.2) is equivalent to ρ A (u)ρ A (v) ≤ ρ A (u + v), ρ A (u) + ρ A (v) i.e., is equivalent to 1 1 1 ≤ + . ρ A (u + v) ρ A (u) ρ A (v)

Let us observe that Theorem 7.3.5 and Corollary 7.3.6, concerning convex bodies, remain valid for star bodies with 0 in the kernel. We mention the counterpart of Corollary 7.3.6: 14.2.4. THEOREM. For every star body A in Rn with 0 ∈ A, 1 Vn (A) = ρ A (u)n dσ (u). n S n−1 In the literature, the name “star body (with respect to 0)” is often used in a more restrictive sense than in Deﬁnition 14.2.1. For example, the radial function is assumed to be continuous or continuous on its support, or 0 is assumed to be an internal point of the set (compare [64], [20], [47]). We shall often restrict our consideration to certain classes of star bodies. The ﬁrst one, S n , is deﬁned by means of the support, S A , of the function ρ A |S n−1 ; let us recall that S A := cl{u ∈ S n−1 | ρ A (u) = 0}. (14.3) 14.2.5. DEFINITION. A ∈ S n if A is a star body, 0 ∈ ker A, and ρ A |S A is continuous.

Figure 14.1.

14.2.6. EXAMPLE. Figure 14.1 presents three star bodies in R2 , only the ﬁrst two of which, A1 and A2 , belong to S 2 (Exercise 14.7).

14.3 Radial metric

179

In Chapter 5 we proved the Brunn–Minkowski inequality for the Minkowski sum of convex bodies (Corollary 5.2.2). Lutwak in [43] proved its analogue for the radial sum of star bodies; it is called the dual Brunn–Minkowski inequality. 14.2.7. THEOREM ([20], p. 374). For every A1 , A2 ∈ S n , 1

1

1

˜ 2 ) n ≤ Vn (A1 ) n + Vn (A2 ) n . Vn (A1 +A Equality holds if and only if either n = 1 or n ≥ 2 and A2 = λA1 for some λ > 0.1

14.3 Radial metric The family S n is not closed in the space of sets star-shaped at 0 with the Hausdorff metric (compare Exercise 14.8). There is another metric, more suitable for this space. 14.3.1. DEFINITION. For any compact subsets A, B of Rn star-shaped at 0, δ(A, B) := sup |ρ A (u) − ρ B (u)|. u∈S n−1

It is easy to check that the function δ is a metric. It is called the radial metric. 14.3.2. PROPOSITION. For any subsets A1 , A2 of Rn star-shaped at 0, ˜ B n and A2 ⊂ A1 +α ˜ B n }. δ(A1 , A2 ) = inf{α > 0 | A1 ⊂ A2 +α

(14.4)

Proof. Denote by α0 the right hand side of (14.4) and let ˜ Bn } αi := inf{α > 0 | Ai ⊂ A j +α for i, j ∈ {1, 2}, i = j. It is easy to see that α0 = max{α1 , α2 }, and by 14.1.9, αi = inf{α > 0 | ρ Ai ≤ ρ A j + α}. Thus α0 = inf{α > 0 | |ρ A1 − ρ A2 | ≤ α} = sup |ρ A1 (u) − ρ A2 (u)| = δ(A1 , A2 ).

u∈S n−1

As a direct consequence of 14.3.2 combined with 14.1.11, we obtain the following. 1 In [20] (B.28) the case n = 1 is omitted.

180

14. Star Sets. Star Bodies

14.3.3. COROLLARY. For every compact subsets A1 , A2 of Rn star-shaped at 0, δ(A1 , A2 ) ≥ H (A1 , A2 ). We shall prove 14.3.4. THEOREM. The radial metric is topologically stronger than the Hausdorff metric. Proof. From 14.3.3 it follows that radial convergence implies Hausdorff convergence. It remains to prove that the converse implication does not hold. To this end, let us consider the following example.2 Let L = linen (where en = (δn1 , . . . , δnn )) and let a = 14 en . Consider the following sequence (Sk )k∈N of (n − 2)-dimensional spheres contained in S n−1 : the 1 sphere Sk has center (1 − k+1 )en , and the hyperplane affSk is orthogonal to L. For every k ∈ N, let Ck be the cone with vertex a containing Sk . The set Ak is now deﬁned by Ak := cl(B n \ convCk )

(14.5)

(Figure 14.2). Then (Ak )k∈N is Hausdorff convergent to B n , but it is not convergent in the sense of the radial metric, because δ(Ak , B n ) = 34 for every k.

Figure 14.2.

Let us observe that each of the sets deﬁned by (14.5) is a star body with continuous radial function; i.e., it belongs to S n . Hence even the restrictions of the Hausdorff metric and the radial metric to S n ×S n are not topologically equivalent. 2 The two-dimensional version of this example is due to K. Rudnik.

14.4 Star metric

181

14.4 Star metric The radial metric has been deﬁned for sets star-shaped at 0. For star sets in the sense of Deﬁnition 14.1.3, which are not “ﬁxed at 0,” a metric invariant under translations is suitable. Such a metric was introduced in [48]: 14.4.1. DEFINITION. For any compact star sets A, B in Rn , let # δ(A, B) := sup

inf δ(A − x, B − y)

x∈ker A y∈ker B

and

# # δst (A, B) := max{δ(A, B), δ(B, A)} + H (ker A, ker B).

14.4.2. THEOREM. δst is a metric. Proof. (a) For every A, δst (A, A) = δ#(A, A) = sup

inf δ(A − x, A − y)

x∈ker A y∈ker A

= sup δ(A − x, A − x) = 0. x∈ker A

# # (b) If δst (A, B) = 0, then ker A = ker B and δ(A, B) = 0 = δ(B, A). Thus, since ker B is compact (compare Exercise 14.1), ∀x ∈ ker A ∃y ∈ ker B δ(A − x, B − y) = 0. But δ(A − x, B − y) = 0 ⇒ A − x = B − y ⇒ ker A − x = ker(A − x) = ker(B − y) = ker B − y = ker A − y ⇒ x = y. Hence A = B. (c) Evidently, δst (A, B) = δst (B, A). (d) It remains to check the triangle inequality. Let A, B, C be compact star sets. Since for every x ∈ ker A, y ∈ ker B, z ∈ ker C, δ(A − x, C − z) ≤ δ(A − x, B − y) + δ(B − y, C − z), it follows that and similarly,

# # # δ(A, C) ≤ δ(A, B) + δ(B, C), # # # δ(C, A) ≤ δ(B, A) + δ(C, B);

hence, by the triangle inequality for H , δst (A, C) ≤ δst (A, B) + δst (B, C). The function δst is called the star metric.

182

14. Star Sets. Star Bodies

Let us note two simple relationships between the star, the radial, and the Hausdorff metrics (Exercises 14.10 and 14.11): 14.4.3. PROPOSITION. If ker A = {a} and ker B = {b}, then δst (A, B) = δ(A − a, B − b) + a − b. 14.4.4. PROPOSITION. For all compact star sets A, B in Rn , the following conditions are equivalent: # (i) δ(A, B) = 0; (ii) B = A + v for some v ∈ Rn ; (iii) δst (A, B) = H (A, B). We are going to prove that the star metric is topologically stronger than the Hausdorff metric. We need two lemmas. 14.4.5. LEMMA. If A and Ak for k ∈ N are compact star sets, A = limst Ak , and A = lim H Ak , then for every x ∈ ker A there exists a sequence (xk )k∈N such that (i) xk ∈ ker Ak for every k and δ(Ak − xk , A − x) → 0; (ii) x = lim xk . Proof. Let x ∈ ker A. By Deﬁnition 14.4.1, there exists a sequence (xk )k∈N satisfying condition (i). Since the radial metric is stronger than the Hausdorff metric (Theorem 14.3.4), it follows that H (Ak − xk , A − x) → 0. But x − xk = H (A− x, A− xk ) ≤ H (A− x, Ak − xk )+H (Ak − xk , A− xk ) → 0, because H (Ak − xk , A − xk ) = H (Ak , A). Thus (ii) is satisﬁed as well.

14.4.6. LEMMA. Let B and Bk for k ∈ N be compact star sets in If B = lim H Bk , then for every increasing sequence of indices, (i k )k∈N , if (ker Bik )k∈N is Hausdorff convergent, then Rn .

lim ker Bik ⊂ ker B. H

Proof. Assume that (ker Bik )k∈N is Hausdorff convergent and let y ∈ lim H ker Bik . Then there exists a sequence (yk )k∈N such that yk ∈ ker Bik for every k and y = lim yk (compare Exercise 1.3). We shall show that y ∈ ker B. Take arbitrary z ∈ B; since B = lim H Bik , it follows that z = lim z k for some sequence (z k )k∈N with z k ∈ Bik . Thus (y, z) = lim H (yk , z k ). Since (yk , z k ) ⊂ Bik for every k, it follows that (y, z) ⊂ B. 14.4.7. THEOREM. The star metric is topologically stronger than the Hausdorff metric.

