Introduction to Geometric Probability This is the first modern introduction to geometric probability, also known as integral geometry. The subject is presented at an elementary level, requiring little more than first year graduate mathematics. The theory of intrinsic volumes due to Hadwiger, McMullen, Santal6 and others is presented, along with a complete and elementary proof of Hadwiger's characterization theorem of invariant measures in Euclidean n-space. The theory of the Euler characteristic is developed from an integralgeometric point of view. The authors then prove the fundamental theorem of integral geometry, namely the kinematic formula. Finally the analogies between invariant measures on polyconvex sets and measures on order ideals of finite partially ordered sets are investigated. The relationship between convex geometry and enumerative combinatorics motivates much of the presentation. Every chapter concludes with a list of unsolved problems. Geometers and combinatorialists will find this a stimulating and fruitful tale.
Daniel A. Klain is Assistant Professor of Mathematics at Georgia Institute of Technology. Gian-Carlo Rota is Professor of Applied Mathematics and Philosophy, Massachusetts Institute of Technology.
Lezioni Lincee Sponsored by Foundazione IBM Italia Editor: Luigi A. Radicati di Bmzolo, Scuola Normale Superiore, Pisa
This series of books arises from lectures given under the auspices of the Accademia Nazionale dei Lincei and is sponsored by Foundazione IBM Italia. The lectures, given by international authorities, will range on scientific topics from mathematics and physics through to biology and economics. The books are intended for a broad audience of graduate students and faculty members, and are meant to provide a 'mise au point' for the subject with which they deal. The symbol of the Accademia, the lynx, is noted for its sharp-sightedness; the volumes in this series will be penetrating studies of scientific topics of contemporary interest.
Already published Chaotic Evolution and Strange Attractors: D. Ruelle Introduction to Polymer Dynamics: P. de Gennes The Geometry and Physics of Knots: M. Atiyah Attractors for Semigroups and Evolution Equations: O. Ladyzhenskaya Asymptotic Behaviour of Solutions of Evolutionary Equations: M. 1. Vishik Half a Century of Free Radical Chemistry: D. N. R. Barton in collaboration with S. 1. Parekh Bound Carbohydrates in Nature: L. Warren Neural Activity and the Growth of the Brain: D. Purves Perspectives in Astrophysical Cosmology: M. Rees Molecular Mechanisms in Striated Muscle: S. V. Perry Some Asymptotic Problems in the Theory of Partial Differential Equations: O.Oleinik
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Cambridge University Press 1997
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A catalogue record for this book is available from the British Libmry ISBN 0 521 59362 X hardback ISBN 0 521 59654 8 paperback
Contents
Preface Using this book 1
1.1 1.2 1.3 2
2.1 2.2 2.3 3
3.1 3.2 3.3 3.4 4
4.1 4.2 4.3 5
5.1 5.2 5.3 5.4 5.5 5.6
page xi xiv
The Buffon needle problem The classical problem The space of lines Notes
1 1 3 5
Valuation and integral Valuations Groemer's integral theorem Notes
6 6 8 11
A discrete lattice Subsets of a finite set Valuations on a simplicial complex A discrete analogue of Helly's theorem Notes
13 13 21 28 29
The intrinsic volumes for parallelotopes The lattice of parallelotopes Invariant valuations on parallelotopes Notes
30 30 35 41
The lattice of polyconvex sets Polyconvex sets The Euler characteristic Helly's theorem Lutwak's containment theorem Cauchy's surface area formula Notes
42 42 46 50 54 55 58
viii 6
6.1 6.2 6.3 6.4 6.5 6.6 6.7 7
7.1 7.2 7.3 7.4 7.5 8
8.1 8.2 8.3 8.4 8.5 8.6 8.7 9
9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10
10.1 10.2 10.3 10.4
Contents Invariant measures on Grassmannians The lattice of subspaces Computing the flag coefficients Properties of the flag coefficients A continuous analogue of Sperner's theorem A continuous analogue of Meshalkin's theorem Helly's theorem for subspaces Notes
60 60 63 70 73 77 81 83
The intrinsic volumes for polyconvex sets The affine Grassmannian The intrinsic volumes and Hadwiger's formula An Euler relation for the intrinsic volumes The mean projection formula Notes
86 86 87 93 94 95
A characterization theorem for volume Simple valuations on polyconvex sets Even and odd valuations The volume theorem The normalization of the intrinsic volumes Lattice points and volume Remarks on Hilbert's third problem Notes
98 98 106 109 111 112 115 117
Hadwiger's characterization theorem A proof of Hadwiger's characterization theorem The intrinsic volumes of the unit ball Crofton's formula The mean projection formula revisited Mean cross-sectional volume The Buffon needle problem revisited Intrinsic volumes on products Computing the intrinsic volumes Notes
118 118 120 123 125 128 129 130 135 140
Kinematic formulas for polyconvex sets The principal kinematic formula Hadwiger's containment theorem Higher kinematic formulas Notes
146 146 150 152 153
Contents Polyconvex sets in the sphere 11.1 Convexity in the sphere 11.2 A characterization for spherical area 11.3 Invariant valuations on spherical polytopes 11.4 Spherical kinematic formulas 11.5 Remarks on higher dimensional spheres 11.6 Notes Bibliography Index of symbols Index 11
ix
154 154 156 159 162 164 166 168 174 176
Preface
If we were allowed to rename the field of geometric probability - sometimes already renamed integral geometry - then we would be tempted to choose the oxymoron 'continuous combinatorics.' On more than one occasion the two fields, geometric probability and enumerative combinatorics, are brought together by mathematical analogy, that most effective breaker of barriers. Like combinatorial enumeration, where sequences of objects bearing a common feature are unified by the idea of a generating function, geometric probability studies sets of geometric objects bearing a common feature, which are unified by the idea of an invariant measure. The basic idea is extremely simple. When considering straight lines, pairs of points, or triangles in space, one determines the invariant measure on the variety of straight lines, of pairs of points, of triangles. This idea is strangely reminiscent of the underlying idea of enumerative geometry, with one major difference: whereas enumerative geometry is bound to the counting of finite sets, geometric probability is given greater freedom, by extending the concept of enumeration to allow the assigning of invariant measures. Invariant measures are far easier to compute and, we dare add, more useful than the curiously large integers that are computed in enumerative geometry. This basic idea goes back to Crofton's article in the ninth edition of the Encyclopaedia Britannica, an article that created the subject from scratch and that is still worth reading today. The one other brilliant contribution to geometric probability in the past century was Barbier's solution of the Buffon needle problem, which remains to this day the basic trick of the subject, still being secretly exploited in ever unsuspected ways. Geometric probability has suffered in this century the fate of other fields that would have enjoyed a healthy autonomous development, had
xii
Preface
it not been for the overpowering development of representation theory. One can reduce integral geometry to the study of actions of Lie groups, to symmetric spaces, to the Radon transform; in so doing, however, the authentic problematic of the subject is lost. Geometric probability is a customer of representation theory, in the same sense that mechanics is a customer of the calculus. The purpose of this book is to present the three basic ideas of geometric probability, stripped of all reliance on group-theoretic techniques. First, we investigate measures on polyconvex sets (i.e., finite unions of compact convex sets) in Euclidean spaces of arbitrary dimension that are invariant under the group of Euclidean motions. A great many mathematicians are still basking in the illusion that there is only one such measure, namely, the volume. We merrily destroy this illusion by proving what is at present the fundamental result of the field (due to Hadwiger), stating that the space of such invariant measures is of dimension n + 1 in a Euclidean space of dimension n. The proof of this fundamental result given in the text is new, due to the first author. It becomes clear, on reading the applications of the fundamental theorem, that the basic invariant measure to be singled out from such a bounty is not the volume, but the Euler characteristic (as Steve Schanuel was first to realize). Here again we meet with wide ignorance on the part of the mathematical public: the fundamental fact that the Euler characteristic is an invariant measure (in fact, it is the only integer-valued invariant measure) is not as well known as it should be. It leads to one-line proofs of most of the fundamental theorems on convex sets. We develop the theory of the Euler characteristic from scratch, in a way that makes it look like an ordinary integral. Second, we prove the fundamental formula of integral geometry, viz., the kinematic formula. Here we displace the common device of Minkowski sums from its typically central role, not merely as a display of mathematical machismo, but with an ulterior motive. Third, we try to bring out from the beginning the striking analogy between the computation of invariant measures and certain combinatorial properties of finite partially ordered sets. The second author pointed out in 1967 that the notion of Euler characteristic could be extended to such partially ordered sets by means of the Mobius function. We now go one step further and show that an analogue of the theory of invariant measures in Euclidean space can be worked out in partially ordered sets, including finite analogues of the kinematic formula and even of Helly's theorem. This analogy brings out in stark contrast the
Preface
xiii
unexplored terrain of classical geometric probability, namely, a thorough understanding of the integral geometric structure of the lattice of subspaces of Euclidean space under the action of the orthogonal group. It also brings us closer to the current outer limits of mathematics, to the theory of Hecke algebras, to Schubert varieties and to the quantum world. We hope that the reading of this introduction to the field of geometric probability will encourage further development of these analogies. The text is based on the 'Lezioni Lincee' given by the second author in 1986 at the Scuola Normale Superiore in Pisa. The authors wish to thank Ennio De Giorgi, Edoardo Vesentini, and Luigi Radicati for providing an interested audience for the original lectures. Thanks are also due to Stefano Mortola for his careful reading of the initial draft, and to Beifang Chen, Steve Fisk, Joseph Fu, Steven Holt, Erwin Lutwak, and three anonymous referees for their valuable comments and suggestions.
Using this book
Although parts of this book assume a knowledge of basic point-set topology, measure theory, and elementary probability theory, the greater part of the text should be accessible to advanced undergraduates. Proofs are either given in full or else stretched to the point from which the reader will be able to reconstruct them without effort. Only on certain technical measure-theoretic points have we felt the need to omit details that, although indispensable in a detailed treatment, are of questionable relevance in an exposition that is meant to stress geometric insight and combinatorial analogy. Some notions that appear vague in the early sections will be revisited later on, after language has been developed for a treatment in clear and rigorous terms. References and open problems are deferred to the notes at the end of each chapter.
1 The Buffon needle problem
We begin with what is probably the best-known problem of geometric probability, the Buffon needle problem. This solution of the needle problem via the characterization of an additive set functional serves to motivate the study of valuations on lattices, the topic of Chapter 2. Variations and generalizations of the Buffon needle problem are presented in Chapters 8 and 9.
1.1 The classical problem
Parallel straight lines are drawn on the plane R2, at a distance d from each other. A needle of length L is dropped at random on the plane. What is the probability that the needle shall meet at least one of the lines? This problem can be solved by computations with conditional probability (Feller, for example, solved it in this way in his well known treatise [23, p. 61]). It is, however, more instructive to solve it by another method, one that minimizes the amount of computation and maximizes the role of probabilistic reasoning. Let Xl be the number of intersections of a randomly dropped needle of length Ll with any of the parallel straight lines. If the needle is long enough, the random variable Xl can take several integer values, whereas if the needle is short, it can take only the values 0 or 1. If Pn is the probability that the needle meets exactly n of the straight lines, and if E(Xd denotes the expectation of the random variable Xl, then we have
E(X l ) =
L npn· n2:0
2
1 The Buffon needle problem
Thus, if L1 < d, then
and P1 is the probability we seek. Therefore, it is sufficient to compute the expectation E(X1)' Suppose that another needle of length L2 is dropped at random. The number of intersections of this second needle with any of the parallel straight lines drawn on R 2 is another random variable, say X 2 . The random variables Xl and X 2 are independent, unless the needles are welded together. Suppose that the needles are rigidly bound at one of their endpoints. They may form a straight line, or they may be at an angle. In either case, if the two rigidly bound needles are simultaneously dropped on R 2 , their total number of intersections will still be Xl + X 2 . The random variables Xl and X 2 will no longer be independent, but their expectation will remain additive: (1.1) The same reasoning applies to the random variable Xl + X 2 + ... + X k , for the case in which k needles are welded together to form a polygonal line of arbitrary shape. Since E(X 1) clearly depends on the length L 1, we can write E(Xd = I(L 1 ), where 1 is a function to be determined. By welding together two needles so that they form one straight line we find that E(X1 + X 2) = I(L1 + L 2), and we infer from (1.1) I(L 1 + L 2)
=
I(L 1) + I(L2).
It then follows that 1 is linear when restricted to rational values of L. Since 1 is clearly a monotonically increasing function with respect to L, we infer that I(L) = rL for all L E R, where the constant r is to be determined. If C is a rigid wire of length L, dropped randomly on R2, and if Y is the number of intersections of C with any of the straight lines, then C can be approximated by polygonal wires, so that Y is approximately equal to Xl + X 2 + ... + Xk. Passing to the limit, we find that
E(Y)
= rL.
(1.2)
This allows us to determine the value of the constant r, by choosing a wire of suitable shape. Let C be a circular wire of diameter d. Obviously E(Y) = 2, and L = 7rd. It then follows from (1.2) that 2
= r7rd,
1.2 The space of lines
3
whence r = 2j(7rd). Thus, for a short needle, we have
E(X 1 ) =
2L
= 7rd·
P1
This result has been used (rather inefficiently) to compute the value of 7r. Instead, we shall use it as the theorem leading into the heart of geometric probability, following the ideas of Crofton and Sylvester.
1.2 The space of lines
Let Graff(2,1) denote the set of all straight lines in R2 (the reason for this notation shall be made clearer in Chapter 7). It is well known that this set enjoys some notable properties. To this end, denote by Z1 the number of intersections of a straight line taken at random with a straight line segment of length L 1 , and let Ai denote the invariant measure on Graff(2,1). The integral
r
Z1
JGraff(2,1)
dAi
depends only on L 1. Since Z1 takes only the values 0 or 1, this integral is equal to the measure of the set of all straight lines that meet the given straight line segment. Since the value of the integral depends only on the length L1 of the straight line segment, denote this value by f(L 1 ). We can now repeat the argument we used for the Buffon needle problem: given a polygonal line consisting of segments of length L 1 , L 2 , ••. , the number of intersections of a randomly chosen straight line with the polygonal line is
r
(Z1+Z2+ ... )dAr=f(L1+L2+ ... ).