14.4 Star metric

183

Proof. We shall ﬁrst prove the implication A = lim Ak ⇒ A = lim Ak . st

H

(14.6)

Let A = limst Ak and x ∈ ker A. Then ker A = lim H ker Ak and there exists a sequence (xk )k∈N such that xk ∈ ker Ak and δ(Ak − xk , A − x) → 0; hence by 14.3.4, A − x = lim(Ak − xk ). (14.7) H

In view of the ﬁnite compactness of the space (Kn , H ) (see 3.2.14), the sequence (Ak )k∈N has a convergent subsequence Aik (because of course, it is bounded); let A = lim H Aik . There exists a subsequence ( jk )k∈N of (i k ) such that (x jk ) j∈N is convergent; let x = lim x jk . By Lemma 14.4.6 combined with (14.7), there exists a subsequence (lk ) of ( jk ), such that lim ker(Alk − xlk ) ⊂ ker(A − x); H

hence ker A − x ⊂ ker A − x, and therefore x = x. Consequently, by (14.7), A − x = A − x, whence A = A. This completes the proof of (14.6). To prove that the implication converse to (14.6) does not hold, we can use the example from the proof of Theorem 14.3.4. Indeed, suppose limst Ak = B n ; then in view of Lemma 14.4.5, for the point x := en there exists a sequence (xk )k∈N such that xk ∈ ker Ak for every k and x = lim xk . But as is easy to see, such a sequence does not exist.

15 Intersection Bodies

15.1 Dual intrinsic volumes In Chapter 7 we were concerned with basic functionals for Kn , which are also called intrinsic volumes (compare Deﬁnition 7.2.4). Erwin Lutwak in [43] introduced dual intrinsic volumes. To avoid confusion, let us stress that the term “dual” used here has nothing in common with the duality considered in Chapter 13. Dual intrinsic volumes V˜i are analogues of the classical intrinsic volumes Vi ; they are deﬁned in terms of radial functions. These analogies are presented, e.g., in [64] pp. 385–6. 15.1.1. DEFINITION ([20], (A.55)). For every A ∈ S n and i ∈ R , 1 V˜i (A) := n

S n−1

(ρ A (u))i dσ (u).

In view of Theorem 14.2.4, V˜n (A) = Vn (A).

(15.1)

By Proposition 14.1.8 and invariance of the spherical measure σ under O(n), we obtain the following. 15.1.2. THEOREM. All the dual intrinsic volumes are invariant under linear isometries.

186

15. Intersection Bodies

15.2 Projection bodies of convex bodies. The Shephard problem The notion of intersection body (for star bodies), which is the main subject of this chapter, is an analogue of the much older notion of projection body (for convex bodies). Hence we begin with projection bodies. 15.2.1. DEFINITION. For any convex body A in Rn , its projection body, A, is the convex body deﬁned by the condition ∀u ∈ S n−1 h(A, u) = Vn−1 (πu ⊥ (A)).

(15.2)

(Let us recall that every convex body is uniquely determined by its support function restricted to S n−1 (compare Corollary 3.4.6).) Directly from Deﬁnition 15.2.1 it follows that 15.2.2. Every projection body is symmetric with respect to 0. We mention the following two of many well-known characterizations of the class of projection bodies (compare [20], Theorem 4.1.11 and Corollary 4.1.12): 15.2.3. THEOREM. For every A ∈ K0n symmetric with respect to 0 the following conditions are equivalent: (i) A is a projection body of a convex body; (ii) A = lim H Ai for some sequence of sets Ai , with each Ai being the Minkowski sum of a ﬁnite number of segments centered at 0; (iii) A = lim H E i for some sequence of sets E i , with each E i being the Minkowski sum of a ﬁnite number of rotation ellipsoids with center 0. Condition (ii) (without the assumption of symmetry) deﬁnes the so-called zonoids. In (iii) rotation ellipsoid is understood as an ellipsoid with a one-dimensional axis of rotation. In 1964 Shephard ([65]) asked the following question, usually referred to as the Shepard problem (compare [43], Introduction): Is it true that for any two convex bodies A, B in Rn symmetric with respect to 0, if n ∀H ∈ Gn−1 Vn−1 (π H (A)) < Vn−1 (π H (B)), then Vn (A) < Vn (B) ? In 1967, Petty and Schneider independently ([55], [59]) proved that generally the implication is not true, but it is true under the additional assumption that B is a projection body. Let n be the class of projection bodies.

15.3 Intersection bodies of star bodies. The Busemann–Petty problem

187

15.2.4. THEOREM. Let A ∈ Kn and B ∈ n . If n ∀H ∈ Gn−1

Vn−1 (π H (A)) ≤ Vn−1 (π H (B)),

then Vn (A) ≤ Vn (B), and equality holds if and only if B is a translate of A. A proof of Theorem 15.2.4 can be found in [43].

15.3 Intersection bodies of star bodies. The Busemann–Petty problem In 1988, Erwin Lutwak introduced the notion of the intersection body of a star body. In the paper cited above, [43], he gave a survey of the theory of projective bodies and in parallel the theory of intersection bodies, and showed analogies between the two concepts. Lutwak deals with star bodies with the origin in the kernel, with a continuous radial function. The class of such star bodies will be denoted by S1n . 15.3.1. DEFINITION. For every A ∈ S1n , let I A be the star body with 0 ∈ ker A and with radial function deﬁned by the formula ρ I A (u) := Vn−1 (A ∩ u ⊥ )

for every u ∈ S n−1 .

The set I A is called the intersection body of A . The following statement is a consequence of Deﬁnition 15.3.1, Theorems 2.5.5 and 2.5.6, and the continuity of Vn−1 . 15.3.2. PROPOSITION. A ∈ S1n ⇒ I A ∈ S1n . Evidently, 15.3.3. Every intersection body is symmetric with respect to 0. Goodey and Weil in [23] proved that condition (iii) in Theorem 15.2.3 characterizing projective bodies has its counterpart for intersection bodies. They deal with the class S0n of star bodies that are symmetric with respect to 0 and have continuous radial functions: S0n := {A ∈ S1n | A = −A}. 15.3.4. THEOREM ([23]). A star body A ∈ S0n is the intersection body of a star body if and only if A is the radial limit of a sequence of ﬁnite radial sums of ellipsoids (with center 0). Hence condition (iii) in Theorem 15.2.3 is here replaced by an analogous con˜ instead of the Minkowski dition, with limδ instead of lim H and radial addition + addition +.

188

15. Intersection Bodies

Let us mention that Goodey and Weil used Theorem 15.3.4 to prove that for star bodies, Hausdorff convergence does not imply radial convergence (compare Theorem 14.3.4): in view of Lemma 3 in [23], every A ∈ S0n is the Hausdorff limit of a sequence of radial sums of ellipsoids; thus if Hausdorff convergence implied radial convergence, then from Theorem 15.3.4 it would follow that every A ∈ S0n is the intersection body of a star body; however, it is not true (see [20]). In 1956, Busemann and Petty asked the following question, usually referred to as the Busemann–Petty problem: Is it true that for all convex bodies A, B in Rn centered at 0, if n ∀H ∈ Gn−1

Vn−1 (A ∩ H ) ≤ Vn−1 (B ∩ H ),

then Vn (A) ≤ Vn (B) ? Lutwak in [43] deals with the class S1n . He proved that the implication does not hold for arbitrary A, B ∈ S1n , but it holds under the additional assumption that A is the intersection body of a star body. Let I n be the class of intersection bodies of members of S1n . 15.3.5. THEOREM (Theorem (10.1) in [43]). Let A ∈ I n and B ∈ S1n . If ∀u ∈ S n−1

Vn−1 (A ∩ u ⊥ ) ≤ Vn−1 (B ∩ u ⊥ ),

then Vn (A) ≤ Vn (B), and equality holds if and only if A = B. The intersection body of a convex body need not be convex: 15.3.6. THEOREM ([20], 8.1.8). For every A ∈ K0n there exists a vector v such that A + v has a nonconvex intersection body. The following theorem answers the question, when does the function I preserve convexity? 15.3.7. THEOREM ([20], 8.1.11). If a convex body A in Rn is symmetric with respect to 0, then I A is convex. The next three theorems concern the question, when is a convex body (symmetric with respect to 0) the intersection body of a star body? 15.3.8. THEOREM ([20], 8.1.15). If A is a convex body in R3 , symmetric with respect to 0, with ρ A |S n−1 of the class C ∞ , then A ∈ I n . 15.3.9. THEOREM ([20], 8.1.16). Let C be a rotation cylinder in Rn (that is, C is the direct sum of an (n − 1)-dimensional ball B with center 0 and a segment with center 0 contained in (linB)⊥ ). Then C ∈ I n ⇐⇒ n ≤ 3. 15.3.10. THEOREM ([20], 8.1.18). Let Q be an n-cube in Rn with center 0. Then

15.4 Star duality

189

Q ∈ I n ⇐⇒ n ≤ 3. The Busemann–Petty problem in its original version (i.e., for convex bodies) has a long and complicated history (see the paper of F. Barthe [3]). Until recently, it has been solved without additional assumptions for n = 4: the answer to the question is negative for n ≤ 3 and positive for n ≥ 5. G. Zhang in [70], applying results obtained by A. Koldobski, proved the following (Theorem 3 in [22]). 15.3.11. THEOREM. If A is a convex body in R4 , symmetric with respect to 0, then A ∈ I 4 . Earlier, in [69], he found the following characterization of those dimensions for which the answer to the Busemann–Petty question is afﬁrmative (Theorem A in [22]): 15.3.12. THEOREM. The solution of the Busemann–Petty problem for Rn is positive if and only if every convex body in Rn symmetric with respect to 0 is the intersection body of a star body. Of course, these two results of Zhang yield the solution for the case n = 4. 15.3.13. COROLLARY. The Busemann–Petty problem has the positive solution for n ≤ 4 and the negative one for n ≥ 5.