JGraff(2,1)
Since integrals are linear, this becomes
r
JGraff(2,1)
Z1 dAr
+
r
JGraff(2,1)
Z2 dAr
+ ... =
f(L 1) + f(L 2) + ... ,
and we again conclude that f(L) = rL. We shall not normalize the measure Ai by setting r = 1; rather, we shall decide later what the 'right' normalization should be. Again we may pass to the limit. Recall that a subset K of the plane is convex if any two points x and y in K are the endpoints of a line segment lying inside K. A curve C in the plane is called convex if C encloses a convex subset. Let C be a convex curve in the plane of length L, and let
4
1 The Buffon needle problem
Zc be the number of intersections of C with a randomly chosen straight line. Then ( zcdAi = rL. lGraff(2,1) In particular, let K1 and K2 be compact convex sets in the plane with non-empty interiors, and with boundaries C 1 = oK1 and C 2 = oK2 of length L1 and L 2. For each i, we have (
ZCi
lGraff(2,1)
dAi
= rLi'
On the other hand, since Ki is convex, a straight line meets Ki either twice or not at all (excluding the limiting cases of tangents, which can be shown to have measure zero). Thus, the function ZC i takes either the value 2 or the value O. If we denote by Di the set of all straight lines in R2 that meet K i , then we have (
ZCi
dAi = 2Ai(Di ).
lGraff(2,1)
To re-state these results in terms of probability, assume that K1 ~ K 2. The conditional probability that a straight line shall meet the compact convex set K 1 , given that it meets K 2 , is the ratio
AI(D 1 ) AI(D2 ) • The computation above shows that this ratio is equal to
L1 length(oK1) = :----"'--=,--7:::-::-::'C:L2 length(oK2)" Note that the value of the normalization constant r is irrelevant to the computation of this conditional probability. The results above (sometimes designated Sylvester's theorem) can be compared to the analogous result for points: if K1 ~ K 2 , the conditional probability that a point taken at random shall belong to K 1 , given that it belongs to K 2 , is area(K1 ) area(K2)' Thus, we see a striking analogy: replacing every occurrence of the word 'point' by the word 'line' corresponds to replacing the word 'area' by the word 'perimeter.' This analogy suggests that a generalization of Sylvester's theorem to arbitrary dimension may prove worthwhile.
1.3 Notes
5
1.3 Notes The solution to Buffon's needle problem presented here is due to Barbier [5], and was later generalized still further by Crofton in [14, 15, 16]. Crofton's main paper, which set geometric probability on its modern footing, is the Encyclopaedia Britannica article [17]. It is still an excellent reference. In [95] Sylvester considered a variation of the Buffon needle problem in which the needle is replaced by a finite rigid collection of compact convex (and possibly disjoint) sets K 1 , .•. , Km tossed randomly into a plane tiled by evenly spaced lines. Sylvester then considered the cases in which a line meets one, some, or all of the sets K i . In the previous section we measured the set of all lines meeting a compact convex set K in the plane. When dealing with multiple convex sets Sylvester was led to consider also the measure of the set of lines that separate two disjoint compact convex regions of the plane. This theme has also been pursued extensively in the work of Ambartzumian [1, 2]. Buffon's result gives a very inefficient means of approximating the number 11"; for a history of this technique, see [30]. For additional modern treatments of geometric probability in the plane, see also [1, 2, 49, 82, 90].
2 Valuation and integral
In Chapter 1 we expressed the Buffon needle problem in terms of a set functional (1.1) on a certain collection of sets in the plane satisfying a certain kind of additivity. We then solved the problem by characterizing this additive functional in (1.2), using in this case the fact that the functional was monotonically increasing and invariant with respect to certain motions of sets in the plane. In this chapter we make more precise the notion of 'additive set functional', or valuation, on a lattice of sets. The abstract notions developed in this chapter will then be specialized to several different specific lattices in the chapters following, leading in turn to similarly elegant solutions to generalizations and analogues of Buffon's original problem. Section 2.2 is devoted to Groemer's integral theorem, which is needed to prove Groemer's extension theorems in Sections 4.1, 5.1, and 11.1.
2.1 Valuations We now introduce a class of set functions that comprise the most basic and important tools of geometric probability, namely valuations. We begin with partially ordered sets and lattices. A partial ordering S; on a set L is a relation satisfying the following conditions for all x, y, z E L.
(i) x S; x. (ii) If x S; y and y S; x then x = y. (iii) If x S; y and y S; z then x S; z. The partially ordered set L is called a lattice if, for all x, y E L, there exist a greatest lower bound (or meet) x 1\ y ELand a least upper bound (or join) x V y E L. A lattice L is said to be distributive if, for all x, y, z E L, we have the following.
2.1 Valuations
(i) x V (y 1\ z) = (x (ii) x 1\ (y V z) = (x
y) 1\ y) V
1\ V
7
(x V z). (x 1\ z).
Let S be a set, and let L be a family of subsets of S closed under finite unions and finite intersections. Such a family is clearly a distributive lattice, in which the partial ordering is given by subset inclusion, while the meet and join are given by intersection and union of sets, respectively. A valuation on a lattice L of sets is a function JL defined on L that takes real values, and that satisfies the following conditions: JL(A U B) = JL(A)
+ JL(B)
- JL(A
n B),
0 is the empty set.
JL(0) = 0, where
(2.1) (2.2)
By iterating the identity (2.1) we obtain the inclusion-exclusion principle for a valuation JL on a lattice L, namely JL(A 1 U A2 U ... U An) = L i
JL(Ai) - L JL(Ai i<j
n Aj) + L
JL(Ai
n Aj n A k) + ... (2.3)
i<j
for each positive integer n. If A is any subset of S, the indicator function (or simply the indicator) of A, denoted by lA, is the function on S given by • lA(S) = 1; sEA, • lA(S) = 0; s tJ. A.
A finite linear combination k
f = L(XilAi'
(2.4)
i=l
where (Xi E R, and Ai E L, is said to be an L-simple function, or a simple function for short. The set of all L-simple functions forms a ring under the usual operations on functions. Indicator functions satisfy the following properties:
By iteration of the identities (2.5) and (2.6) we obtain the inclusionexclusion formula for indicators,
2 Valuation and integral
8
IA 1UA 2 U... UA n = 1 - (1 - IAJ(1 - IA 2 ) · · · (1 - IAJ
+
= L1Ai - LIAinAj
i<j
L IAinAjnA k i<j
+....
(2.7)
A subset G of L that is closed under finite intersections is said to be a generating set of L when every element of L is a finite union of elements of G. Using the inclusion-exclusion formula for indicators, it can be shown that every L-simple function can be written as a finite linear combination r
f
= L{3i I B i
(2.8)
,
i=l
where Bi E G. A real-valued function v on G is called a valuation on G provided that v satisfies identities (2.1) and (2.2) for all sets A, BEG such that AUB E G as well. Note that, since G need not be closed under unions, identity (2.1) does not make sense for all pairs of sets A, BEG. Hence, there is no reason to assume that the identities (2.3) should hold for v if n > 2. Since every element BEL can be expressed as a union B = Bl U ... U Bn with B l , ... , Bn E G, we can attempt to extend v to a valuation p, on all of L by setting
p,(B)
=
Lv(Bi) - L V(Bi n Bj) i<j
+ ... ,
(2.9)
as is suggested by (2.3). There remains to check that p,(B) is well defined, in the case that B could be expressed as a union of elements of G in more than one way. Given a valuation p, on G, define the integral with respect to p, as follows. For an L-simple function f = alIA 1 + ... + akIA1, with Ai E G for 1 :S i :S k, define
Jf
k
dp,
=
L aiP,(A).
(2.10)
i=l
In general, a simple function f has infinitely many expressions of the form (2.4), for Ai in G. Consequently we must check that the integral in (2.10) is well defined.
2.2 Groemer's integral theorem
The existence of the extension (2.9) and the integral (2.10) turn out to be equivalent properties of p" a nontrivial fact stated formally as follows.
2.2 Groemer's integral theorem
9
Theorem 2.2.1 (Groemer's integral theorem) Let G be a generating set for a lattice L, and let f.L be a valuation on G. The following statements are equivalent.
(i) f.L extends uniquely to a valuation on L. (ii) f.L satisfies the inclusion-exclusion identities f.L(B 1UB 2 U···UB n ) = Lf.L(Bi )- Lf.L(BinBj )+ ... , (2.11) i<j
whenever Bi E G and B1 U B2 U··· U Bn E G, and for all n :2: 2. (iii) f.L defines an integral on the vector space of linear combinations of indicator functions of sets in L. Proof We prove the implications (i) ::::} (ii) ::::} (iii) ::::} (i). If f.L extends uniquely to a valuation on all of L, then (ii) follows from an iteration of identity (2.1). Therefore, (i) implies (ii). To show that (ii) implies (iii), suppose there exist non-empty distinct K 1, ... , Km E G and nonzero real numbers a1, ... , am such that m
LaiIKi =0,
(2.12)
i=l
while m
L aiJt(Ki)
=I 0.
(2.13)
i=l
Let L1 = K 1,···, Lm = K m, Lm+1 = K 1nK2, Lm+2 = K 1nK3 , and so on, to define a list L 1, L 2, ... , L p , comprising all possible intersections of the sets K i . Since G is closed under intersections, Li E G for all i. Note also that the collection {Li} is closed under intersections. Suppose that p
LaiILi =0,
(2.14)
i=q
while p
L
aif.L(Li ) =I 0,
(2.15)
i=q
where a q =I 0. Choose an instance of these equations such that q is maximal. It follows from (2.12) and (2.13) that q :2: 1, while the conditions (2.14) and (2.15) imply that q < p.
10
2 Valuation and integral Suppose that x E Lq -
U;=q+l L j •
Then (2.14) implies that
P
O'.q = LO'.ihi(x) = 0, i=q
contradicting our assumption. It follows that
so that
For i > q, note that Lq n Li = L j , where j > q. Using the principle of inclusion-exclusion (ii) we obtain
so that p
L
(3ij.l(L i )
-I- 0
(2.16)
i=q+l
by (2.15), where each (3i is obtained by collecting the terms containing j.l(Li). Meanwhile, application of the same inclusion-exclusion procedure to the indicator functions yields
so that p
L
f3ih i =0
(2.17)
i=q+l
by (2.14). Together (2.16) and (2.17) contradict the maximality of q. This completes the proof that (ii) implies (iii). To show that (iii) implies (i), suppose that the function j.l defines an integral on the space of L-simple functions. For A E L define j.l(A)
=
J
fA
dj.l.
The linearity of the integral together with the identity (2.6) implies that this extension of j.l is a valuation on L. D
2.3 Notes
11
A linear functional T on the vector space of simple functions determines a valuation JL by setting
for every A E L. It is easily verified that, for a simple function
T(f)
=
Jf
f,
dJL.
Thus, insofar as simple functions are concerned, there is a bijective correspondence between linear functionals and valuations. Let B (L) be the relative Boolean algebra generated by the distributive lattice L; that is, the smallest family of subsets of S containing L that is closed under finite unions, finite intersections, and relative complements. Note that for A, BEL (2.18) Let I( L) denote the algebra of simple functions generated by finite sums, products, and differences of indicator functions of sets in L. If follows from (2.5), (2.6), and (2.18) that Ie E I(L) for all C E B(L). Corollary 2.2.2 A valuation JL defined on a distributive lattice L has a uniquely defined extension to the Boolean algebra B (L ).
Proof By Theorem 2.2.1, JL defines an integral on the space of indicator functions I(L). For C E B(L) define JL(C)
=
J
Ie dJL.
The linearity of the integral together with identity (2.7) implies that this extension of JL is a valuation on B(L). D
2.3 Notes
The study of valuations, while natural enough on its own as a finitely additive precursor to the measure theory of modern probability, was invigorated especially by interest in dissection problems on polytopes, and Hilbert's third problem in particular [8, 71, 72, 81] (see also Section 8.6). In the sections that follow we shall see that most of the interesting functionals of geometric probability satisfy the valuation property in some respect.
12
2 Valuation and integral
The integral and extension theorems of Groemer may be found in [32). McMullen and Schneider gave a thorough survey of the modern theory of valuations on convex bodies in [72), later updated by McMullen in
[71).
3 A discrete lattice
In this chapter we focus on combinatorial properties of the lattice of subsets of a finite set, properties which carryover in analogous forms to the lattice of parallelotopes in Chapter 4, of subspaces in Chapter 6, of polyconvex sets in Chapters 5, 7-10, and of spherical polyconvex sets in Chapter 11. An especially important result of this chapter is the characterization of valuations invariant under the permutation group. The idea of characterizing valuations invariant with respect to a group action or a set of symmetries is central to our treatment of geometric probability; this theme will recur frequently in the chapters following.
3.1 Subsets of a finite set Let S be a non-empty set with n elements, and denote by P(S) the set of all subsets of S, partially ordered by subset inclusion. The set P( S) is a (finite) Boolean algebra of subsets. Recall that the union and intersection of sets coincide with the least upper bound and greatest lower bound in the partially ordered set P(S). We denote the elements of P(S) by lower case letters x, y, etc. A segment of P(S), denoted by [x, y], where x ::::: y, consists of all elements Z E P(S) such that x::::: Z ::::: y. Every segment [x, y) is naturally isomorphic to the Boolean algebra P(y - x). A chain in P(S) is a linearly ordered subset; that is, a subset in which, for every pair x, y, either x ::::: y or y ::::: x. An antichain is a subset A <:;; P(S) such that, if x, YEA, then neither x < y nor y < x. A flag F in P(S) is a maximal chain; that is, a chain such that if G ;;;? F and G is a chain, then G = F. These notions apply to all partially ordered sets.
14
3 A discrete lattice
For x E P(S), the rank r(x) is the number of elements of the set x. The antichain consisting of all elements of P(S) of rank k shall be denoted by Pk(S). The size, or number of elements, of Pk(S) is the binomial coefficient
A flag in P(S) is naturally identified with a linear order (Sl' S2,"" sn) on the one-element subsets Si of S. Hence, there are n! flags in P(S). A flag contains an element x E Pk (S) whenever x = Sl U S2 U ... U Sk. Thus, there are k!(n - k)! flags containing a particular x E Pk(S). This elementary argument gives the classical expression for the binomial coefficient:
n! ( n) k - k!(n - k)!' Actually, the same argument can be made to yield a much stronger result. Denote by IAI the size of a finite set A. Theorem 3.1.1 (Sperner's theorem) Let A be an antichain in P(S).
Then
IAIS
((n~2))'
Here the expression (n/2) denotes the greatest integer smaller than or equal to n/2. Evidently equality is attained in Theorem 3.1.1 when A = P(n/2) (S). Proof The proof of this theorem depends on a more precise result, known as the Lubell-Yamamoto-Meshalkin (L.Y.M.) inequality. Let A be an antichain in P(S), and let Ak consist of all elements of A of rank k. Then (3.1) To prove (3.1), notice that every flag meets A in at most one element of P(S). Therefore, the number p of flags meeting A is n
p
=
L k!(n k=O
k)!/Akl·
15
3.1 Subsets of a finite set
Since there are n! flags in P( S), we have p :::; nL Dividing by n! proves the L.Y.M. inequality (3.1). To complete the proof of Sperner's theorem, recall that
for all 0 :::; k :::; n. The L.Y.M. inequality (3.1) now gives
t~
k=O (~) -
((n/2)) -
,
that is,
D
Suppose that 1 :::; r :::; n + 1 is a integer. A subset F ~ P(S) is called an r-family if chains in F contain no more than r elements. For example, an antichain is a I-family. Given an r-family F in P(S) let Fk = FnPk(S). Since every flag in P(S) meets F in at most r elements, we have n
Lk!(n - k)!/Fk/ :::; n!· r. k=O
We then obtain the following generalization of (3.1): n
/Fk /
k=O
k
(3.2)
L-(n) :::;r.
This inequality leads in turn to a generalization of Sperner's theorem to r-families: Theorem 3.1.2 Let F be an r-family in P(S). Then
In order to prove Theorem 3.1.2 we make use of the following lemma.
Lemma 3.1.3 Suppose that 0:::; i :::; n, and if Xo
Co
2:
+ Xl + ... + Xn
CI
2:
2: ... 2:
Co
Cn
>
o.
If Ci 2:
+ CI + ... + Cr-l,
Xi
2: 0 for (3.3)
16
3 A discrete lattice
then (3.4) If Co > ... > Cn > 0, then equality holds in (3·4) if and only if Xi for 0 :::; i :::; r - 1 and Xi = 0 for r :::; i :::; n. Proof Suppose that Xo, . .. ,X n
:::::
= Ci
0 minimize the sum (3.5)
subject to the condition (3.3). If this minimum value of (3.5) is less than r then Xi < Ci for some i :::; r - 1. Let i be the smallest index such that Xi < Ci, so that Xk/Ck = 1 for all 0:::; k < i (unless i = 0). The inequality (3.3) then implies that there is an index j ::::: r such that Xj > O. Let j be the largest such index, so that Xk = 0 for k > j (unless j = n). In particular, note that j > i. Suppose Ci = Cj. Since Ci ::::: CHI··· ::::: Cj, we have Ci = CHI = ... = Cj. It then follows that Co
+ ... + Ci-I + Xi + ... + Xj
+ ... +xn > Co + ... + Cr-I Co + ... + Ci-I + (r Xo
i)Ci,
since r - 1 < j. Therefore, Xi
It follows that
+ ... + Xj
:::::
(r -
i)Ci.
(3.6)
3.1 Subsets of a finite set
17
by (3.6). In other words, n
L
Xk
2:
(3.7)
r.
k=O Ck
This contradicts the assumption that Xo, .•. , Xn Illllllmizes (3.5) at a value less than r. Therefore, inequality (3.4) follows. Now suppose instead that Ci > Cj. If Xi + Xj :s; Ci set Yi = Xi + Xj and Yj = O. Otherwise, set Yi = Ci and Yj = Xj - (Ci - Xi). In either case set all other Yk = Xk· Since Ci > Cj it follows in both cases that Yi
+ Yj <
Xi
Ci
Cj
Ci
+ Xj, Cj
so that n
L
n Yk
k=O Ck
Xk
,
k=O Ck
contradicting the assumption that Xo, • .• ,X n minimizes (3.5). Inequality (3.4) now follows again for this case. If Co > ... > Cn, then Ci > Cj is guaranteed, and so it follows that (3.5) is minimized only if Xi = Ci for 0 :s; i :s; r - 1 and Xi = 0 for r :s; i :s; n. D
Proof of Theorem 3.1.2 Relabel the binomial coefficients Co, CI,···, Cn in descending order, so that Co 2: CI 2: ... 2: en > 0; then relabel the numerators /Fk / in (3.2) by xo, Xl, •.. , X n , so that each Xk is the numerator of that term of (3.2) having Ck as denominator. The inequality (3.2) now becomes
It then follows from Lemma 3.1.3 that Xo
+ Xl + ... + Xn :s; Co + CI + ... + Cr-l·
In other words,
D
The theory of binomial coefficients and antichains generalizes nicely to a theory of multinomial coefficients and special collections of ordered
3 A discrete lattice
18
partitions, known as s-systems. Sperner's result can also be generalized from a bound on the size of an antichain to a bound on the size of an s-system. A map 15 : {I, ... ,r} --+ P(8) is called an r-decomposition of 8 if (i) 8(i) n 8(j) = 0 for i i= j, and (ii) 15(1) U··· U 8(r) = 8. Denote by Dec(8, r) the set of all r-decompositions of 8. Note that for each 15 E Dec(8, r) 115(1)1
+ ... + 18(r)1 =
n.
Given non-negative integers aI, a2, ... , ar such that al + ... + ar = n we denote by Pal , ... ,a r (8) the set of all r-decompositions 15 such that 18(i)1 = ai for i = 1, ... ,r. In other words, Pal , ... ,ar (8) is the set of all (ordered) partitions of 8 into disjoint unions of subsets having sizes aI, ... , a r . Evidently the set Dec( 8, r) can be expressed as the finite disjoint union Dec(8,r) The size of
Pal, ... ,ar
=
(8) is given by the multinomial coefficient
An s-system of order r (or an s-system in Dec(8, r)) is a subset (J Dec( 8, r) such that the set {15 (i) : 15 E (J}
~
(3.8)
is an antichain in P(8) for each 1 ::::; i ::::; r. An obvious example of an s-system of order r is Pal , ... ,a r (8) for some admissible selection of aI, ... , a r . If 15, ( E Pal, ... ,a r (8) then 15 (i) and ((i) both have size ai, so that either 8(i) = ((i) or the two sets are incomparable in the subset partial ordering on P(8). This holds for i = 1, ... , r, and so the antichain condition on (3.8) is satisfied. Other disguised examples with which we have already worked are the s-systems of order 2. Let A be an antichain in P(8). For each x E A we can express 8 as the disjoint union x l±J 8 - x, so that the pair (x, 8 - x) is a 2-decomposition in Dec(8, 2). Moreover, the set
{8-x:xEA}
19
3.1 Subsets of a finite set
is also an antichain in P(S), so that the set IT
= {(x, S - x) : x E A}
is an s-system of order 2. Thus the notion of s-system is a generalization ofthe notion of an antichain. Similarly, the collection Pk(S) can also be viewed as Pk,n-k(S) through the bijection x f-+ (x, S - x). For 15 E Pa1, ... ,ar(S) we say that a flag (XO,XI,""X n ) in P(S) is compatible with 15 if (i) x a1 = 15(1), and (ii) x a1 +'+ ai - xal+.+ai_l = 8(i), for i :2: 2. Here the difference X a1 +.+ai - X a1 +.+ai-l denotes the complement of the set xal+.+ai_l inside the larger set x a1 +.+ ai . For A <:;;: Pa1, ... ,a r , let Flag(A) be the set of all flags (xo, Xl,.'" Xn) compatible with some 15 E A. Note that for each 15 E Pa1, ... ,ar(S), there are exactly al!a2!'" ar! flags compatible with 15; that is, to choose a flag compatible with 15 one must choose a permutation of the al elements of 15(1), of which there are all, and then a permutation of the a2 elements of 15(2), of which there are a2!, and so on up through 8(r). Since each of the n! flags is compatible with one (and only one) r-decomposition in Pa1, ... ,ar(S), it follows that
Just as Sperner's Theorem 3.1.1 gives the maximum possible measure for an antichain A in P(S), a generalization of this theorem gives the maximum possible size for an s-system in Dec( S, r). En route to such a generalization we prove a multinomial version of the L.Y.M. inequality.
Theorem 3.1.4 (The multinomial L.Y.M. inequality) Let Dec(S, r) be an s-system. For
so that IT=
al
+ ... + ar =
IT
C
n let
u
is a disjoint union. Then
(3.9)
20
3 A discrete lattice
Proof For al + .. ·+ar is given by
=
n the number of flags compatible with aal, ... ,a r
IFlag(aal, ... ,ar )1
=
laal, ... ,ar lal"" ar·, ·
Suppose that a flag (XO,Xl, ... ,Xn ) is compatible with both ,,(,8 E a. Then "(1) = x a1 and 8(1) = Xbll where al = h(l)1 and bl = 18(1)1. Since (XO, Xl,"" Xn) is a flag, we have Xa1 ~ Xbl or vice versa. However, a is an s-system, so that either "(1) = 8(1) or the two sets are incomparable. Therefore "(1) = 8(1) and al = bl . Continuing, we have "(2) = x a1 +a2 - Xa1 and 8(2) = Xbl+b2 - x a1 (since al = bI). A similar argument then implies that "(2) = 8(2) and a2 = b2 . Continuing in this manner we conclude that "(i) = 8(i) for each 1 ~ i ~ r, so that "( = 8. In other words, every flag in P(S) is compatible with at most one r-decomposition 8 E a. It follows that
ar', la al, ... ,a r lall.·· .
L
IFlag(aal, ... ,aJI
=
IFlag(a)I ~ n!
al+···+ar=n so that
D
We are now ready to prove a multinomial generalization of Sperner's Theorem 3.1.1. Theorem 3.1.5 (Meshalkin's theorem) Let a be an s-system in Dec(S, r). Then
lal
~ ((n/r) , ... , (n/r), (n7r) + 1"", (n/r) + 1) , , v
r-b
",
v
(3.10)
",
b
where n == b mod r. Here (n/r) denotes the largest positive integer less than or equal to n/r.
Proof It is not difficult to show that, for all compositions al + .. ·+ar = n of a positive integer n, we have
3.2 Valuations on a simplicial complex
21
For a sketch of a proof, see the analogous Proposition 6.5.2. Let aal, ... ,a r = an Pa1 ,... ,ar (8). It now follows from (3.9) that
L
al
(
L
/aal, ... ,arl ) < /za1,;;.,a)/ ::; 1, +·+ar=n (n/r), .. ,(n/r),(n/r)+l, .. ,(n/r)+l al +·+ar=n al,···,ar
so that
/a/
al+~r=laal, ... ,ar/ ::; Cn/r) , .'.. , (n/r) , (n7r) + 1"", (n/r) + 1)' D
Several other results of extremal set theory follow from the L.Y.M. inequality or its variants, but we reluctantly move on to the next topic.
3.2 Valuations on a simplicial complex Define a simplicial complex to be a subset A of P( 8) such that if x E A and y ::; x then YEA. A simplicial complex is a partially ordered set in the order induced by P(8). The set of maximal elements of a simplicial complex is an antichain. A simplicial complex having exactly one maximal element is called a simplex. A simplex whose unique maximal element is a set of size k is called a k-simplex. The (set-theoretic) union and intersection of any number of simplicial complexes is again a simplicial complex. Thus, the set L(8) of all simplicial complexes in P(8) is a distributive lattice, and we can study valuations on L(8). For x E P(8), denote by x the simplex whose maximal element is X; that is, the set of all y E P(8) such that y ::; x. It follows from Groemer's integral Theorem 2.2.1 and Corollary 2.2.2 that every valuation fJ on L(8) extends uniquely to a valuation, again denoted by fJ, on the Boolean algebra P(P(8)) of all subsets of P(8), which is generated by L(8). Such a valuation is evidently determined by its value on the one-element subsets of P( 8); that is, by arbitrarily assigning a value fJ( {x}) for each X E P(8). ,Let x be of rank k, and let A 1, A 2 , ••. ,Ak be the maximal simplices Ai E x such that Ai i=- x. (These simplices are sometimes called the facets of x.) Then
3 A discrete lattice
22
The right-hand side can be computed in terms of simplices of lower rank, by the inclusion-exclusion principle. Thus, by induction on the rank, we have the following theorem.
Theorem 3.2.1 Every valuation J-l on the distributive lattice L(8) of all simplicial complexes is uniquely determined by the values J-l(X) , x E P(8). The values J-l(x) may be arbitrarily assigned. D A valuation J-l on L(8) is called invariant if it is invariant under the group of permutations of the set 8; that is, if J-l(A) = J-l(gA) for every simplicial complex A and for every permutation 9 of the set 8 (which induces a permutation on L(8), also denoted by g). We next establish the existence of the Euler characteristic. The following is an immediate consequence of Theorem 3.2.1.