15.4 Star duality Polarity ∗, deﬁned in Chapter 13 for arbitrary nonempty subsets of Rn (Deﬁnin of convex bodies with 0 tion 13.2.1), has interesting properties for the class K00 n n in the interior: ∗ : K00 → K00 is an involution and reverses inclusion. Metric polarity : K0n → K0n is deﬁned in terms of the minimal ring, and thus only for convex bodies (Deﬁnition 13.7.1). It is an involution too. In [51] both functions were extended to functors on some categories with sets of n and Kn , respectively. Those functors were called dualities. Here, we objects K00 0 put aside the categorical approach, but we preserve the term duality for involution. n of star bodies with 0 in the kernel and positive continuous For the class S+ radial function, such a duality ◦ was introduced in [47]; it is called the star duality. Let i : Rn \ {0} → Rn \ {0} be inversion with respect to S n−1 : i(x) :=

x . x2

n , 15.4.1. DEFINITION. For every A ∈ S+

A◦ := cl(Rn \ i(A)). n, 15.4.2. PROPOSITION. For every A ∈ S+

(15.3)

190

15. Intersection Bodies

ρ A◦ =

1 . ρA

Proof. Since inversion i is a homeomorphism of Rn \ {0} onto itself, it follows that bd i(A) = bd(Rn \ i(A)) = i(bdA). Hence for every u ∈ S n−1 , if a ∈ posu ∩ bdA, then ρ A (u) = (a) and ρ A◦ (u) = i(a). In view of (15.3), this completes the proof. 15.4.3. COROLLARY. The function A → A◦ is an involution that reverses inclusion. For convex bodies with 0 in the interior, there is the following relationship between polarity and star duality: n . Then 15.4.4. THEOREM. Let A ∈ K00

κn ¯ ∗ V˜1 (A◦ ) = · b(A ). 2 Proof. By Proposition 15.4.2 combined with Theorem 13.3.5, for every u ∈ S n−1 , ρ(A◦ , u) = h(A∗ , u). (15.4) Using 15.1.1, 2.2.8, and 3.4.9, we obtain 1 1 V˜1 (A◦ ) = ρ(A◦ , u)dσ (u) = h(A∗ , u)dσ (u) n S n−1 n S n−1 κn ¯ ∗ = b(A ). 2

Generally, for an arbitrary convex body, its polar body is different from the dual star body (Figure 15.1):

Figure 15.1. n the following conditions are equiva15.4.5. THEOREM. For every A ∈ K00 lent: (i) A◦ = A∗ ; (ii) A = α B n for some α > 0.

15.4 Star duality

191

Proof. The implication (ii) ⇒ (i) is obvious. (i) ⇒ (ii): Let A◦ = A∗ . Then, by (15.4), ρ(A∗ , u) = h(A∗ , u) for every u ∈ S n−1 ; thus for every u, posu ∩ bdA∗ ⊂ H (A∗ , u). Therefore, as can be shown, bdA∗ is of the class C 1 (Exercise 15.1). Since for every linear subspace E of dimension 2, E ∩ A◦ = (E ∩ A)◦ , without any loss of generality we may assume that n = 2. Consider a parametrization r : R → bdA∗ of the boundary of A∗ : r (t) := ρ(t) ¯ · u(t), where ρ(t) ¯ = ρ(A∗ , u(t)) and u(t) = (cos t, sin t). Since r (t) ◦ r (t) = 0 for every t, it follows that ρ¯ · ρ¯ = 0. Thus ρ¯ = 0, whence ρ A∗ = const. Since A∗ is a disk, so is A (compare 13.2.3 (b) and 13.2.5 (i)). The question arises whether there are any connections between star duality and the notion of intersection body. Some results related to this question are presented in [47]; they concern connections between star duality and the (slightly more general) notion of the intersection body of order k. 15.4.6. DEFINITION ([20], Note 8.3). For every A ∈ S1n and k ∈ R, let Ik A be the set star-shaped at 0 with radial function deﬁned by ρ Ik A (u) := V˜k (A ∩ u ⊥ ). The set Ik A is called the intersection body of order k of A. Let Ikn be the class of intersection bodies of order k. The following problem is open for arbitrary k as well as for k = n: 15.4.7. PROBLEM. Prove or disprove the implication A ∈ Ikn ⇒ A◦ ∈ Ikn .

16 Selectors for Star Bodies

Since properties of star bodies deﬁned in terms of radial functions generally are not invariant under translations, the problem arises what is a proper position of a given star body with respect to 0, or equivalently, how to choose a point that should play the role of the origin. Of course, one has to decide what position of the body with respect to 0 is good. Some solutions of this problem were suggested in [48].

16.1 Radial centers of a star body The problem raised above can be formulated as the problem of looking for selectors of the family S n or some of its subfamilies. 16.1.1. DEFINITION. For any family F of star bodies in Rn , a function s : F → Rn is a selector for F if s(A) ∈ ker A for every A ∈ F. Similarly, s : F → n 2R is a multiselector if ∅ = s(A) ⊂ ker A for every A ∈ F. We begin with some multiselectors. For every star body A and function ϕ : [0, ∞) → [0, ∞), let us deﬁne A : ker A → R by the formula A (x) := ϕρ A−x (u)dσ (u). (16.1) S n−1

We shall consider the subset Mϕ (A) of ker A consisting of the points at which A attains its upper bound, i.e., the set of maximizers of A . The points of Mϕ (A) will be called radial centers of A associated with ϕ.

194

16. Selectors for Star Bodies

Let us note that if ϕ(t) = t α for some α ∈ (0; 1), then A (x) is the dual volume of A − x of order α: A (x) = V˜α (A − x) (compare [20] A.55 and [43]). Thus for any ϕ, the function A is a generalized dual volume. Let us deﬁne a family T n as follows: 16.1.2. DEFINITION. Let A be a star body in Rn . A ∈ T n if A ∈ S n and there exists S0 ⊂ S n−1 such that σ (S0 ) = 0 and for every u ∈ S n−1 \ S0 the function ker A x → ρ A−x (u) ∈ R is continuous. It can be proved (see Theorem 2.5 in [48]) that K0n ⊂ T n

(16.2)

and (A ∈ S n

ker A ⊂ int A) ⇒ A ∈ T n .

and

(16.3)

16.1.3. THEOREM. If ϕ : [0, ∞) → R is continuous, then for every A ∈ T n (i) the function A is continuous; (ii) the set Mϕ (A) is nonempty and compact. Proof. It is easy to see that for every A ∈ T n the function A is continuous. Thus is attains its upper bound because ker A is compact. In view of 16.1.3, every continuous function ϕ : [0, ∞) → R induces a multiselector Mϕ : T n → Rn . Generally, Mϕ is not a selector. Moreover, it may happen that it does not choose any proper subset of ker A. 16.1.4. EXAMPLE. If ϕ(t) = t n for every t ≥ 0, then for every star body A, Mϕ (A) = ker A, because A is constant: for every x ∈ ker A, A (x) = ρ A−x (u)n dσ (u) = nVn (A) S n−1

(see Theorem 14.2.4). As Example 16.1.4 shows, to ensure that the multiselector Mϕ is a selector, it is not enough to restrict the class of star bodies, but also additional assumptions on ϕ have to be admitted. The following problem is open. 16.1.5. PROBLEM. Is it possible to formulate conditions on ϕ ensuring that Mϕ is a selector for T n ?

16.2 Radial centers of a convex body

195

The following example shows that generally for star bodies the radial centers associated with the identity are not unique. 16.1.6. EXAMPLE (see [33]). Let A be the subset of R2 bounded by four arcs of hyperbolas (Figure 16.1): H1 := {(x1 , x2 ) | (x1 + 1)(x2 + 1) = 4, 0 ≤ xi ≤ 3 for i = 1, 2}, H2 := σ L 2 (H1 ), H3 := σ0 (H1 ), H4 := σ L 1 (H1 ), where L i = aff(0, ei ) for i = 1, 2. Then ker A is an octagon with one of its vertices a = (0, 34 ). Suppose A has a unique radial center associated with the identity, rid (A). Since 0 is the center of symmetry of A, it follows that rid (A) = 0. However, it can be computed that, approximately, A (0) = 11,046, while A (a) = 21,054; hence A (0) < A (a), a contradiction.

Figure 16.1.

16.2 Radial centers of a convex body We are going to prove that for n ≥ 2, under some assumptions on ϕ every convex body in Rn has a unique radial center associated with ϕ (Corollary 16.2.2). 16.2.1. THEOREM ([35]). Let n ≥ 2. If A ∈ K0n and ϕ : R+ → R+ is concave and strictly increasing, then the function A deﬁned by (16.1) is strictly concave. Proof. Let x0 , x1 ∈ A, x0 = x1 , t ∈ (0; 1), and let x := (1 − t)x0 + t x1 . Let Ai := A − xi for i = 0, 1. Then

196

16. Selectors for Star Bodies

A − x = (1 − t)A0 + t A1 , because A is convex. Hence by 14.1.11 and by the assumptions on ϕ, A (x) ≥ (1 − t) A (x0 ) + t A (x1 ).

(16.4)

It remains to prove that the inequality in (16.4) is sharp. By the dual Brunn–Minkowski inequality (Theorem 14.2.7), it follows that 1

1

1

Vn (A) n = ((1 − t)Vn (A0 ) n + t Vn (A1 ) n 1

˜ A1 ) n , ≥ Vn ((1 − t)A0 )+t

(16.5)

and equality holds if and only if there exists λ > 0 such that t A1 = λ(1 − t)A0 . t Comparing the n-volumes, we obtain λ = 1−t , which is equivalent to A − x0 = A − x1 ; but this contradicts the assumption x0 = x1 . Hence by (16.5), ˜ A1 ), Vn (A − x) > Vn ((1 − t)A0 +t

that is,

S n−1

ρ nA−x >

S n−1

((1 − t)ρ A0 + tρ A1 )n .

Thus for some S ⊂ S n−1 of positive measure, ρ A−x (u) > (1 − t)ρ A0 (u) + tρ A1 (u) for u ∈ S, whence for u ∈ S, ϕρ A−x (u) > (1 − t)ϕρ A−x0 (u) + tϕρ A−x1 (u). Integrating both sides over S n−1 , we obtain A ((1 − t)x0 + t x1 ) > (1 − t) A (x0 ) + t A (x1 ), i.e., A is strictly concave.

16.2.2. COROLLARY. Let n ≥ 2. If ϕ is concave and strictly increasing, then Mϕ (A) is a singleton for every A ∈ K0n . Proof. Let A ∈ K0n . Then by (16.2), A ∈ T n , whence by Theorem 16.1.3, the function A is continuous. Thus by 16.2.1, A has a unique maximizer. Let rϕ (A) be the unique radial center of a convex body A, induced by ϕ. In view of Corollary 16.2.2, if n ≥ 2 and ϕ is concave and strictly increasing, then rϕ is a selector for K0n . 16.2.3. PROPOSITION. The selector rϕ is equivariant under isometries. (Compare Exercise 16.1.)

16.2 Radial centers of a convex body

197

We shall prove the continuity of the radial centers for convex bodies with respect to the Hausdorff metric. For every A ∈ K0n , let U (A) := {u ∈ S n−1 | ∃a, b b − a ∈ linu and (a, b) ⊂ bdA}. We start from the following. 16.2.4. LEMMA ([35]). Let Ak ∈ K0n and xk ∈ Ak for every k ∈ N ∪ {0}. If A0 = lim H Ak and x0 = lim xk , then for every u ∈ S n−1 \ ∞ k=0 U (Ak ), ρ A−x0 (u) = lim ρ Ak −xk (u). Proof. Without any loss of generality we may assume that xk = x0 for every k. Indeed, let Ak := Ak + x0 − xk ; then lim H Ak = A = lim H Ak and ρ Ak −x0 = ρ Ak −xk . Hence, let x0 ∈ A ∩ k≥0 Ak , u ∈ S n−1 \ k≥0 U (Ak )), and L + := x0 + posu, L − := x0 − posu. Further, let

xk+ ∈ L + ∩ bdAk and xk− ∈ L − ∩ bdAk

for every k ∈ N ∪ {0}. It sufﬁces to prove that

x0+ = lim xk+ .

(16.6)

For every k ≥ 0 there are the following four possibilities: (1)k (2)k (3)k (4)k

xk− xk− xk− xk−

= x0 = xk+ , = x0 = xk+ , = x0 = xk+ , = x0 = xk+ .

Passing to subsequences of (Ak )k∈N , we may assume that (xk+ )k∈N is convergent and that exactly one of (1)k –(4)k is satisﬁed for all k ≥ 1, i.e., for all members of the sequence (Ak )k∈N . Consequently, 16 conjunctions (i)k ∧ ( j)0 for i, j ∈ {1, . . . , 4} are to be considered. If L + ∩ bdA0 is a singleton, i.e., one of the conditions (i)k ∧ (1)0 , (i)k ∧ (2)0 , and (i)k ∧ (4)0 is satisﬁed, then (16.6) holds. Thus the 16 conditions are reduced to (i)k ∧ (3)0 for i ∈ {1, . . . , 4}. But in view of Theorem 2.5.7, by backward induction on the dimension of the afﬁne subspace, the above conjunction may hold only for i = 3, which means that the only possibility is xk− = x0 = xk+ which implies (16.6).

and

x0− = x0 = x0+ ,

198

16. Selectors for Star Bodies

16.2.5. THEOREM ([35]). For every n ≥ 2 and every concave and strictly increasing function ϕ, the selector rϕ : K0n → Rn is continuous with respect to the Hausdorff metric. Proof. Let A = lim H Ak for a sequence (Ak )k∈N of convex bodies in Rn , and let xk = rϕ (Ak ) for every k. We may assume that (xk )k∈N is convergent. Let x = lim xk . From Lemma 16.2.4 we deduce that A (x) = lim Ak (xk ).

(16.7)

We shall prove that x = rϕ (A). Suppose, to the contrary, that there exists y ∈ A such that A (y) > A (x). (16.8) Since A = lim H Ak , there exists (yk )k∈N such that yk ∈ Ak for every k and lim yk = y. Since Ak (xk ) ≥ Ak (yk ) for every k, applying again Lemma 16.2.4 to the sequence (yk )k∈N we infer from (16.7) that A (x) ≥ A (y), contrary to (16.8).

16.3 Extended radial centers of a star body Observe that in the proof of Theorem 16.1.3 we did not make use of the fact that A has all of ker A as its domain; the only thing we needed was that the domain was a compact, convex, and nonempty subset of ker A. Also, in Theorem 16.2.1 and Corollary 16.2.2 we needed only the fact that the domain of A was a compact, convex, and nonempty subset of a convex body A. Hence the statements 16.1.3 (ii) and 16.2.2 can be easily generalized as follows: 16.3.1. THEOREM. If ϕ : [0, ∞) → R is concave and strictly increasing, then for every A ∈ T n and every C ∈ Kn contained in ker A, the set of maximizers of A |C is nonempty and compact. 16.3.2. THEOREM. Let n ≥ 2. If ϕ : [0, ∞) → R is concave and strictly increasing, A ∈ K0n , C ∈ Kn , and C ⊂ A, then the function A |C has a unique maximizer. Let us denote by rϕ (A, C) the unique point of C whose existence is ensured by Theorem 16.3.2. In view of Theorem 16.3.2, the selector rϕ deﬁned for K0n can be extended to a selector for S n : 16.3.3. THEOREM. Let n ≥ 2. If ϕ is concave and strictly increasing, then the formula r˜ϕ (A) := rϕ (convA, ker A) for A ∈ S n (16.9) deﬁnes a selector r˜ϕ for the class S n . It is an extension of rϕ :

16.3 Extended radial centers of a star body

199

∀A ∈ K0n r˜ϕ (A) = rϕ (A). Proof. In view of Theorem 16.3.2 applied to the convex body conv A and its subset C := ker A, the function r˜ϕ is a selector. If A ∈ K0n , then convA = A = ker A, whence by (16.9), r˜ϕ (A) = rϕ (A, A) = rϕ (A).

16.3.4. PROPOSITION. The selector r˜ϕ is equivariant under isometries. (Exercise 16.2.) It is easy to show that r˜ϕ is not continuous with respect to the Hausdorff metric (Exercise 16.3). However, as we shall see, it is continuous with respect to the star metric (compare [48]). We need two lemmas (16.3.6 and 16.3.7) and the following result obtained by I. Herburt. 16.3.5. THEOREM ([31]). The restriction of the function conv to the class {A ∈ S n | 0 ∈ ker A ∩ int A} is continuous with respect to the radial metric. 16.3.6. LEMMA. Let A, Ak ∈ S n , xk ∈ ker Ak , ak ∈ ker A for every k ∈ N, and δ(Ak − xk , A − ak ) → 0. If ϕ : [0, ∞) → R is a Lipschitz function, then lim | Ak (xk ) − A (ak )| = 0. Proof. Let λ be the Lipschitz constant of ϕ. Then | Ak (xk ) − A (ak )| ≤ λ |ρ Ak −xk (u) − ρ A−ak (u)|dσ (u) S n−1

≤ λnκn δ(Ak − xk , A − ak ) → 0.

The other lemma concerns the class T0n deﬁned as follows: T0n := K0n ∪ {A ∈ T n | ker A ⊂ int A}.

(16.10)

Observe that, by (16.2), T0n ⊂ T n .

(16.11)

16.3.7. LEMMA. Let A, Ak ∈ T0n for k ∈ N, A = limst Ak , and let ϕ : [0, ∞) → [0, ∞) be a Lipschitz function. If for every k the function convAk | ker Ak attains its upper bound at pk and convA | ker A attains its upper bound at p, then lim convAk ( pk ) = convA ( p). Proof. Since A = limst Ak , there exists a sequence (ak )k∈N in ker A such that δ(Ak − pk , A − ak ) → 0,

(16.12)

200

16. Selectors for Star Bodies

whence by Theorem 14.3.4, H (Ak − pk , A − ak ) → 0. By Theorem 14.4.7, A = lim H Ak ; thus pk − ak → 0. In view of Theorem 16.3.5, from (16.12) it follows that δ(convAk − pk , convA − ak ) → 0. Applying Lemma 16.3.6 to the convex hulls of the sets Ak and A, we infer that lim |convAk ( pk ) − convA (ak )| = 0.

(16.13)

Assume ( pk ) to be convergent (we may do so because otherwise, we can repeat this reasoning for every convergent subsequence). Let x = lim pk . Then x = lim ak . Since |convAk ( pk ) − convA (x)| ≤ |convAk ( pk ) − convA (ak )| + |convA (ak ) − convA (x)|, from (16.13) and Theorem 16.1.3 (i) it follows that convA (x) = lim convAk ( pk ). Hence convA ( p) ≥ lim convAk ( pk ).

(16.14)

On the other hand, there exists a sequence (xk ) ∈ P∞ k=1 ker Ak such that p = lim xk and δ(Ak − xk , A − p) → 0; thus by Theorem 16.3.5, δ(convAk − xk , convA − p) → 0. Applying again Lemma 16.3.6 to convex hulls, we obtain the inequality convA ( p) ≤ lim convAk ( pk ), which, together with (16.14), completes the proof.