Theorem 3.2.2 (The existence of the Euler characteristic) There exists a unique invariant valuation J-l on L(8), called the Euler characteristic, such that J-lo(x) = 1 for every simplex x with rex) > 0, and such D that J-lo(0) = 0. Next, we derive the classical alternating formula for the Euler characteristic. Define a valuation on P(P(8)), denoted J-l~, by setting
J-l~(0) = 0, and J-l~({x})
= (-I)k-l,
if rex) = k. Then
J-l~(X)
=
L
J-l~( {y})
y<;'x
L
J-l~({Y}) +
y<;,x,r(y)=l
Since the simplex
L
J-l~({Y})
+ ... + J-l~({x}).
y<;,x,r(y)=2
x contains
elements of rank j, the right-hand side simplifies to
G) -G) + G) -... +
(_I)k+l
G)
=
1,
3.2 Valuations on a simplicial complex
23
so that JL~(x) = JLo(x), for all simplices x. It now follows from Theorem 3.2.1 that JL~ = JLo, and that the following formula holds.
Theorem 3.2.3 (The discrete Euler formula) Let A be a simplicial complex, and let fk be the number of elements {or 'faces'} of rank k. Then
JLo(A) =
h - h + h - ....
(3.11) D
For i > 0, set
and extend JLi to all of L(8) by Theorem 3.2.1. Clearly, for every simplicial complex A,
The discrete Euler formula can now be rewritten
JLo(A)
=
JL1(A) -1L2(A)
+ JL3(A) -
... ,
(3.12)
for any simplicial complex A. The valuations JLk can also be expressed in terms of symmetric functions. Let P 1(8) = {a1' a2, ... , an}. Given the symmetric function
let ti(X) = 1 if ai E x and ti(X) = 0 if ai fj. x. Evaluated at x, for x f 0, we have til ti2 ... tik = 1 if {ail' ... ,aik} E x, and til ti2 ... tik = 0 otherwise. It follows that ek(t1, t2, ... , tn) = JLk(X) for every simplex x other than {0}. Note also that if x E P(8) and r(x) = j then JLi({X}) = 1 if i = j, while JLi({X}) = 0 ifi fj.
Theorem 3.2.4 (The discrete basis theorem) The invariant valuations JLo, JL1, ... ,JLn span the vector space of all invariant valuations JL on L(8) such that JL( {0}) = O. The only linear relation among them is formula {3.12}.
Proof Suppose JL is an invariant valuation on L(8) such that JL( {0}) = o. Extend JL to all of P(P(8)). Note that the extended valuation, which is still denoted JL, is again invariant. If x and y have the same rank in P(8), say r(x) = r(y) = i, then there exists a permutation g of 8 such
3 A discrete lattice
24
that gx = y. Therefore, JL( {x}) Thus, the valuation
= JL( {y}) = Ci, for some constant Ci'
vanishes on all singleton sets {x}, for all x E P(8), and therefore vanishes on all of P(P(8)). D As an application of the discrete basis theorem, we shall derive a discrete analogue of the kinematic formula (whose classical geometric version appears in Chapter 10). One way to construct invariant valuations on L(8) is the following. Start with any valuation JL on L(8) such that JL( {0}) = 0, and let B be any simplicial complex. For any simplicial complex A, set
JL(A; B)
1
= , L JL(A n gB), n.
9
where 9 ranges over all permutations of the set 8 of size n. For fixed A, the set function JL(A; B) is a valuation in the variable B; in fact, it is an invariant valuation. It can therefore be expressed as a linear combination of the valuations JLi, with coefficients ci(A) depending on A: n
JL(A; B)
L
=
ci(A)JLi(B).
(3.13)
i=l
Meanwhile, for fixed B, the set function JL(A; B) is a valuation in the variable A. From this it follows that each of the coefficients ci(A) is a valuation in the variable A. The coefficient ci(A) can be given an explicit expression in terms of JL. This follows from the fact that if x E P(8) and r ( x) = j then JLi ({x}) = 1 if i = j and is zero if i =f- j. Consider the case in which JL is an invariant valuation. If so, then
JL(A;B)
1
,LJL(AngB) n. 9
1
, LJL(gA n B) n.
=, 1
n.
LJL(g-lAnB) 9
= JL(B; A).
9
Moreover, the coefficients ci(A) are now invariant valuations in the variable A. Therefore, Theorem 3.2.4 implies that n
JL(A; B)
=
L i,j=l
CijJLi(A)JLj(B).
25
3.2 Valuations on a simplicial complex
Since JL(A; B) = JL(B; A), it is evident that Cij = Cji. It turns out that most of the constants Cij are equal to zero. In order to compute the constants Cij explicitly, extend the valuation JL to the Boolean algebra P(P(8)) generated by L(8), and let (Vi denote the value of JL on a singleton set in P(P(8)) whose element is a subset of 8 of size i.
Theorem 3.2.5 (The discrete kinematic formula) Suppose that JL is an invariant valuation on L( 8). For all A, B E L( 8),
Proof Suppose that Xi, Yj C 8 with size i and j respectively. Let A = {Xi} and B = {Yj}. For any permutation 9 of 8, the set An gB = 0 if i =f- j. If i = j then An gB = 0 if Xi =f- gYj. Since there are i!(n - i)! permutations 9 of 8 such that Xi = gYj (if i = j), we have
JL(A;B)
1 i'(n - i)' =, LJL(AngB) =. , 'JL(A) =
n.
n.
9
(n)-l
.
(Vi·
2
Meanwhile, JLk(A) = 1 if k = i and is equal to zero otherwise. Similarly, JLk(B) = 1 if k = j and is equal to zero otherwise. Hence, n
JL(A; B) = L CijJLi(A)JLj(B) = Cij' i,j=l Therefore,
if i
=j
and is equal to zero otherwise.
D
We shall be particularly concerned with the case JL = JLo. The discrete Euler formula (3.11) implies that JLo({xd) = (-I)i+I, so that
~! ~ JLo(A n gB) =
t(
_1)i+l
(~)
-1 JLi(A)JLi(B),
for all A, BE L(8). If x and 'fJ are simplices, then either x n 'fJ is a smaller (non-empty) simplex, in which case JLo(x n 'fJ) = 1, or x n 'fJ = 0, in which case
26
3 A discrete lattice
J-lo (x
n y) = O. The probability that a randomly chosen k-simplex
Xk
shall meet a fixed l-simplex Yl can now be computed as follows:
so that 1
n!
~J-lO(YI n gXk) =
t;( _1),+1 (n) (l) (k) n.
-1
iii .
(3.14)
The equation (3.14) leads to a notable example of how the discrete kinematic formula can be used to generate identities for the binomial coefficients. To generate such an identity, we use elementary probabilistic reasoning to compute instead the probability that Yl n gXk = 0, for a random permutation g. Label the elements of S by {Sl, .. . , Sn} so that Xk = {Sl, ... ,Sk}. In order for Yl ngxk = 0 to hold, we require gSl E S-Yl, of which there are n-l choices. There then remain n-(l+I) possible values for gS2, etc., so that there are (n -l)(n -l-I)··· (n - l - k
+ 1)
choices of values for gSl, ... , gSk. Having chosen these values, there are n - k possible choices remaining for gSk+1, then n - k -1 possible values for gSk+2, and so on, up to one possible value remaining for gSn. It follows that there are (n -l)··· (n -l - k
+ 1)(n _
k) ... 1 = (n -l)!(n - k)! (n - k -l)!
permutations g of S such that Yl n gXk = 0. Hence the probability that Yl n gXk = 0 for a random permutation g is given by
~ (n -l)!(n - k)! n!
(n-k-l)!
=
k!(n - k)!(n -l)! = (n)-l (n -l) n!k!(n-k-l)! k k'
It now follows from (3.14) that
(3.15) By adding the term corresponding to i = 0 to both sides of (3.15) and multiplying by -1 we obtain the following identity:
3.2 Valuations on a simplicial complex
27
Theorem 3.2.6
for all positive integers 0 :::; k, l :::; n. Note that, if k
0
+ l > n, then
In the preceding argument this corresponds to the case in which the two sets Yl and gXk have non-empty overlap for any permutation g, i.e. the case in which Yl n gXk = 0 with probability zero. We conclude this discussion of subsets and simplices with an application of the results of Section 3.1 to a question posed by Spemer. Let A E £(8), and suppose that all ofthe maximal elements of A have rank kj i.e. suppose that every face of A is contained in a k-face of A. Let [All denote the collection of all l-faces of A, for 0 :::; l :::; k. Is there a lower bound for the number of l-faces I[Altl, given the number of k-faces (maximal faces) I[Alkl? The L.Y.M. inequality gives one answer to this question. Theorem 3.2.7 Suppose A E £(S) such that every maximal element of A has rank k. For 0 :::; l :::; k,
I[Alti ~
k!(n - k)! l!(n -l)! I[Alkl·
Proof Let Bl = PI(S) - [A]t. If Y E Bl then Y cannot be contained in any x E A. In other words, the set [Alk UBI is an antichain. It then follows from (3.1) that
Meanwhile, IBtI
so that
=
IPI(S) - [Altl
=
(7) -1[Altl,
28
3 A discrete lattice
It follows that
I[All! ~
(n) -1 k!(n - k)! ( n) l k I[Alkl = l!(n -l)! I[Alkl·
o For example, in the case l
=k-
1, we have
3.3 A discrete analogue of Helly's theorem We turn next to a special case and discrete analogue of Helly's theorem, whose classical geometric version is given in Chapter 5.
Theorem 3.3.1 (The discrete Helly theorem) Let S be a finite set of size lSI = n, and let F be a family of subsets of s. Suppose that, for any subset G <:;;: F such that IGI ::; n (that is, every subfamily of cardinality at most n of F),
n
A#0.
n
A#0.
AEG
Then
AEF
In other words, if every n elements of F have non-empty intersection, then the entire family F of subsets has non-empty intersection. Proof If IFI ::; n the result is trivial. Suppose that Theorem 3.3.1 holds for IFI = m, for some m ~ n. We show that the theorem also holds for IFI = m + l. Write S = {Sl, ... , sn} and F = {AI, ... ,Am+ 1 }, where Ai <:;;: S. For each 1 ::; j ::; m + 1 denote by L j the intersection
Our induction assumption for the case IFI = m implies that each L j is non-empty. Therefore, there exists Sij E L j for each j. However, since m + 1 > n there must be s E S such that s E Ljl n Liz for some j1 # j2. Since s E Ljp we have s E Ai for all i # j1. Similarly, s E Ai for all
3.4 Notes
29
i =f- j2. Hence, m+l
sE
n A= n A.
i=l
AEF
o 3.4 Notes For a general reference to the theory of lattices and partially ordered sets, see [92] (also [3, 27,42, 80]). For a graph-theoretic viewpoint, see [7]. In [84], Schanuel developed the Euler characteristic from a categorytheoretic perspective. In [12], Chen extended the Euler characteristic to linear combinations of indicator functions of unbounded closed convex sets and unbounded relatively open convex sets. A thorough treatment of the combinatorial theory of the Euler characteristic appeared in [79, 80]. The face enumerators J-li playa role analogous to that of the intrinsic volumes on parallelotopes and polyconvex sets (see Sections 4.2 and 7.2). The discrete basis theorem and kinematic formulas can be extended to the more general setting of finite vector spaces; see [53]. Sperner's Theorem 3.1.1 first appeared in [91], as did Theorem 3.2.7. The proof presented in this section is due to Lubell [63]. Meshalkin proved Theorem 3.1.5 in [73]. In this section we follow a more simplified approach due to Hochberg and Hirsch [45]. The generalization of Sperner's theorem to r-families (Theorem 3.1.2) was originally due to Erdos [22], but the proof given in this section is due to Harper and Rota [42]. While Theorem 3.2.7 gives a lower bound to the number of l-faces of a simplicial complex A, all of whose maximal elements have dimension k, Katona [48] and Kruskal [57] independently obtained a much stronger result, giving the exact minimum size of [A]l, a minimum that is independent of the size of the ambient set S. Katona's original paper may also be found in [27]. For a shorter proof of the Katona-Kruskal theorem, see [3, 7]. A survey of results in extremal set theory, including generalizations of Sperner's theorem and the L.Y.M. inequality, was presented in [29]. The discrete Helly Theorem 3.3.1 is a special case of a more general theorem of convex geometry. This geometric result is treated in greater detail in Chapter 5 (see also [85, pp. 3-5]). Helly's original theorem motivated considerable developments in the field of combinatorial geometry, and 'Helly-type' theorems are now common in many branches of mathematics [7, 18, 21].
4 The intrinsic volumes for parallelotopes
We develop next a theory of invariant valuations for the lattice of finite unions of orthogonal parallelotopes, having edges parallel to a fixed frame. Many of the central results in geometric probability can be stated and proven easily in the context of parallelotopes. For this reason the lattice of parallelotopes will serve as a model for the more difficult task of developing a lattice theory for finite unions of compact convex sets in Rn. The Euler characteristic, intrinsic volumes, and valuation characterization theorems for the lattice of parallelotopes will serve as prototypes in Chapters 5-9 for analogous constructions and characterization theorems in the more general context of polyconvex sets.
4.1 The lattice of parallelotopes Choose a Cartesian coordinate system in Rn, which shall remain fixed throughout this chapter, and let Par(n) denote the family of sets that are obtained by taking finite unions and intersections of orthogonal parallelotopes (i.e. rectilinear boxes), with sides parallel to the coordinate axes. If P E Par(n), we shall say that P is of dimension n (or has full dimension) if P is not contained in a finite union of hyperplanes of Rn; that is, if P has a non-empty interior. Otherwise, we shall say that P is of lower dimension. (Recall that a hyperplane in R n is a plane of dimension n - 1, not necessarily through the origin.) In general, a set P E Par( n) has dimension k if P is contained in a finite union of k-planes in R n, but is not contained in any finite union of k - 1 planes. Note that Par(n) is closed under finite unions and intersections. This follows from the basic fact that the intersection of two parallelotopes is a parallelotope. In other words, Par(n) is a distributive lattice.