16.3.8. THEOREM. Let n ≥ 2. For every concave, strictly increasing Lipschitz function ϕ : [0, ∞) → [0, ∞), the selector r˜ϕ |T0n is continuous with respect to the star metric. Proof. Let A, Ak ∈ T0n and A = limst Ak . Let pk = r˜ϕ (Ak ) and p = r˜ϕ (A). On the one hand, by Lemma 16.3.7, convA ( p) = lim convAk ( pk ). On the other hand, there exists a sequence (ak ) in ker A such that δ(Ak − pk , A − ak ) → 0.

(16.15)

16.3 Extended radial centers of a star body

201

As before, we may assume that (ak ) is convergent. Let x = lim ak . Then x = lim pk by 14.4.7 combined with 14.3.4. Since, by 16.1.3 (i) combined with 16.3.5, the function convA is continuous, from Lemma 16.3.6 it follows that convA (x) = lim convAk ( pk ). Hence by (16.15), we infer that convA (x) = convA ( p), whence x = p because p is the unique maximizer. Therefore, p = lim pk .

Exercises to Part I

Chapter 1 1.1. Prove that for any metric space (X, ) the function (·, A) : X → R is a weak contraction: |(x, A) − (y, A)| ≤ (x, y). 1.2. Explain why the proof of Theorem 1.1.11, which concerns Rn , does not work for Kn . 1.3. Prove that for every bounded sequence of nonempty compact subsets A, A1 , A2 , . . . of a ﬁnitely compact metric space (X, ), the condition A = lim Ak H

is equivalent to the conjunction of the following two conditions: (i) for every increasing sequence of indices (i k )k∈N and every sequence (xik )k∈N ∈ P∞ k=1 Ai k convergent in (X, ), lim xik ∈ A; (ii) for every x ∈ A there exists a sequence (xk )k∈N ∈ P∞ k=1 Ak such that x = lim xk . (Compare [64], note 3, p. 57.)

204

Exercises to Part I

1.4. Prove that for arbitrary A, B ∈ C(Rn ), H (A, B) = sup |(x, A) − (x, B)|. x∈Rn

1.5. Prove that every ﬁnitely compact metric space is complete. 1.6. Prove or disprove the following statement: If (X, ) is ﬁnitely compact, then so is (C n (X ), H )). 1.7. Prove that for nonempty subsets A, B of Rn and δ, ε > 0, ((A)δ )ε = (A)δ+ε . (Compare 1.1.6.) Give an example of a metric space (X, ) for which the inclusion ⊃ generally fails.

Chapter 2 2.1. Prove that if A1 , A2 , A1 ∪ A2 ∈ Kn , then (A1 ∪ A2 ) + (A1 ∩ A2 ) = A1 + A2 . 2.2. Prove that for every A ∈ C n the function f : R → C n deﬁned by f (t) := t · A is continuous. 2.3. Prove that (i) if A1 , A2 ∈ C n and A1 ∩ A2 = ∅, then for every x ∈ Rn , (x, A1 ∩ A2 ) ≥ max (x, Ai ); i=1,2

(ii) if A1 , A2 , A1 ∪ A2 ∈ Kn , then for every x ∈ Rn , (x, A1 ∩ A2 ) = max (x, Ai ) i=1,2

(see the proof of Theorem 7.3.2). 2.4. Prove that if a subset A of Rn is convex, then also int A and clA are convex. 2.5. Prove Theorem 2.3.11.

Exercises to Part I

205

2.6. (a) Prove that the limit in the set E n of hyperplanes (Deﬁnition 2.5.2) is induced by some metric in E n . (Hint: Notice that if φ is the parametric representation of E n (formula (2.7)), then the function h deﬁned for E = φ(v, t) by h(E) = ((t − 1)v, (t + 1)v) is a bijection.) (b) Show that X is open (closed) in E n if and only if φ −1 (X ) is open (closed) in S n−1 × R+ . 2.7. Prove that if E, E k ∈ E n for every k ∈ N, and f : Rn → Rn is an isometry, then E = lim E k ⇒ f (E) = lim f (E k ) (Theorem 2.5.3). 2.8. Prove that if H, E, E k ∈ E n and H ∩ E = H = H ∩ E k for every k ∈ N, then lim E k = E ⇒ lim(E k ∩ H ) = E ∩ H. 2.9. Check whether the set of hyperplanes parallel to a given one is a) closed in E n , b) open in E n , c) dense in E n . 2.10. Check whether the set of hyperplanes passing through a given point is a) nowhere dense in E n , b) closed in E n , c) open in E n . 2.11. Let E be a hyperplane in Rn . Find compact convex sets A1 , A2 with π E (A1 ) ∩ π E (A2 ) = π E (A1 ∩ A2 ).

Chapter 3 3.1. Draw a picture that illustrates the proof of 3.3.1. 3.2. Prove the converse theorem to 3.3.6. 3.3. Let B A be the family of balls in Rn that contain a bounded set A. Prove that A is closed and convex if and only if A = B A . 3.4. Check whether in 3.4.7 the assumption that A is closed is essential. 3.5. Prove that for every afﬁne automorphism f : Rn → Rn and every subset A of Rn , f (convA) = conv f (A).

206

Exercises to Part I

3.6. Prove that for every A, B ⊂ Rn , conv(A ∩ B) ⊂ convA ∩ convB

(1)

convA ∪ convB ⊂ conv(A ∪ B).

(2)

and Give an example of A, B for which equality holds neither in (1) nor in (2). 3.7. Prove that for every X ∈ C n , ¯ ¯ ), diam(convX ) = diamX, d(convX ) = d(X ). b(convX ) = b(X 3.8. Prove that for any convex subsets A, B of Rn , conv(A ∪ B) = (1 − t)A + t B. t∈[0;1]

3.9. Find an example of A ∈ Kn such that b(A, u) = const and A is not a ball (a) for n = 2, (b) for n = 3. 3.10. Prove that for every u ∈ S n−1 the function h(·, u) is continuous with respect to the Hausdorff metric. 3.11. Let E be an afﬁne subspace of Rn with dim E ∈ {0, . . . , n − 1}. Prove that for every subset A of Rn , if A is symmetric with respect to E, then so is convA.

Chapter 4 4.1. Prove that the function conv is not induced by any transformation of Rn . 4.2. Complete the proof of Theorem 4.2.7. 4.3. Prove that for any afﬁne subspace E of dimension 1, the function S E preserves Kn and K0n . (See Deﬁnition 4.3.1.) 4.4. Prove that for any afﬁne subspace E of dimension 1, the symmetrization S E preserves volume. 4.5. Prove that for every line E, the symmetrization S E is not induced by any transformation of Rn . 4.6. Prove Proposition 4.4.5.

Exercises to Part I

4.7. Prove that for any f 1 , . . . , f m ∈ O(n) the map m1 and only if m = 1 or f 1 = · · · = f m .

m

i=1 f i

207

is an isometry if

˙ 1 Let P be a convex polygon in R2 with k vertices (k ≥ 3). Evaluate 4.8. (T.Z.) the maximal and minimal numbers of vertices of S H (P) for a hyperplane H . 4.9. Generalize Example 4.2.12 to arbitrary n. 4.10. Generalize Example 4.2.15 to arbitrary n.

Chapter 5 5.1. Prove that κn > 1 for every n ∈ N. (Hint: Apply the Bieberbach Theorem 5.4.1.)

Chapter 6 6.1. Give an example of a set of simplices that is not a complex, though its union is a geometric polyhedron. 6.2. Give an example of two different triangulations of a polyhedron. 6.3. Let P ∈ P0n . Prove that every simplex of any triangulation of P is a face of some n-dimensional simplex in this triangulation. 6.4. Give examples that illustrate the ﬁrst part of the proof of Theorem 6.2.4. 6.5. Prove that for any afﬁne subspace E of Rn and any convex polytope P, if P ∩ E = ∅, then P ∩ E is a convex polytope. 6.6. Prove that P n is an afﬁne invariant (compare 6.2.6). 6.7. Prove that Minkowski addition does not decrease dimension of convex polytopes: dim(P + Q) ≥ max{dim P, dim Q}. Check whether the same is true for arbitrary A, B = ∅. 6.8. Prove that in 6.2.9 the sets on the right-hand side of the equality have pairwise disjoint interiors. 1 “(T.Z.)” ˙ means that the exercise was suggested by Tomasz Zukowski. ˙

208

Exercises to Part I

6.9. Complete and illustrate the proof of Theorem 6.4.4. 6.10. Prove that if n-dimensional convex polytopes P1 , P2 in Rn have convex union and disjoint interiors, then dim(P1 ∩ P2 ) = n − 1. 6.11. Prove Corollary 6.4.5. 6.12. Prove Proposition 6.5.2. (Hint: Notice that it sufﬁces to prove it for n = 2.) 6.13. Prove that there exists a cylindric polytope in Rn for n > 3 that is not equivalent by dissection to a cube. (Hint: See remarks following Theorem 6.4.2.) 6.14. Prove Theorem 6.1.4. To this end, prove ﬁrst that if a complex T0 is a subdivision of a triangulation T (that is, every simplex S ∈ T is a union of some simplices in T0 ), then χ (T0 ) = χ (T ). 6.15. Applying 6.1.5 and 6.1.6, prove that χ (A) = 1 for every convex body A in Rn . (Hint: Every convex body in Rn is homeomorphic to B n .) ˙ An A ∈ Kn is said to be Minkowski decomposable if there exist 6.16. (T.Z.) A1 , A2 ∈ Kn such that A = A1 + A2 and A2 is not a homothet of A1 (i.e., there is no λ > 0 and v ∈ Rn with A2 = λA1 + v). Otherwise, A is said to be Minkowski indecomposable. (a) Prove that every triangle is Minkowski indecomposable (compare [64] Theorem 3.2.11.) (b) Prove that every convex polygon is the Minkowski sum of some triangles and segments. ˙ Prove that mean width b¯ and perimeter l are additive functions on P 2 : 6.17. (T.Z.) for every P1 , P2 ∈ P 2 , ¯ 1 + A2 ) = b(A ¯ 1 ) + b(A ¯ 2 ), l(A1 + A2 ) = l(A1 ) + l(A2 ). b(A (Compare Corollary 9.1.2, for n = 3). ˙ Apply 6.16 and 6.17 to calculate the mean width of the following 6.18. (T.Z.) subsets of R2 : (a) segment, (b) triangle, (c) parallelogram, (d) hexagon with pairwise parallel and congruent opposite sides. ˙ Prove that b(P) ¯ 6.19. (T.Z.) =

1 π l(P)

for every P ∈ P 2 .