31
4.1 The lattice of parallelotopes
Denote by Tn the group generated by translations and permutations of coordinates in Rn. For A ~ Rn and 9 E Tn, write
gA = g(A) = {g(a) : a E A}. A valuation f..L defined on Par( n) is said to be invariant when
f..L(gP)
=
(4.1)
f..L(P)
for all 9 E Tn and all P E Par(n). If f..L(gP) = f..L(P) is known to hold only for translations 9 of R n, then we shall say that f..L is translation invariant. The object of this section is to determine all invariant valuations defined on Par(n). To avoid pathological cases, we shall impose a continuity condition on the valuations to be considered. For A ~ Rn and x E R n , the distance d(x, A) from the point x to the set A is given by
d(x, A)
inf d(x, a),
=
aEA
where d(x, a) = Ix - al is the usual distance between points in Rn. Note that d(x, A) = 0 if x E A or if x is a limit point of A. For K,L ~ Rn, the Hausdorff distance 8(K,L) is defined by
(4.2)
8(K, L) = max (sup d(a, L), supd(b, K)) . aEK
bEL
If K and L are compact, then 8(K,L) = 0 if and only if K = L. A sequence of compact subsets Kn of Rn converges to a set K, or Kn --+ K, if 8(Kn' K) --+ 0 as n --t 00. Let Bn denote the unit ball in Rn. For K ~ Rn compact and f > 0, define
K
+ fB n =
{x
+ EU
:
x E K and u E Bn}.
The following lemma gives an easier and more practical way to think about the distance 8. Lemma 4.1.1 Let K, L ~ Rn be compact sets. Then 8(K, L) ::::; only if K ~ L + fB n and L ~ K + fB n .
f
if and
Proof Suppose that K ~ L + fBn. For x E K there exist y ELand u E Bn such that x = y + EU. In other words, Ix - yl : : ; f, so that d(x, L) ::::; f. Similarly, if L ~ K + fBn, then d(y, K) ::::; f for all y E L. Therefore, 8(K, L) ::::; f.
32
4 The intrinsic volumes for parallelotopes
Meanwhile, if there exists x E K such that x 1:. L + fBn (or vice versa), then Ix - yl > f for all y E L, so that 8(K, L) ;::: d(x, L) > f. 0 In view of Lemma 4.1.1, we see that a sequence of compact subsets Kn -----t K, if for f > 0 there exists N > 0 such that K c:;:: Ki + fBn and Ki c:;:: K + fBn whenever i > N. Theorem 4.1.2 The distance 8 defines a metric on the set of all compact subsets ofRn. Proof The distance 8 is clearly symmetric and non-negative. To verif:Y the triangle inequality, suppose that K, L, M c:;:: Rn are compact. Let fl = 8(K, M) and f2 = 8(L, M). By Lemma 4.1.1, K c:;:: M + f1B n and M c:;:: L + f2Bn' so that K c:;:: L + f2Bn + f1B n
= L + (f2 +
fdB n .
,
Similarly, L c:;:: K +(f2+fl)Bn. Lemma 4.1.1 then implies that 8(K, L) :s;
fl + f2.
0
We now focus once again on the lattice Par( n) of finite unions of parallelotopes. A valuation f..L is said to be continuous on Par(n), provided that
whenever Pi, P are parallelotopes (and not just finite unions) and Pi -----t P. Another condition that will prove useful is monotonicity. A valuation f..L is said to be increasing on Par(n), provided that f..L(P) :s; f..L(Q), whenever P, Q E Par(n) and P c:;:: Q. Similarly one defines decreasing valuations. A valuation f..L is said to be monotone on Par(n), if f..L is either an increasing valuation or a decreasing valuation. When studying valuations on Par(n) we may restrict our attention to the generating set of parallelotopes in R n with edges parallel to the coordinate axes. Spe~ifically, we have the following extension theorem. Theorem 4.1.3 (Groemer's extension theorem for Par(n)) A valuation f..L defined on parallelotopes with edges parallel to the coordinate axes admits a unique extension to a valuation on the lattice Par( n). Proof In view of Groemer's integral Theorem 2.2.1, it is sufficient to show that f..L defines an integral on the space of indicator functions of parallelotopes.
4.1 The lattice of parallelotopes
33
The proposition is trivial in dimension zero. Assume that the proposition holds in dimension n - 1. Suppose that there exist distinct parallelotopes PI, ... , Pm such that m
(4.3)
LaJPi =0 i=l
while m
L aiJ.l(Pi ) =
(4.4)
1.
i=l
Let k be the number of parallelotopes Pi in the expressions above of full dimension n, and suppose that k is minimal over all possible such expressions. If k = 0 then PI, ... , Pm are each contained inside a hyperplane. Let l denote the (finite) number of hyperplanes containing the parallelotopes PI"'" Pm, and suppose that l is minimal over all such expressions. If l = 1 then PI,'" , Pm are all contained in a single hyperplane. It then follows from the induction assumption (on the dimension of the ambient Euclidean space) that we have a contradiction. Therefore l > 1, and there exist hyperplanes HI"'" HI, orthogonal to the coordinate axes, such that Pi ~ HI U ... U Hz for i = 1, ... ,m. Suppose, without loss of generality, that PI ~ HI. Since IpinHl = IpJHll it follows from (4.3) that m
(4.5)
L aJpinHl = O. i=l
Meanwhile, m
L aiJ.l(Pi n Hd =
0,
(4.6)
i=l
by the induction assumption on dimension, since each Pi n HI ~ HI, a hyperplane. Subtracting equations (4.5) and (4.6) from (4.3) and (4.4) respectively, we have m
L ai(Ipi - IpinHJ = 0,
(4.7)
i=l
and m
L ai(J.l(Pi ) ,-- J.l(Pi n HI)) = 1. i=l
(4.8)
34
4 The intrinsic volumes for parallelotopes
Since PI n HI = PI, equations (4.7) and (4.8) take the form of (4.3) and (4.4), where the nonzero terms involve parallelotopes Pi in at most l - 1 hyperplanes, contradicting the minimality of l. It follows that k 2: 1. Suppose then that PI has dimension n. Choose a hyperplane H, with associated closed half-spaces H+ and H- such that PI n H is a facet of PI, oriented so that PI c H+. Since I Pi nH + = Ip,!H+, it follows from (4.3) that m
= O.
L <Xi1pinH+ i=l
Similarly, m
m
L<XilpinH
=
0 and
i=l
L<xJpi nH -
=
o.
i=l
Meanwhile, since JL is a valuation, m
m
m
m
i=l
i=l
i=l
i=l
Since the sets Pi n H lie inside a space of dimension n - 1, the sum
2::1 <XiJL(Pi n H) = 0 by the induction assumption. Because PI n Hhas dimension n -1, the sum 2::1 <XiJL(Pi nH-) = 0 by the minimality of k. From (4.4) we have m
m
i=l
i=l
There are n hyperplanes HI"'" Hn such that n
By iterating the preceding argument, we have m
L <XiJL(Pi n Hi n··· n H:}:) i=l
so that m
L <XiJL(Pi n Pd
=
i=l
while a similar argument using (4.3) gives m
L<XiIPinPl i=l
= O.
1,
=
1,
4.2 Invariant valuations on parallelotopes
35
After repeating this argument with parallelotopes P2 , ••. , Pm we have
~aiJ-l(PI n .. · nPm) = (~ai) J-l(PI n .. · nPm) =
°
This implies that al + .. ·+a m =f- and that PIn·· ·npm =fa similar argument using (4.3) gives
f
aJp1n ... nPm
i=l
so that either al either case.
=
(f
a i ) Ipln ... nPm
1.
0. Meanwhile
= 0,
i=l
+ ... + am =
°or PI n··· n Pm = 0, a contradiction in 0
4.2 Invariant valuations on parallelotopes To begin the classification of invariant valuations on Par(n), consider the problem in RI. An element of Par(l) is a finite union of closed intervals. Set
J-l6(A) = number of connected components of A, J-l~(A) = length of A.
One easily verifies that J-l6 and J-l~ are both continuous invariant valuations on Par(l). We shall prove that every continuous invariant valuation on Par(l) is a linear combination of J-l6 and J-ll. Suppose that J-l is a continuous invariant valuation on Par(l). Let c = J-l(A) , where A is a set consisting of a single point in Rl, and let J-l' = J-l - CJ-l6· Note that the invariant valuation J-l' vanishes on points. Define a continuous function f : [0, +00) ----t [0, +00) by the equation
f(x)
=
J-l'([0, x]).
If A is a closed interval of length x, then the invariance of J-l' implies that J-l'(A) = f(x). If A and B are closed intervals of length x and y, such that An B is a point, then
f(x + y) = J-l'(A U B) = J-l'(A) + J-l'(B) - J-l'(A n B) = f(x) + f(y), so that f(x) = rx for some constant r. Hence, J-l' = rJ-lL and our assertion is proved. We now turn to Rn. There is one well known continuous invariant valuation defined on Par(n), namely, the volume. Denote by J-ln(P) the volume of a finite union P of parallelotopes of dimension n.
4 The intrinsic volumes for parallelotopes
36
Recall that the elementary symmetric functions of the polynomials
eo
=
Xl,
X2, ... ,Xn are
1,
n
ek(XI, X2,··· ,Xn ) =
L
Xi1Xi2 .. , Xik' 1 :S k :S n.
l:'Oh < .. -
We shall prove the following theorem. Theorem 4.2.1 For 0 :S k :S n, there exists a unique continuous valuation fJk on Par(n), invariant under translations and permutations of coordinates, such that
(4.9) whenever P is a parallelotope with sides of length
Xl,
X2, ... ,Xn-
For example, fJn(P) is the volume of P, if P has dimension n. Proof Let fJ6 and fJ~ be the valuations previously defined on R I, and let fJf = fJ6 + tfJ~, where t is a variable. Consider the product valuation on Rn, given by the n-fold product (4.10) Because a parallelotope in Par(n) is a Cartesian product of line segments, fJf defines a valuation on parallelotopes. By Groemer's extension Theorem 4.1.3, fJf then extends to a valuation on all of Par(n). Let us compute the value of fJf on the parallelotope P with edges of length Xl, X2, ... , Xn . Since P = h X 12 x··· x In, where Ij is an interval of length Xj, the definition (4.10) implies that (4.11) Meanwhile
fJ}(Ij) = fJ6(Ij)
+ tfJ~(Ij) = 1 + tXj.
By expanding the right-hand side of (4.11), we obtain fJ~(P)
(1 + txd(l
+ tX2) ... (1 + tXn) 1 + el(xI, ... xn)t + e2(xI, ... xn)t 2 + ... + en(xI, ... Xn)t n .
Now let Q E Par(n). The set Q can be expressed as a union Q = PI U P 2 U ... U Pn , where the Pi are parallelotopes. (Recall that the
4.2 Invariant valuations on parallelotopes
37
intersection of a collection of parallelotopes is a parallelotope.) By the incl usion-excl usion principle,
JL~(Q)
=
LJL~(Pi) - LJL~(Pi i
n Pj ) + ... - ....
i<j
Collecting terms in each power of t, we conclude that there exist valuations JLo, JLI, ... ,JLn such that
where the valuations JLi are uniquely determined by the identities (4.9).
o Note that, if the parallelotope P has dimension k < n, then the valuation JLi(P) has an ambiguous sense. It may indicate the value of JLi(P) in Rn as defined in the previous theorem, but it may also denote the value of JLi(P) when computed within a lower dimensional plane containing P (and isomorphic to R m , where k :s; m < n). It is a notable consequence of the preceding theorem that the two valuations coincide. In other words, JL7'(P) = JL7(P).
Therefore, there will be no need to indicate the dependence of JLi(P) on the space R n in which the parallelotope P is embedded. We summarize this remarkable fact as follows. Theorem 4.2.2 The valuations JLi on Par(n) are normalized independently of the dimension n. 0
In other words, the value JLk(P) is 'intrinsic' to the set P, and independent of the dimension of the ambient space. For this reason the valuation JLk is called the kth intrinsic volume. The valuation JLo is called the Euler characteristic. As a corollary of Theorem 4.2.1, we see that the Euler characteristic is the only valuation on Par(n) that takes the value 1 on all (non-empty) parallelotopes: Let HI and H2 be complementary orthogonal subspaces ofRn spanned by subsets of the given coordinate system and having dimensions hand n - h, respectively. Let Pi be a parallelotope in Hi and let P = PI X P2 . The intrinsic volumes satisfy the following property with regard to orthogonal Cartesian products.
4 The intrinsic volumes for parallelotopes
38
Proposition 4.2.3 f.li(PI x P 2 ) =
L
(4.12)
f.lr (Pdf.ls (P2 ).
r+s=i
The identity (4.12) is therefore valid when PI and P2 are finite unions of parallelotopes. Proof Suppose that PI has sides of length Xl, ... ,Xh and P2 has sides of length YI, ... , Yn-h. Then we have r+s=i
Let jr+l = kl Yn-h. Then
+ h, ... ,ji = jr+s = ks + h, and let Xh+1 = YI,···, Xn
=
r+s=i
X·Jl ···X·Jr X·Jr+l ···X·Ji
°
o
A valuation f.l on Par(n) is said to be simple if f.l(P) = for all P of dimension less than n. The restriction of the volume f.ln to the lattice Par( n) is characterized by the following theorem.
Theorem 4.2.4 (The volume theorem for Par(n» Let f.l be a translation invariant simple valuation defined on Par(n), and suppose that f.l is either continuous or monotone. Then there exists C E R such that f.l(P) = Cf.ln(P) for all P E Par(n); that is, f.l is equal to the volume, up to a constant factor.