Exercises to Part I

209

6.20. Let P1 and P2 be polyhedra contained in afﬁne subspaces E 1 and E 2 , respectively, and let E 1 ∩ E 2 be a singleton. Prove that F (0) (P1 + P2 ) = F (0) (P1 ) + F (0) (P2 ). (Let us recall that F (0) (P) is the set of vertices of the polytope P.)

Chapter 7 7.1. Prove that mean width is a valuation, while the minimal width, diameter, and the functional r0 are not (compare 7.1.6). 7.2. Prove that if a functional 0 : Pn → R is invariant under an isometry g of Rn , then also and deﬁned by 7.1.9 are invariant under g. 7.3. Prove that our deﬁnition of measure of outer normal angle, γ (P, F) (Deﬁnition 7.2.1), agrees with that given by Schneider ([64], p. 100). 7.4. Prove Theorem 7.2.7 on the monotonicity of Vn−1 |P n for polytopes of dimension less than n. 7.5. Apply Theorem 7.2.10, which concerns the class P n , to prove that for every k ∈ {0, . . . , n} the functional Vk : Kn → Kn is increasing.

Chapter 8 8.1. Prove the Hadwiger Theorem 8.1.5 for n = 2. 8.2. Prove Theorem 8.2.2 modifying the proof of Theorem 8.1.5. 8.3. Let = 0, . . . , n.

n

i=0 αi Vi .

Prove that if is increasing, then αi ≥ 0 for i =

Chapter 9 9.1. Calculate the mean width of the sets A1 = ((0, 0), (1, 1), (0, 1)) and A2 = I 2 in R2 and of the sets A1 × {0} and A2 × {0} in R3 : (a) applying Deﬁnition 2.2.8, (b) applying Theorem 9.1.1. (Compare Exercise 6.18.) 9.2. Prove Proposition 9.2.2.

210

Exercises to Part I

9.3. Prove that µ{φ(v, 0) | v ∈ S n−1 } = 0 (compare Deﬁnition 9.2.1). 9.4. Prove the Crofton formulae for n = 3 (Theorem 9.2.7). 9.5. According to traditional notation (compare [29]), for any A ∈ K2 , l(A) is the perimeter of A if int A = ∅, and double length of A if int A = ∅; f (A) is the area of A. Thus, l(A) = 2V1 (A), f (A) = V2 (A). In view of Proposition 9.2.6 for n = 2, there exist α, β such that f (A) = α l(A ∩ E)dµ(E) and l(A) = β dµ(E) = βµ(E A ). EA

Calculate α and β.

EA

Exercises to Part II

Chapter 10 10.1. Let A be the regular triangle in R2 with vertices a, b, c and with sides of length 1. Calculate the curvature measures i (A, X ) for i = 0, 1, 2 (a) if X = {a}, (b) if X = (a, b). 10.2. R. Schneider in [64] (p. 70 and 77) deﬁnes the spherical image σ (A, X ) as follows: σ (A, X ) := σ (A, x), x∈A∩X

where σ (A, x) = {u ∈ S n−1 |x ∈ H (A, u)} for x ∈ A ∩ X . Prove that this deﬁnition is equivalent to 10.1.8. 10.3. Prove Theorem 10.1.9. 10.4. Prove that for any P ∈ P n and F ∈ F 0 (P), γ0 (P, F) = γ (P, F) (compare Deﬁnition 7.2.1 and 10.1.10). 10.5. Verify conditions (i) and (ii) in Theorem 10.1.12.

212

Exercises to Part II

10.6. Prove that the set of weakly continuous, locally deﬁned, and invariant valuations from Kn into the set of Borel measures on Rn is closed under addition and multiplication by nonnegative scalars. 10.7. Prove that if n−1 (A1 , X ) = n−1 (A2 , X ) for every X ∈ B(Rn ), then A1 = A2 . 10.8. Explain why Theorem 10.1.18 gives a partial solution of Problem 10.1.17. 10.9. Prove Proposition 10.3.2. 10.10. Let A ∈ Kn . Prove that A is strictly convex if and only if bdA does not contain any segment. 10.11. Verify the formulae for Uε (A, X j ), j = 1, 2, in Example 101..20.

Chapter 11 11.1. Give example of a set A with reachA = inf{reach(A, a) | a ∈ A}. 11.2. Find reachA (a) for A = S 1 in R2 , (b) for A = {(x1 , x2 ) ∈ R2 | x12 = x22 }. 11.3. Prove Proposition 11.2.4. 11.4. Let a = (1, 0), b = (0, 1), and A = ((−a, a) + (−b, b)) \ int B 2 . Find χ (A) (a) proving that A is homeomorphic to a polyhedron and applying 6.1.3 and 6.1.6, (b) proving that A is homeomorphic to a member of the family U 2 and applying (11.10) and 11.2.7.

Chapter 12 12.1. Give and example of a subset A of Rn such that the set of symmetry centers, C0 (A), (a) is countable, (b) has the cardinality continuum. 12.2. Prove (by induction) formula (12.2).

Exercises to Part II

213

12.3. Give examples of selectors for the family of triangles in R2 . 12.4. Prove Theorem 12.2.6. 12.5. Describe the relationship between the notion of center of gravity of a ﬁnite set (formula (12.5)) and the notion of convex combination (Deﬁnition 3.1.1). 12.6. Prove 12.3.7. 12.7. Prove that if f : R2 → R2 is a similarity and points p1 , . . . , pk , q1 , . . . , ql satisfy (12.16) for a set A and a point x0 , then f ( p1 ), . . . , f ( pk ), f (q1 ), . . . , f (ql ) satisfy this condition for the set f (A) and the point f (x0 ). 12.8. Prove that the Steiner point map s : Kn → Rn is equivariant under similarities and continuous. 12.9. Find the Steiner point of the polygon conv{a, b, c, d} for (a) a = (0, 0), b = (4, 0), c = (1, 1), d = (3, 1), (b) a = (−2, 0), b = (2, 0), c = (0, 1), d = (0, −4). 12.10. Prove that for every A ∈ K0n the function A x → A ∩ σx (A) ∈ Kn is continuous. 12.11. Let A ∈ K0n . For every x ∈ A, let A x := A ∩ σx (A) (compare (12.18)). Prove that the family (A x )x∈A is concave, i.e., for every t ∈ [0, 1], A(1−t)x1 +t x2 ⊃ (1 − t)A x1 + t A x2 . 12.12. Prove that (under the notation of Exercise 12.11) A x = A y ⇒ x = y. 12.13. Prove that the pseudocenter of a convex body belongs to its interior: p(A) ∈ int A for every A ∈ K0n . 12.14. Prove Proposition 12.6.3. 12.15. Prove Proposition 12.6.6. 12.16. Prove that for any subgroup G of O(n) and any convex body A the set of G-pseudocenters PG (A) is convex. 12.17. Prove that for every A ∈ K0n and every subgroup G of O(n), the family (A x,G )x∈A is concave (compare Exercise 12.11 and formula (12.22)).

214

Exercises to Part II

12.18. Complete the proof of Theorem 12.7.5: (a) prove that H ∩ C has the properties required, (b) prove that B0 ⊃ C. 12.19. Let A be a regular m-gon in R2 . Prove that for m odd the set of quasicenters Q(A) is inﬁnite, while for m even it is a singleton. 12.20. Prove that the Chebyshev point map cˇ is equivariant under similarities. 12.21. Let A ∈ Kn and x ∈ Rn . Prove that H (A, {x}) = R A (x), the radius of the smallest ball with center x containing A.

Chapter 13 13.1. Find the polar lines of p = ( 12 , 12 ) and q = (1, 2) with respect to S 1 (a) analytically: applying Deﬁnition 13.1.2, (b) geometrically: applying Theorem 13.1.3. 13.2. Prove 13.2.3 (b). 13.3. Generalize the notion of combinatorial duality on arbitrary convex bodies n . and prove that Theorem 13.4.3 remains valid for every A ∈ K00 13.4. Find combinatorial dualities for the pairs of polytopes in 13.4.2. 13.5. Prove (13.12). 13.6. Prove that for every convex body A in Rn symmetric with respect to 0 and every x ∈ int A \ {0}, (A − x)∗ = A∗ − x. 13.7. Find a convex body A in Rn and a point x ∈ int A such that (a) Vn ((A − x)∗ ) = Vn (A∗ ); (b) Vn ((A − x)∗ ) = Vn (A∗ ). 13.8. Prove that

cλn (A) = 0 ⇐⇒ s0 (A∗ ) = 0.

(Compare (13.14).) 13.9. Prove that if a ∈ S n−1 ∩ bd(A∗ ), then B(a) is a support hyperplane of A∗ at a. 13.10. Prove that for every A ⊂ Rn , (clA)∗ = (convA)∗ = A∗ . 13.11. Find conditions sufﬁcient for a subset A of Rn to satisfy (bdA)∗ = A∗ .