Proof Let [0,1]n denote the unit cube in Rn, and let C = f.l([0,1]n). Recall that f.l is translation invariant and vanishes on lower dimensions. Since f.l([0, 1]n) = c, a simple cut-and-paste argument shows that
39
4.2 Invariant valuations on parallelotopes
fL([O, 1/k]n) = c/k n for all integers k > O. Therefore, fL(C) = CfLn(C) for every box C of rational dimensions, with sides parallel to the coordinate axes. This follows from the fact that such a box can be built up by stacking cubes of the form [0,1/k]n for some k > O. Since fL is either continuous or monotone, it follows that fL( C) = CfLn (C) for every box C of positive real dimensions, with sides parallel to the coordinate axes. It then follows from the inclusion--exclusion principle that fL(P) = cfLn(P) for all P E Par(n). 0 The condition of either continuity or monotonicity is necessary to the characterization given by Theorem 4.2.4. If we omit these conditions then counterexamples to Theorem 4.2.4 can be found even in the case of Par(1)! To see this, recall that R is a vector space of infinite dimension over the field Q of rational numbers. Denote this vector space RQ. Since the dual space is also of infinite dimension, there exists a nontrivial map f E RQ; i.e., a linear map f : RQ ---., Q such that
RQ
f(1)
= 1 and f(x) E Q for all x E R. A parallelotope P E Par(1) is just a closed bounded interval of the form [a, b], having length b - a. Define a valuation TJ on parallelotopes (intervals) in Par(1) by the formula
TJ([a, b]) = f(b - a). Evidently TJ is invariant, depending only on the length of the closed interval. Moreover, if a ~ C ~ b ~ d, then
TJ([a, b]
U
[c, dj)
+ TJ([a, b] n [c, dj)
= TJ([a, d]) + TJ([c, b]) = f(d - a) + f(b - c)
+ f(b -
c) =f(b-a)+f(d-c)
= (f(c - a) = TJ([a, b])
+ f(d -
b))
+ f(b -
c)
+ TJ([c, dj),
by the linearity of f. It now follows from Groemer's extension Theorem 4.1.3 that TJ extends to an invariant valuation on Par(1) that vanishes on lower dimensions. However, TJ is not equal to length (onedimensional volume), since TJ takes only rational values. In other words, invariance alone is insufficient to characterize the volume - either continuity or monotonicity is also required. The reasoning that underlies this counterexample to Theorem 4.2.4 is easily extended to provide counterexamples for Par( n), where n 2: 1.
40
4 The intrinsic volumes for parallelotopes
We are now able to determine all continuous valuations on Par(n) that are invariant under translation and permutations of coordinates. We shall not yet prove that they are also rotation invariant. Theorem 4.2.5 The valuations /-Lo, /-Ll, .•• ,/-Ln form a basis for the vector space of all continuous invariant valuations defined on Par(n).
Proof Let /-L be a continuous invariant valuation on Par(n). Denote by Xl, X2, ... ,Xn the standard orthonormal basis for R n, and let H j denote the (n -I)-hyperplane in Rn spanned by the coordinate vectors Xl, ... , X j -1 , X j +1, ... , Xn- The restriction of /-L to H j is an invariant valuation on parallelotopes in H j . Proceeding by induction, we may assume that n-l
= L Ci/-Li(A),
/-L(A)
i=O
for all A E Par( n) such that A ~ H j . Moreover, the coefficients <;i are the same for each choice of H j , since the valuations /-L; /-Lo, ... , /-Ln-l are invariant under permutation of the coordinates Xl, •.• , X n . Thus, the valuation n-l
/-L- LCi/-Li i=O
vanishes on all lower dimensional parallelotopes in Par(n), since any such parallelotope is contained in a hyperplane parallel to one of the hyperplanes H j . By Theorem 4.2.4, n-l
/-L - L
Ci/-Li
=
Cn/-Ln,
i=O
where /-Ln is the volume on Rn, and where Cn is a real constant. In other words, n
/-L = LCi/-Li. i=O
o A valuation /-L on Par(n) is said to be homogeneous of degree k > 0 if
/-L(aP) for all P E Par(n) and all a 2:
o.
=
ak/-L(P)
4.3 Notes
41
Corollary 4.2.6 Let f-l be a continuous invariant valuation defined on
Par(n) that is homogeneous of degree k, for some 0::; k ::; n. Then there exists C E R such that f-l(P) = Cf-lk(P) for all P E Par(n). Proof By Theorem 4.2.5 there exist
Cl,""
Cn
E
R such that
n
f-l = LCif-li. i=O
If P
=
[0, l]n then, for a > 0,
f-l(aP) =
~Cif-li(ap) = ~Ciaif-li(p) = ~ (7)Ciai
Meanwhile,
f-l(aP) = akf-l(P) = a k Therefore,
Ci
~Cif-li(P) = ~ (7)Ciak
°
= if i =f. k, and f-l = Ckf-lk·
o
4.3 Notes
For a more complete discussion of the Hausdorff topology on the space of compact subsets of Rn and the subspace of compact convex sets, see [85, pp. 47-61]. Theorem 4.1.3 is a special case of a more general extension theorem of Groemer, in which the lattice Par(n) is replaced with the lattice of polytopes in Rn (see [32]). Theorem 4.2.5 can be generalized to a classification of all continuous translation invariant valuations on the lattice Par(n) , omitting the requirement that valuations be invariant under permutation of coordinates. The result is a 2n -dimensional space of valuations, with a basis indexed by the collection of all coordinate subspaces of Rn with respect to the fixed basis for parallelotope edges in Par(n); that is, by the set of all 2n subsets of that n-element basis. For a detailed treatment, see [54].
5 The lattice of polyconvex sets
We turn now to the lattice of polyconvex sets, which is a natural setting for the study of classical geometric probability. In Section 5.2 we define the Euler characteristic on polyconvex sets, which is an important tool for the extension in Chapter 7 of the intrinsic volumes of Section 4.2 to polyconvex sets. The Euler characteristic will also reappear in Chapter 10, in which we generalize the discrete kinematic formula of Chapter 3 to polyconvex sets. Section 5.5, while interesting in its own right, points to the correct normalization of the rotation invariant measures on Grassmannians in Section 6.1.
5.1 Polyconvex sets A subset K of Rn is said to be convex if any two points x and y in K are the endpoints of a line segment lying inside K. Denote by Kn the collection of all compact convex subsets of R n. A finite union of compact convex sets will be called a polyconvex set (a term suggested by E. De Giorgi). If A is a polyconvex set in Rn, we shall say that A is of dimension n if A is not contained in a finite union of hyperplanes of R n; that is, if A has a non-empty interior. Otherwise, we shall say that A is of lower dimension. The union and intersection of polyconvex sets are polyconvex. This follows from the basic fact that the intersection oftwo convex sets is convex. In other words, the family of polyconvex sets in Rn is a distributive lattice. We denote this lattice Polycon(n). Note that Par( n) is a sub lattice of Polycon( n). The lattice Polycon( n) is also sometimes called the convex ring. A non-empty compact convex set K E Kn is determined uniquely by its support function hK : sn-l --+ R, defined by hK(U) =
5.1 Poly convex sets
43
maxXEK{ x . u}, where· denotes the standard inner product on Rn. For example, if vERn and v denotes the line segment with endpoints v and -v, then hv(u) = lu· vi, for all u E sn-l. More generally, suppose that h: sn-l ----+ R, and consider the radial extension h : Rn ----+ R given by h(au) = ah(u), for all u E sn-l and a 2': O. The original function h is a support fUEction of a compact convex set in Rn if and only if the radial extension h is sublinear; that is,
h(x + y)
~
h(x)
+ h(y),
for all x,y ERn. Note that, for u E sn-l, a compact convex set K lies entirely on one side (the '-u' side) of the hyperplane H(K, u) determined by the equation X· u = hK(U). The hyperplane H(K,u) is called the support plane of K in the direction u. If H(K, u)- denotes the closed half-space x· u ~ hK(U) bounded by H(K, u), then we have
K =
n H(K,u)-. uES n
For compact convex sets K and L the Minkowski sum K by
K
+L
=
{x
+y : x
E
K and y E L}.
+ L is defined (5.1)
It is not difficult to show that hK + L = hK + h L . Recall from Lemma 4.1.1 that for compact sets K and L in Rn, the Hausdorff metric satisfies d(K, L) ~ E if and only if K ~ L + EB and L ~ K + EB. It follows easily from (5.1) that
8(K, L)
=
IhK(U) - hL{u)l.
sup uES n
(5.2)
- 1
In other words, the Hausdorff topology on Kn is also given by the uniform metric topology on the set of support functions of compact convex sets. Denote by En the Euclidean group on Rn; that is, the group generated by translations and (proper or improper) rotations. If A C Rn and 9 E En, write gA
= g(A) =
{g(a) : a E A}.
The subgroup of translations (relative to a fixed Cartesian coordinate system) shall be denoted by Tn. A valuation J-l defined on polyconvex sets in Rn is said to be rigid motion invariant (or simply invariant, when no confusion is possible) if J-l(A)
= J-l(gA)
(5.3)
44
5 The lattice of polyconvex sets
for all 9 E En and all A E Polycon(n). If the equality (5.3) holds only when 9 E Tn, we say that J.L is translation invariant. Our objective is to determine all invariant valuations defined on polyconvex sets in Rn. As with Par(n), we impose a continuity condition on the valuations to be considered. A valuation J.L is said to be convexcontinuous (or simply continuous, when no confusion is possible) provided that
whenever An, A are compact convex sets and An --+ A with respect to the metric (5.2). Examples of continuous invariant valuations on Polycon(n) include volume and surface area. The following proposition shows that we can restrict our attention to convex-continuous valuations defined on the generating set Kn. Theorem 5.1.1 (Groemer's extension theorem for Polycon(n» A convex-continuous valuation J.L on Kn admits a unique extension to a valuation on the lattice Polycon( n).
Proof Suppose that J.L is a continuous valuation. In view of Groemer's integral Theorem 2.2.1, it is sufficient to show that J.L defines an integral on the space of indicator functions. The theor,em is trivial in dimension ~ero. Assume that the theorem holds in dimension n-1. Suppose that there exist distinct K 1 , ••. ,Km E Kn such that LaJKi =0
(5.4)
i=l
while Tn
L aiJ.L(Ki) =
1.
(5.5)
i=l
Let m be the least positive integer for which such expressions (5.4) and (5.5) exist. Choose a hyperplane H, with associated closed half-spaces H+ and H- such that Kl C lnt H+. Since I KinH + = IKJH+, it follows from (5.4) that Tn
LaiIKinH+ i=l
= O.
5.1 Polyconvex sets
45
Similarly, m
m
L cxiIK,nH = 0 and
LCXJK,nH- = O.
i=l
i=l
Meanwhile, since J-l is a valuation, m
m
m
m
i=l
i=l
i=l
i=l
Since the sets Ki n H lie inside a space of dimension n - 1, the sum cxiJ-l(KinH) = 0 by the induction assumption. Because KlnH- = 0, the sum L~l cxiJ-l(Ki n H-) = 0 by the minimality of m. From (5.5) we have L~l
m
m
i=l
i=l
Choose a sequence of hyperplanes HI, H 2 , ... such that Kl C lnt Hi and
By iterating the preceding argument, we have m
LCXiJ-l(Ki n Hi n··· n H:) = 1 i=l
for all q 2: 1. Since J-l is continuous, the limit as q
~
(Xl
gives
m
L cxiJ-l(Ki n K l ) = 1, i=l
while a similar argument using (5.4) gives m
L cxiIK,nK1
=
O.
i=l
After repeating this argument with the bodies K 2 , . .• ,Km we have
tCXiJ-l(Kl n··· n Km) = (tCXi) J-l(Kl n··· n Km) = l. This implies that CXl + .,. + CXm =I- 0 and that Kl n .. , n Km =I- 0. Meanwhile a similar argument using (5.4) gives
f
i=l
CXJK1n ... nKm =
(f ,=1
CX i )
IKln ... nKm =
0,
46
5 The lattice of polyconvex sets
so that either a1 in either case.
+ ... + am = 0 or K1 n ... n Km = 0, a contradiction 0
5.2 The Euler characteristic Next, we shall extend the valuation J-lo to the entire distributive lattice Polycon(n). We have seen that J-lo is a well-defined functional on sets that are finite unions and intersections of parallelotopes, and that J-lo(P) = 1 if P is a non-empty parallelotope. These results motivate the following theorem.
Theorem 5.2.1 (The existence of the Euler characteristic) There exists a unique convex-continuous invariant valuation J-lo defined on the family Polycon(n) of all polyconvex sets in Rn, such that J-lo(K) = 1 whenever K is a non-empty compact convex set. The valuation J-lo is again called the Euler characteristic. Proof We proceed by induction on the dimension n, the case n = 1 having been established previously. By Theorems 2.2.1 and 5.1.1, it will suffice to establish the existence of a linear functional Ln defined on Kn-simple functions, such that Ln(IK) = 1 whenever K is a non-empty compact convex set. For n = 1, set L 1 (1) = L(I(x)-f(x+O)), xER
where f(x + 0) = lima--+o+ f(x + a). The sum on the right-hand side is finite, and, for f = I K , where K is an interval [a, bj, we have
Thus, L 1 (IK
)
= J-l6(K),
so that L 1 (1) =
J
f dJ-l6·
For arbitrary n, choose an orthogonal coordinate system Xl, X2,"" x n . Given the first coordinate x, let Hx be the hyperplane parallel to the coordinates X2, ... , Xn and passing through the point (x, 0, ... ,0). Let f = f(X1, X2,"" x n ) be a simple function. The function fx(X2,' .. ,xn ) = f(x, X2,' .. ,xn ) is a simple function in H x , and we
5.2 The Euler characteristic
47
assume that L n - l (fx) has been defined in H x , since Hx is isomorphic to Rn-l. Set F(x) = Ln-l(fx), and set
Note that the function F is simple, so that the right-hand side is well defined. If f = I K , where K is a compact convex set, then fx is the indicator function of the slice of K by the hyperplane at Xl = X, and F is the indicator function of the projection of K onto the xl-coordinate axis. It follows that Ll (F) = 1. Since Ln(f) =
Jf
d'/a,
for some valuation f.lo, it follows that f.lo is the desired valuation.