Exercises to Part III

Chapter 14 14.1. Prove that for every subset A of Rn the set ker A is closed in Rn and convex. 14.2. Choose several subsets of Rn and ﬁnd their kernels. 14.3. For any A ⊂ Rn , let the extended kernel of A (in symbols exkerA) consist of the points of Rn at which A is star-shaped (see 14.1.2). (a) Find the extended kernels of chosen sets. (b) Show that ker A ⊂ exker A for every A ⊂ Rn and there exists A with ker A = ∅ = exkerA. 14.4. Prove 14.1.7. 14.5. Prove 14.1.11. 14.6. Prove 14.2.2. (Hint: Show that cl intA = A for every A ∈ K0n .) 14.7. Justify Example 14.2.6. 14.8. Give an example of a Hausdorff convergent sequence (Ak )k∈N such that Ak ∈ S n for every k but lim H Ak ∈ S n . 14.9. Prove that for any compact star set B, δst ({a}, B) = sup δ({0}, B − y) + sup a − y. y∈ker B

y∈ker B

216

Exercises to Part III

14.10. Prove 14.4.3. 14.11. Prove 14.4.4. 14.12. Prove that if a star body A in Rn is not convex, then there exist two points x, y ∈ bdA with relint(x, y) ⊂ Rn \ A. 14.13. Let E be an afﬁne subspace of Rn with dim E ∈ {0, . . . , n − 1}. Prove that for every subset A of Rn , if A is symmetric with respect to E, then so is ker A. 14.14. Find a star body A in R2 with intA connected and ker A being (a) a singleton; (b) a segment.

Chapter 15 n and posu ∩ bdA ⊂ A(u), then bdA is of the class C 1 15.1. Prove that if A ∈ K00 (of course, the set on the left-hand side of the inclusion is a singleton).

Chapter 16 16.1. Prove that the selector rϕ : K0n → R is equivariant under isometries. 16.2. Prove that the selector r˜ϕ : S n → R is equivariant under isometries. 16.3. Prove that the selector r˜ϕ is not continuous with respect to the Hausdorff metric.

References

[1] A.D. Aleksandrov, Zur Theorie der gemischten Volumina von konvexen K¨orpern I: Verallgemeinerung einiger Begriffe der Theorie der konvexen K¨orper (in Russian), Mat. Sbornik, N.S., 2 (1937), 947–972. [2] I. B´ar´any, On the minimal ring containing the boundary of a convex body, Acta Sci. Math. 52 (1988), 93–100. [3] F. Barthe, M. Fradelizi, B. Maurey, A Short Solution to the Busemann– Petty Problem, Positivity 3 (1999), 95–100. [4] P. Billingsley, Probability and Measure, John Wiley & Sons, 1979. [5] W. Blaschke, Kreis und Kugel, 1949. ˇ [6] A. Bogdewicz, M. Moszy´nska, Cebyˇ sev sets in the space of convex bodies, to appear in Rend. Circ. Mat. Palermo (2005). [7] V. Boltianskii, Hilbert’s Third Problem, John Wiley & Sons, 1978. ¨ [8] T. Bonnesen, Uber das isoperimetrische Deﬁzit ebener Figuren, Math. Ann., 91 (1924), 252–268. [9] T. Bonnesen, Les Probl`emes des Isop´erim`etres et des Is´epiphanes, Gauthier-Villars, 1929. [10] T. Bonnesen, W. Fenchel, Theorie der konvexen K¨orper, Chelsea Publishing Company, 1948. [11] K. Borsuk, Multidimensional Analytic Geometry, Polish Scientiﬁc Publishers, 1969. [12] K. Borsuk, Theory of Retracts, PWN, 1967. [13] J.J. Charatonik, W.J. Charatonik, Inducible mappings between hyperspaces, Bull. Pol. Acad. Sci. 46(1) (1998), 5–9. [14] R. Engelking, Theory of dimensions, Finite and Inﬁnite, Heldermann Verlag, 1995.

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[15] R. Engelking, General Topology, Heldermann Verlag, 1989. [16] K.J. Falconer, The Geometry of Fractal Sets, Cambridge University Press, 1985. [17] I. F´ary, L. R´edei, Der zentralsymmetrische Kern und die zentralsymmetrische H¨ulle von konvexen K¨orpern, Math. Ann., 122 (1950), 205–220. [18] H. Federer, Curvature measures, Trans. AMS 93 (1959), 418–481. [19] W. Fenchel, B. Jessen, Mengenfunktionen und konvexe K¨orper, Danske Vid. Selsk. Mat.-Fys. Medd., 16(3) (1938), 1–31. [20] R.J. Gardner, Geometric Tomography, Cambridge University Press, 1995. [21] R.J. Gardner, A. Volˇciˇc, Tomography of convex and star bodies, Advances in Math. 108 (1994), 367–399. [22] R.J. Gardner, A. Koldobsky, T. Schlumprecht, An analytic solution to the Busemann–Petty problem on sections of convex bodies, Annals of Math., 149 (1999), 691–703. [23] P. Goodey, W. Weil, Intersection Bodies and Ellipsoids, Mathematika 42 (1995), 295–304. [24] H. Groemer, On the Euler Characteristic in Spaces with a Separability Property, Math. Ann. 211 (1974), 315–321. [25] P.M. Gruber, The space of compact subsets of E d , Geom. Dedicata 9 (1980), 87–90. [26] P. Gruber, G. Lettl, Isometries of the space of convex bodies in Euclidean space, Bull. London Math. Soc. 12 (1980), 455–462. [27] B. Gr¨unbaum, Measures of symmetry for convex sets, Proc. Symposia Pure Math., Vol. VII, Convexity; Amer. Math. Soc. (1963), 233–270. [28] B. Gr¨unbaum, Convex Polytopes, John Wiley & Sons, 1967. [29] H. Hadwiger, Altes und Neues uber Konvexe K¨orper, Birkh¨auser, 1955. [30] H. Hadwiger, Vorlesungen u¨ ber Inhalt, Oberﬂ¨ache und Isoperimetrie, Springer, 1957. [31] I. Herburt, On convex hulls of star sets, Bull. Pol. Acad. Sci., 49 (4) (2001), 433–440. [32] I. Herburt, On the Lipschitz continuity of the centre of the minimal ring, Rend. Circ. Mat. Palermo, Serie II, Suppl. 70 (2002), 385–393. [33] I. Herburt, Convexity of dual volumes for convex bodies, submitted for publication. [34] I. Herburt, M. Moszy´nska, On metric products, Coll. Math. 62 (1991), 121– 133. [35] I. Herburt, M. Moszy´nska, Z. Peradzy´nski, Remarks on radial centres of convex bodies, Analysis and Geometry, 8(2) (2005), 157–172. [36] J.G. Hocking, G.S. Young, Topology, Addison-Wesley Publ. Comp., London 1961. [37] A. Illanes, S.B. Nadler, Jr., Hyperspaces, Marcel Dekker, Inc., 1999. [38] D. Klain, A short proof of Hadwiger’s characterization theorem, Mathematika, 42 (1995), 329–339. [39] D. Klain, G.-C.Rota, Introduction to Geometric Probability, Cambridge University Press, 1997.

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[40] M. Kordos, Streifz¨uge durch die Mathematik-Geschichte, Ernst Klett Verlag, Stuttgart 1999. [41] K. Leichtweiss, Konvexe Mengen, Springer, 1980. [42] E. Lutwak, Dual mixed volumes, Paciﬁc J. Math. 58 (1975), 531–538. [43] E. Lutwak, Intersection Bodies and Dual Mixed Volumes, Advances in Math. 71(2) (1988), 232–261. [44] P. McMullen, The Polytope Algebra, Advances in Math. 78 (1989), 76–130. [45] P. McMullen, R. Schneider, Valuations on convex bodies; in Convexity and Its Applications, (ed. by P.M. Gruber and J.M. Wills), Birkh¨auser, 1983, 170–247. [46] M. Moszy´nska, Remarks on the minimal rings of convex bodies, Studia Sci. Math. Hung. 35 (1999), 1–20. [47] M. Moszy´nska, Quotient star bodies, intersection bodies, and star duality, J. Math. Anal. Appl. 232 (1999), 45–60. [48] M. Moszy´nska, Looking for selectors of star bodies, Geometriae Dedicata 81 (2000), 131–147. [49] M. Moszy´nska, K. Przesławski, Santal´o points associated with radial densities, Rend. Circ. Mat. Palermo, (2) Suppl. 50 (1997), 273–288. ´ ¸ cicka, Geometry and Linear Algebra (in Polish), [50] M. Moszy´nska, J. Swie Polish Scientiﬁc Publishers, 1987. ˙ [51] M. Moszy´nska, T. Zukowski, Duality of convex bodies, Geometriae Dedicata 58 (1995), 161–173. ˙ [52] M. Moszy´nska, T. Zukowski, On G-pseudo-centres of convex bodies, Glasnik Mat. 33(53) (1998), 251–265. ˙ [53] M. Moszy´nska, T. Zukowski, On G-pseudo-centres of convex bodies (II), Rend. Circ. Mat. Palermo, Serie II, Suppl. 70 (2002), 177–190. [54] T.S. Motzkin, Sur quelques propri´et´es caract´eristiques des ensembles convexes, Atti Real. Accad. Naz. Lincei, Rend. Cl. Sci. Fis., Mat., Natur., Serie VI,21, (1935), 562–667. [55] C.M. Petty, Projection bodies; in: Proceedings, Coll. Convexity, Copenhagen, 1965, Kobenhavns Univ. Mat. Inst., (1967), 234–241. [56] K. Przesławski, Linear and Lipschitz continuous selectors for the family of convex sets in Euclidean vector spaces, Bull. Pol. Acad. Sci. 33 (1985), 31–33. [57] C.H. Sah, Hilbert’s Third Prroblem: Scissors Congruence, Pitman, 1979. [58] A. Sard, Linear Approximation, AMS, 1963. [59] R. Schneider, Zu einem Problem von Shephard u¨ ber die Projektionen konvexer K¨orper, Math. Z. 101 (1967), 71–82. [60] R. Schneider, On Steiner points of convex bodies, Israel J. Math. 9 (1971), 241–249. [61] R. Schneider, Isometrien des Raumes der konvexen K¨orper, Coll. Math. 33 (1975), 219–224. [62] R. Schneider, Curvature Measures of Convex Bodies, Ann. Mat. Pura Appl. 116 (1978), 101–134.

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[63] R. Schneider, Convex surfaces, curvature and surface area measures; in Handbook of Convex Geometry (ed. by P.M. Gruber and J.M. Wills), NorthHolland, 1993. [64] R. Schneider, Convex Bodies: The Brunn–Minkowski Theory, Cambridge University Press, 1993. [65] G.C. Shephard, Shadow systems of convex bodies, Israel J. Math. 2 (1964), 229–236. [66] D.W. Stroock, A Concise Introduction to the Theory of Integration, Birkh¨auser, 1990. [67] H. Weyl, On the volume of tubes, Amer. J. Math. 61 (1939), 461–472. [68] B. Zdrodowski, On minimal rings for convex bodies (in Polish), master thesis, 1995, Dept. of Math., Computer Sci., and Mech., Warsaw Univ. (unpublished). [69] G. Zhang, Centered bodies and dual mixed volumes, Trans. AMS 345 (1994), 777–801. [70] G. Zhang, A positive solution to the Busemann–Petty problem in R4 , Ann. of Math. 149 (1999), 535–543.

List of Symbols

Rn , xiv S n−1 , xiv B n , xiv B(X ), xiv Vn , xiv λn , xiv σn−1 , σ , xiv κn , ωn , xiv card, xv P∞ k=1 , xv cl, int, bd, xv dist, xv lin, aff, pos, xv relint, relbd, xv Gkn , xv E ⊥ , xv v ⊥ , xv E ⊥ (x), xv π E , xv σ E , xv ◦, xv G L(n), O(n), S O(n), xv Tr, xv ≡G , xv

w

→, xv Hm , xvi j δi , xvi ∀, ∃, ∃1 , xvi (x, A), 3 (A)ε , 4 B(a, ε), 4 diam, 4 C(X ), 6 H , 6 lim H , 6 C n , 11 A + B, 11 t A, 12 H (A, v), E(A, v), A(v), 14 b(A, v), 15 d(A), 15 ¯ b(A), 16 r0 (A), 17 (a, b), 18 Kn K0n , 19 E n , 20 aff, 25 c(a1 , . . . , ak ; t1 , . . . , tk ), 25

222

List of Symbols

C(A), 25 (a1 , . . . , ak ), 25 conv, 28 F(A), F0 (A), 30 ξ A , 32 h A , h(A, ·), 35 f ∗ , 39 S H , 41 S E , 48 M H , 49 T (A), 49 S(A), 53 T(A), 56 (a0 , . . . , ak ), 61 dim, 62 T (i) , 62 dim, 62 χ(P), 63 dim, 64 P n , P0n , 64 F(P), F (i) (P), 65 ∼G , 69 nor(P, F), γ (P, F), 77 Vk , 78 Wk , 79 ρ A , 86 V, F, M, C, 87 E A , 99 G n , 102 colε A, 109 Uε , 109 i (A, ·), 112 σ (A, X ), 112 γ0 (A, X ), 113 u A , 117 Vε , 118 i (A, ·), 119 τ (A, Y ), 119 σ A , 120 τ A , 120 Ci (A, ·), 123 reach, 125 Tan(A, a), 126 U(F), 128

χ (A), 130 C0 (A), 135 K1n , 136 c0 (A), 136 cµ (A), 139 pi (A), 141 s(A), 143 R A (x), r A (x), 144 R(A), r (A), 147 ring, 147 c(A), 147 p(A), 150 PG (A), 152 pG (A), 153 c(A), ˇ 156 Q G (A), qG (A), 157 B S (A), 160 A∗ , 161 n , 163 K00 n , 166 P00 s0 (A), 167 A, 170 ker, 176 ρ A (x), ρ(A, x), 176 ˜ 177 A+B, S n , 178 S A , 178 δ(A, B), 179 δ#(A, B), δst (A, B), 181 V˜i , 185 A, 186 S1n , 187 I A, 187 S0n , 187 I n , 188 n , 189 S+ A◦ , 189 Ik A, 191 Ikn , 191 Mϕ (A), 193 T n , 194 rϕ (A, C), 198 r˜ϕ (A), 198

Index

afﬁne combination, 25 hull, 25 subspace generated by a subset, 25 ball, 4 basic functionals, 79 Bieberbach theorem, 58 bijection, xiv Blaschke selection theorem, 31 symmetrization, 49 body, 177 boundary, xv bounded set, 4 Brunn–Minkowski inequality, 55 dual, 179 Carath´eodory theorem, 27 Cauchy formulae, 104 center of gravity, 138 center of minimal ring, 147 centrally symmetric hull, 154 centroid, 140

Chebyshev point, 156 closure, xv collar of a set, 109 combinatorial duality of polytopes, 165 concave family, 213 function, 19 contraction, xiv convex body, 19 combination, 25 cone, 71 function, 19 hull, 28 polytope, 64 set, 18 convexity ring, 127 Crofton formulae, 100 curvature measure, 112, 123 cylindric polytope, 70 Dehn invariant, 69 diameter of, 4

224

Index

dimension of complex, 62 of convex polytope, 64 of simplex, 62 distance of point and set, 3 duality, 168, 189 edge of convex polytope, 65 of polyhedral convex cone, 71 equivalence by dissection, 69 ε-hull of a set, 4 Euclidean metric, 11 Euler–Poincar´e characteristic, 63, 133 extended kernel, 215 face of convex body, 166 of polyhedral cone, 71 of simplex, 62 proper, 62 of spherical polytope, 72 facet of convex polytope, 65 ﬁnite intersection property, 28 ﬁnitely compact metric space, 7 fully additive function, 128 function deﬁned locally, 114 weakly continuous, 114 functional, 73 homogeneous, 74 of degree p, 74 increasing, 74 invariant, 73 with respect to a transformation group, 73 G-invariant set, 151 G-kernel, 152 G-pseudocenter, 152 G-quasi-center, 157 Gaussian curvature, 122 generalized ball, 4 geometric

polyhedron, 62 realization of complex, 62 Hadwiger invariant, 70 theorem, 92, 95 Hausdorff limit, 6 metric, 6 Helly theorem, 28 i-dimensional skeleton of a complex, 62 improper face of convex polytope, 64 injection, xiv integral curvature, 87 integral of mean curvature, 97 interior, xv intersection body of order k, 191 intersection body of a star body, 187 intrinsic volumes, 79 dual, 185 isometric embedding, xv, 5 isometric spaces, 5 isometry, xv, 5 kernel of a star-shaped set, 176 Lipschitz constant, xiv meager set, 154 mean curvature, 97, 122 of rotations, 49 width, 16 mean curvature, 87 measure deﬁned locally, 113, 119 metric, 3 duality, 169 polarity, 169 projection, 32 minimal ring of a convex body, 147 minimal width, 15

Index

Minkowski decomposable, 208 sum, 11 symmetrization, 49 multiselector, 193 nerve, 133 normal section, 121 normed measure of outer normal angle, 77 outer normal angle of polytope P with respect to its face F, 77 outer normal vector of a halfspace, 14 of a support hyperplane, 14 polar hyperplane, 160 set, 161 polarity, 161 polyhedral cone, 71 polyhedron, 62 principal curvature, 122 product of set by scalar, 12 projection body of a convex body, 186 proper face of convex body, 166 of convex polytope, 64 pseudocenter, 150 quasi-center, 154 radial centre, 193 function, 86, 176 metric, 179 sum, 176 regular direction, 120 point, 120 relative boundary, xv interior, xv

225

residual set, 154 retract, xv retraction, xv reverse spherical map, 121 Santal´o point, 167 Schwarz symmetrization, 48 second fundamental form, 122 selector, 137, 193 selector equivariant under a map, 137 separation of sets by halfspaces, 34 set with positive reach, 126 similar spaces, 5 similarity, 5 simple valuation, 89 simplex, 25, 61 of dimension 0, 62 of dimension −1, 62 simplicial complex, 62 singular direction, 120 point, 120 smooth convex body, 121 spherical image of a set, 112 map, 121 polytope, 72 reverse image, 119 star body, 177 duality, 189 metric, 181 set, 176 star-shaped set at a point, 175, 176 Steiner point, 143 symmetrization, 41 theorem, 80, 82 strictly convex body, 121 support function, 35 halfspace, 14 hyperplane, 14 point, 14

226

Index

set, 14 surface area measure, 119 surjection, xiv symmetric kernel, 150 symmetrization, 41 symmetry center, 135

valuation, 73 vertex of convex polytope, 65 of simplex, 25, 61 of spherical polytope, 72 volume, xiv

tangent vector, 126 tangent cone, 160 translation of Kn , 40 triangulation of polyhedron, 62

weak contraction, xiv width, 15 zonoid, 186

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