0
Note that the Euler characteristic f.lo is normalized. In other words, if K is a polyconvex set of dimension k in R n, and if V is a plane of dimension j containing K, then f.l~(K) computed within V is equal to f.lo(K) computed in Rn. This follows from the fact that f.lo(K) can be computed via the inclusion-exclusion principle after K has been expressed as a finite union of compact convex sets, whereas f.lo(K) = 1 for all non-empty compact convex sets K in spaces of any (finite) dimension. For this reason we write f.lo in place of f.lo. The argument in the preceding proof can also be used to compute the Euler characteristic of a polytope. By Corollary 2.2.2, the valuation f.lo extends uniquely to a valuation defined on the relative Boolean algebra generated by Polycon(n), a valuation that is again denoted by f.lo. Consider the (smaller) distributive sublattice ofPolycon(n) generated by compact convex polytopes. Recall that a convex polytope is the intersection of a finite collection of closed half-spaces. A polytope is a finite union of convex polytopes. Given a polytope P, the boundary 8P is also a polytope (which is not the case for an arbitrary compact convex set). Therefore, f.lo(8P) is defined. Theorem 5.2.2 If P is a compact convex polytope of dimension n > 0, then
Proof Using again the notation of Theorem 5.2.1, we note that H x n8P = 8( Hx n P) if X is not a boundary point of 1f P, the orthogonal projection
48
5 The lattice of polyconvex sets
of P onto the line H;:. Let F(x) = J-lo(o(Hx np)), where J-lo is taken in the space Hx. For the case n = 1 we have J-lo(oP) = 2 = 1- (-1), since OP consists of two distinct points. For n > 1 it follows from the induction hypothesis that
when x E nP is not a boundary point of nP. Meanwhile, if x E o(nP), we have
J-lo(Hx n oP) = 1, since Hx n P is a face of P (though possibly a single point). Finally,
J-lo(Hx n oP)
=
0,
when Hx n oP = 0. We can now compute
L1(F(x)) = ~)F(x) - F(x + 0)), x
a sum that vanishes except at the two points, call them a and b (with a < b), where Hx touches the boundary of P. The right-hand side then reduces to
F(a) - F(a + 0)
+ F(b) -
F(b + 0).
We compute
F(b+ 0) F(b)
=
=
0,
1,
F(a) = 1, F(a+O)
= 1-
(_I)n-l.
On adding, we find that
Ll(F(x)) = 1-1 + (_I)n-l as desired
+ 1 = 1 + (_I)n-l = 1- (_I)n, o
If P is a compact convex polytope of dimension k in R n, let V be the k-dimensional plane containing P. We denote by relint(P) the interior of P relative to the topology of V; that is, the relative interior of P.
49
5.2 The Euler characteristic
Theorem 5.2.3 Let P be a compact convex polytope of dimension k in Rn. Then 1L0(relint(P))
= (_l)k.
Proof Since 1L0 is normalized independently of the ambient space, we compute within the k-dimensional plane in Rn containing P. From Theorem 5.2.2 we have
= 1L0(P) -1L0(8P) = (_l)k.
1L0(relint(P»
o We can now generalize Euler's formula to arbitrary (nonconvex) polytopes. To this end, we define the notion of a system of faces F of a polytope P. This will be a family with the following properties: • Every element of F is a convex polytope. UQEF relint(Q) = P . • If Q, Q' E F and Q -=I- Q', then relint(Q) n relint(Q') =
•
0
We can now prove the following result.
Theorem 5.2.4 (The Euler-Schliifli-Poincare formula) Let F be a system of faces of a polytope P, and let fi be the number of elements of F of dimension i. Then
1L0
fo --;-
=
II + h - ...
First proof Place a linear ordering on the elements of F, or 'faces', such that, if Q < Q' then flim(Q) ::; dim(Q'). Evidently, Ip
=
L
(IQ - lQ1)'
QEF
where Q1 = Q n (UQI
l
L
dlLo
J(IQ - lQ1) dlLo +
QEFo
+
L QEF2
L QEF1
J (IQ - l Q1 ) dlLo
+ ...
J (IQ - lQ1) dlLo
5 The lattice of polyconvex sets
50
= 0,
For Q E F o, we have IQl
so that
For Q E F i , where i > 0, we have
UQ
Ql = Q n (
1
)
= 8Q,
Q'
and hence,
f.Lo(Q) - f.Lo(8Q) = 1 - (1 - (_l)i) l-l+(-l)i=(-lt Therefore,
L
J
(fQ - IQ,) df.Lo = (_l)i Ii,
QEFi
so that
f.Lo(P)
=
J
Ip df.Lo
=
fo - h
+ h - .... o
Second proof We use the relative interior theorem. This requires extending the valuation f.Lo to the relative Boolean algebra generated by polyconvex sets. If F is any system of faces, then
f.Lo(P)
= f.Lo (
U relint(Q)) = L QEF
L
f.Lo(relint(Q))
QEF
(_l)dimQ
QEF
by Theorem 5.2.3, since each Q E F is a convex polytope. We collect terms of the same dimension, and the theorem then follows. 0
5.3 Helly's theorem The following is a remarkable application of the Euler characteristic.
5.3 H elly's theorem
51
Theorem 5.3.1 (Klee's theorem) Let F be a finite family of compact convex sets such that
is convex. Let i < IFI, and suppose that, for any subset that IGI = i (that is, every subset of cardinality i of F),
G~
F such
n K=I 0. KEG
Then there exists a subset H of F with cardinality i
+ 1,
such that
n K=I 0. KEH
Proof Let n > 1 be a positive integer. Recall that 1-
(7) + (;) - (~) + ... +(-l)j(;) =1O,
(5.6)
for all positive integers j < n. To see why (5.6) holds, suppose that j ~ (n/2). In this case the left-hand side of (5.6) is an alternating sum of strictly increasing terms, and is therefore unequal to zero. Since
and
it also follows that (5.6) holds if we replace j ~ (n/2) by j ::::: (n/2). Now let n = IFI. From Theorem 5.2.1 and the inclusion-exclusion formula we have l=f-lO
(U K) KEF
L KEF
f-lo (K) -
L
f-lo (K
n L) + ...
K#-LEF
whenever nKEG K = 0 for all G ~ F such that IGI this is impossible, by virtue of the inequality (5.6).
= i + 1.
However, 0
Given a set A ~ Rn, the convex hull of A is the smallest convex set in R n that contains A; that is, the intersection of all convex sets containing
52
5 The lattice of polyconvex sets
A. The following is a simple and yet fundamental property of convex hulls in Rn. Theorem 5.3.2 (Caratheodory's theorem) Let T be the convex hull of a family of compact convex sets K 1 , K 2 , ..• ,Km in Rn. For each x E T, there exists a subfamily Kjl' ... ' K jp ' with convex hull T x , such that p ~ n + 1 and x E T x. Proof If n = 0 the result is trivial. Assuming that the theorem holds for dimension n - 1, we prove the theorem for dimension n. Let x E T. If x lies on the boundary of the compact convex set T, let H denote a support plane of T at x. Because H supports the convex set T, all of T lies inside one of the closed half-spaces bounded by H. Since x E T n H, it follows that x lies in the convex hull of the convex sets K j n H. By the induction assumption on dimension, there exists a subfamily Kl1 n H, ... ,Kjp n H with convex hull T* such that p ~ n and x E T*. It follows that x lies in the convex hull of the subfamily K j1 , ... ,Kjp. If x lies in the interior of T, suppose that x rj. K j for any j (otherwise the proof is finished). Let £ denote a line through x that also meets K m , and let x' be the point of intersection of £ with the boundary of T. Since £ meets the boundary of T at two points, choose x' so that x lies between x' and £ n Km. It follows from the argument in the previous paragraph that x' lies in the convex hull of a subfamily Kjl , ... ,Kjp of F, where p ~ n. It then follows that x must lie the convex hull of the sets K j1 , ... ,Kjp ' K m , a collection of at most n + 1 sets in F. 0 Combining CaratModory's Theorem 5.3.2 with Klee's Theorem 5.3.1 we obtain the following celebrated theorem of Helly. Theorem 5.3.3 (Helly's theorem) Let F be a finite family of compact convex sets in Rn. Suppose that, for any subset G ~ F such that IGI ~ n + 1 (that is, every subset of cardinality at most n + 1 of F),
n K#0. KEG
Then
n K#0. KEF
5.3 Helly's theorem
53
In other words, if every n + 1 elements of F have non-empty intersection, then the entire family F of convex sets has non-empty intersection. Proof If IFI ~ n + 1 the result is trivial. Suppose that Theorem 5.3.3 holds for IFI = m, for some m 2': n + 1. We show that the theorem also holds for IFI = m + 1. Let F = {K 1 , ... , K mH }. For each 1 ~ j ~ m + 1 denote by L j the intersection
Our induction assumption for the case IFI = m implies that each L j is non-empty. Let M denote the convex hull of the union L1 u··· u L m +1. If x E M then Caratheodory's Theorem 5.3.2 implies that x lies in the convex hull of the union Lil U Li2 U ... U Lin+l for some 1 ~ i1 ~ ... ~ in+ 1 ~ m + 1. For each i ¢:. {iI, ... , in+ I},
so that x E K i . In other words, m+1
UK
M S;;
i·
i=l
Let Mi = Ki
n M for 1 ~ i
n
Mi
=
i-f-j
Since M1 U ... U Mm+1 rem 5.3.1 that
~
m
+ 1.
n
Ki n M
Note that, for each j,
= Lj n M = Lj
-=I-
0.
i-f-j
= M
is convex, it follows from Klee's Theo-
Hence,
n
m+1
i=l
n
m+1
Ki;2
Mi -=I- 0.
i=l
o Theorem 5.3.3 is in fact a generalization of Theorem 3.3.1. To see this, suppose that S = {Sl' ... ,sn} is a finite set, and associate to each Si a distinct point Xi E Rn-l, chosen so that the collection {Xl, ... ,x n } is affinely independent. Let ~ denote the geometric simplex in R n - 1 having vertices {Xl, ... , x n }. Subsets of S now correspond to faces ofthe
5 The lattice of polyconvex sets
54
simplex ~, and Theorem 3.3.1 is now a specialization of Theorem 5.3.3 to families of faces of the simplex ~.
5.4 Lutwak's containment theorem We now turn to a beautiful application of Helly's theorem to the question of containment of convex bodies. Given compact convex sets K and L with non-empty interiors, is there a simple condition that guarantees that some translate of K is a subset of L? It turns out that the answer to this question is determined by the relationship of K to the simplices in R n that contain L. For K E Kn and vERn, denote by K + v the set
K In other words, K
+v =
{x + v : x E K}.
+ v denotes the translation of the set K
by the vector
v. Theorem 5.4.1 (Lutwak's containment theorem) Let K, L E Kn with non-empty interiors. The following are equivalent.
(i) For every simplex
~ such that L ~ ~, there exists vERn such that K +v ~~. (ii) There exists Va E R n such that K + Va ~ L.
In other words, if every simplex containing L also contains a translate of K, then L itself contains a translate of K. Proof The implication (ii) '* (i) is obvious. We show that (i) (ii). To begin, suppose first that L is a polytope, with facets L 1 , L 2 , .•. ,Lrn and corresponding facet (outward) unit normal vectors U1, U2,.'" Urn' Assume also that every selection of n distinct unit normals Uj is a linearly independent set. (Were this not the case, a small perturbation of L would make it so.) For each facet L i , let Hi denote the (n - I)-dimensional hyperplane in Rn containing L i , and let Ht denote the closed half-space bounded by Hi and containing the polytope L. Finally, denote by Ti the set of vectors vERn such that K + v c Ht. Since each Ht is a (convex) closed half-space and K is compact, it is clear that each Ti is a nonempty closed convex set. The independence condition on the unit normals {Ui} implies that, for each distinct selection Uil' Ui2' ... ,Uin+l of n + 1 unit normals, either
'*
5.5 Cauchy's surface area formula
55
the corresponding intersection n+1
H-~1,~2""'~n+l . .
-nH+ is
-
(5.7)
8=1
contains a simplex .6. il ,i2, ... ,in +l such that L ~ .6.i l,i2, ... ,i n +1l or this intersection contains a translate of the ball O'.B of radius 0'., for all 0'. > O. (This is the case in which the intersection (5.7) is unbounded.) In the first case, the hypothesis of the theorem implies the existence of a vector vERn such that K + v ~ .6.il ,i2, ... ,i n +l' In the second case there also exists v such that K + v lies in the intersection (5.7). In either case, there exists v E Til n· .. n T in + l . In other words, each collection of n + 1 sets Ti has a non-empty intersection. ReIly's Theorem 5.3.3 then implies the existence of a vector v such that m
In other words, K + v ~ Ht for i = 1, ... ,m. Since L = Hi n· .. n H:!, it follows that K + v ~ L. Now suppose that L is an arbitrary compact convex set. Let {Pi}~l be a decreasing collection of polytopes such that Pi ~ L as i ~ 00, and such that each n of the facet normals to Pi are linearly independent. If .6. is a simplex containing Pi, then L ~ Pi ~ .6., so there exists a vector w such that K + w ~.6.. Since the theorem holds for the polytopes Pi, it then follows that there exists a vector Vi for each i, such that K + Vi ~ Pi' Since the Pi are decreasing (with respect to the relation of subset containment), the sequence {Vi} is bounded and must contain a convergent subsequence. Assume then without loss of generality that Vi ~ v. Since Pi ~ L, it follows that K + v ~ L. 0
5.5 Cauchy's surface area formula We conclude this chapter with the following interpretation of the surface area S(K) of a convex body in Rn, which will be of use to us in the sequel. If K E Kn and V is an (n - I)-dimensional subspace of R n , denote by KIV the orthogonal projection of K onto V. Let W n -1 denote the (n - I)-dimensional volume of the unit ball B n - 1 in Rn-1.
Lemma 5.5.1 For v E sn-l,
f
lSn-l
lu, vi du
= 2Wn -1.
56
5 The lattice of polyconvex sets
Proof Recall from elementary calculus that
where sn-l is partitioned into many small regions Ai, having area S(Ai), and where Ui E Ai for each i. Let Ai denote the orthogonal projection of Ai onto the tangent hyperplane to sn-l at the point Ui. Denote by AilvJ.. the orthogonal projection of Ai onto the hyperplane vJ..; that is, into the disk B n- l in vJ... Because Ai is a flat region lying inside a hyperplane with unit normal Ui, we have S(AilvJ..) = Iu . vIS(Ai)' Meanwhile, for Ai sufficiently small, we have S(Ai) ~ S(Ai) and S(AilvJ..) ~ S(AilvJ..). Therefore,
kn-l
Iu· vi du
~L
S(AilvJ..) .
•
Since the collection of sets {Ai IvJ.. } covers the disk Bi twice, projecting from both of the directions v and -v (that is, from both hemispheres of sn-l), we have { lu, vi du }Sn-l
~ 2Wn - l ,
where the similarities converge to equalities in the limit, as the mesh of the partition {Ai} goes to zero. 0
Theorem 5.5.2 (Cauchy's surface area formula) For all K E Kn,
S(K)
= - I Wn-l
1
Sn-l
!-In-I(Klu J.. ) duo
(5.8)
Here sn-l denotes the (n-I)-dimensional sphere; i.e. the set of all unit vectors in Rn. Note that !-In_I(KluJ..) is just the (n - I)-dimensional volume of the projection of K onto the subspace uJ... Proof Let P be a compact convex polytope with facet unit normal vectors VI, ... ,Vm and corresponding facet surface areas al, ... , am. If a facet Pi of P has unit normal vector Vi and surface area (that is, (n - I)volume) ai, then the projection PiluJ.. has (n - I)-volume
!-In_I(PiluJ..) = ail u , vii· For u E sn-l, the (n-I)-volume !-In_I(PluJ..) is computed by summing the (n-I)-volumes of the projections of the facets of P. This sum gives
5.5 Cauchy's surface area formula
57
twice the desired value, since each point of PluJ.. is hit twice (from above and below) as we sum over projections of all facets of P. Hence, we have
r
JSn-l
/-In-I (PluJ..) du
=
where the third equality follows from Lemma 5.5.1. Since (5.8) then holds for any convex polytope P, the equation holds for any convex body K by continuity. 0 Recall that, for parallelotopes P E Par(n), /-In-I(P) = ~S(P).
We are therefore motivated to define /-In-I(K) = ~S(K),
for all K E
}Cn.
The equation (5.8) now becomes
1
/-In-I(K) = -1/-In-I(Klu J.. ) duo 2Wn-1 Sn-l
(5.9)
Since every line f through the origin in R n meets sn-I at exactly two points, we can rewrite this result as an integral over the projective space Gr(n,l) (Le. the set of all lines f through the origin in Rn) instead of integrating over the sphere. We then renormalize the resultant measure on Gr(n, 1) so that the space Gr(n, 1) has total measure 1. This measure is called the Haar probability measure on Gr(n,l). Integrating with respect to the Haar probability measure on Gr(n, 1) we obtain /-In-I(K)
=a
r
/-In_I(KlfJ..) df,
JGr(n,l)
where a is a constant independent of K. To compute a, set K the unit ball in R n , to obtain nw 2n
= /-In-I(Bn) = a
r
JGr(n,l)
/-In_I(BnlfJ..) df
= aWn-I,
= Bn ,
58
5 The lattice of polyconvex sets
since the total measure of the space Gr( n, 1) is 1. It follows that
nwn 2Wn - l
a=--.
Denote by Tn the rotation invariant measure on Gr(n, 1) normalized so that Tn(Gr(n, 1)) = 2nWn . Wn-l
(5.10)
The equation (5.9) now becomes !-In-l(K)
=
r
JGr(n,l)
!-In_l(KlfJ..) dTn.
(5.11)
5.6 Notes For a general reference to the theory of convex bodies, see the book by Schneider [85] and the survey on valuations by McMullen and Schneider [72]. The normalization for intrinsic volumes is due to McMullen [70]. Groemer's extension Theorem 5.1.1 originally appeared in [32], which contains a comprehensive presentation of extension theorems for valuations on various classes of compact convex sets. For a treatment of the combinatorial theory of the Euler characteristic, see [79, 80]. See also [75] for the connection between the Euler characteristic and algebraic topology. Hadwiger presented numerous geometric applications of the Euler characteristic in [37, 39]. For a modern treatment of the Euler characteristic as a valuation on convex sets, see [71, 72, 85]. Helly's Theorem 5.3.3 first appeared in [43]; the proof given in this section is due to Hadwiger [37]. Theorem 5.3.2 is due to Caratheodory [85]. The proof of Caratheodory's theorem presented above may also be found in [37], where it appeared as a special case of a more general theorem of Steinitz [93]. Klee's Theorem 5.3.1 was noted by Klee in [56], and has many generalizations (see [6, 28, 37, 62]; also [18, pp. 123-128]). A non-numerical form (in terms of cone functions) was given by Chen [11]. A general survey on Helly's theorem, its proofs, variations and applications, may be found in [18] and in [21]. It is conceivable that the generalization of the Euler-SchHifli-Poincare formula (5.2.4) to intrinsic volumes should lead to interesting quantitative generalizations of Klee's Theorem 5.3.1 and thereby of Helly's Theorem 5.3.3. Lutwak's containment Theorem 5.4.1 appeared in [68].
5.6 Notes
59
Cauchy published his surface area formula (Theorem 5.5.2) for dimensions 2 and 3 in [9] and [10]. Generalizations of Cauchy's formula are treated in Sections 7.4 and 9.4 (see also [82, p. 218] and [85, p. 295]).
6 Invariant measures on Grassmannians
Before we extend the notions introduced in Chapter 4 to the lattice of polyconvex sets we shall require a deeper understanding of the lattice of subspaces of Rn. In Section 6.1 we introduce the the flag coefficients, which rely on a crucial choice of normalization for the rotation invariant measures on real Grassmannians. This construction leads in turn to continuous analogues of the extremal combinatorial results of Chapter 3, here in the context of subspaces. The flag coefficients will reappear in Chapters 8-10 as we investigate connections between the integral geometry of Grassmannians and the intrinsic volumes on polyconvex sets (see especially Section 9.4).
6.1 The lattice of subspaces Let Mod(n) denote the set of all linear subspaces of Rn; that is, the set of all linear varieties passing through the origin (having fixed an origin once and for all). The set Mod(n) is a partially ordered set under the relation of inclusion of linear subspaces. Moreover, it is a lattice, whereby the join xVy and the meet x Ay oftwo elements x, y E Mod(n) are defined respectively as the linear subspaces spanned by x and y and as the intersection of x and y. The lattice Mod(n) may be viewed as a continuous analogue of the lattice P(S) of subsets of a set S with n elements. Note, however, that this analogy is only a partial one, since the distributive law governing unions and intersections of subsets of S does not hold in the lattice Mod(n). Nonetheless, this analogy shall carry us as far as we need to go. In the lattice Mod( n) one defines the notions of chain, flag, and rank as for the lattice of subsets of R n. In particular, an element x of Mod( n) has rank k (that is, r(x) = k), whenever x is a linear subspace of dimension
6.1 The lattice of subspaces
61
k. The subspace {O} is the minimal element ofthe lattice Mod(n). Much as the group of permutations of the set S acts naturally on P( S), the orthogonal group O(n) (that is, the group of rotations about the origin and reflections across hyperplanes through the origin) acts naturally on the lattice Mod(n). The set of all elements of Mod(n) of dimension (rank) k, denoted Gr(n, k), is called the Grassmannian. Our objective is to describe the invariant (Haar) measure that acts on Gr(n, k). It is known that this measure is unique up to a common factor (but we will not prove this). If our analogy is not misleading, the total measure of Gr( n, k) should be an analogue of the binomial coefficient. To begin, consider the invariant measure Tn on Gr(n,l); that is, on the set of all straight lines through the origin. Denote [nJ = T n (Gr(n,l)) = 2nWn ,
Wn-l
(6.1)
where Wn denotes the volume of the unit ball En in Rn. The real numbers [nJ will playa role analogous to that of the positive integers n in the discrete case. The measure Tn may be viewed constructively in the following way. Let the invariant measure on the unit sphere sn-l (that is, on the surface of the unit ball) be denoted by an-I. For any measurable subset A of Gr(n,l), let A' be the subset of the unit sphere sn-l defined by A'
U xnsn-l.
=
xEA
It then follows from (5.10) that
Tn(A)
=
an-l (A') . 2Wn -1
To check this, note that the surface area of the unit sphere is an_I(Sn-l)
= nWn
so that T n (Gr(n,l)) =
nWn .
2W n -1
Again it is clear that the measure Tn is invariant under rotations. Let Flag(n) be the set of flags in Mod(n). For x E Mod(n), denote by Flag(x) the set of all flags that contain x; that is, the set of all sequences (XO, Xl, ... , Xn) of Xi E Mod(n), where dim(xi) = i, such that
6 Invariant measures on Grassmannians
62
Xo Xo
~ Xl ~ '"
~
Xn , and such that one of the Xi equals x. Note that = {O} and Xn = Rn. For fixed Xk, the set of all sequences (Xk, Xk+l, ... , Xn) with Xi E Mod(n), such that dim(xi) = i and Xi ~ Xi+l, is isomorphic to Flag(nk). Similarly, the set of sequences (xo, Xl>' .. , Xk) is isomorphic to Flag(k). Denote by
... , xn) is a simple function on the set Flag(n), then let
J
f d
=
JJ
f(xo, Xl> ... , xn)
d
. .. , Xn-l) dTn,
so that the measure ... , Xn) is a flag in Mod(n), let
The sequence (YI, Y2, ... , Yn) is a sequence of orthogonal straight lines, a frame. Conversely, given a frame (Yl> Y2,· .. , Yn), we obtain a flag by setting
Xo = {O},
Xl
= YI, X2 = YI VY2, X3 = YI V Y2 V Y3, ...
This defines a one-to-one correspondence between flags and frames. If f(xo, Xl> ... , Xn) is a real-valued (measurable) function on flags, let /(Yl> Y2, ... , Yn) be the corresponding function on frames. Then
This result can be read in the simpler language of combinatorics. Once the line YI has been chosen, which can be done in Tn ways, the subspace X2 is determined by the choice of a straight line Y2 in the space orthogonal to Xl (through the origin), which can be done in Tn-l ways, etc. The measure of Flag( n) turns out to be
[n]! =
= 1. Note that
n!WnWn-I" 'WI 2nWn-IWn_2'" Wo
(6.2)
6.2 Computing the flag coefficients
63
We now define an invariant measure on Gr(n, k). For A ~ Gr(n, k), let Flag(A) be the set of all flags (xo, XI, ... ,xn ) such that Xk EA. Set (6.3)
To justify this normalization combinatorially, note that, for each Xk E A, there are exactly [kJ![n - kJ! flags containing Xk; that is, to choose a flag containing Xk one must choose a frame for the vector space Xk, of which there are [kJ! choices, and a frame for the (n - k)-dimensional complementary space of which there are [n - kJ! choices. The measure Ilk is evidently invariant under rotations, and we have
xt,
Ilk (Gr(n, k))
=
[kJ![~~ kJ! = [~].
These values, called flag coefficients, are continuous analogues of the binomial coefficients of the discrete lattice in Chapter 3. From (6.2) we obtain
n! Wn [n] k - k!(n - k)! WkWn-k
-
(n) k
Wn WkWn-k·
(6.4)
6.2 Computing the flag coefficients It is not difficult to compute the numerical values of the numbers [nJ and their associated flag coefficients. We begin by deriving the following well known formula for the volume Wn of the unit ball in Rn.
Proposition 6.2.1 For n
~
Wn
1
= r((n/2) + 1)·
Here r(t) denotes the Euler gamma function, given by
Proof We first compute the integral
64
6 Invariant measures on Grassmannians
We can think of the square of this integral as the product of integrals in two independent variables, x and y.
(1:
e- x2 dX) 2
=
1:
e- x2 dx
1:
e- y2 dy
=
1:
2 e-(x +y2)
dxdy
On switching to polar coordinates in the xy-plane, we obtain
1: 121r 2n
e-(x 2+ y2 )
1
00
1
00
dxdy
e _r2 r dr dO
e- r2 r dr
n,
so that
(6.5) Following a similar argument in n-variables,
since the surface area of the sphere sn-l is equal to nwn . On substituting y = r2, we obtain
6.2 Computing the flag coefficients
65
Hence, we have 1fn/2 Wn
r(~
=
+ 1)" o
Proposition 6.2.1 can be given a more computationally pliable form, provided that we examine separately the cases in which the dimension n of the unit ball is even or odd. Proposition 6.2.2 Let k be a non-negative integer. Then and
W2k+l
=
22k+11f k k! (2k + I)!
Proof It immediately follows from Proposition 6.2.1 that W2k
=
1fk f (k + 1)
1fk
= If·
To compute W2k+l first note that
1
00
f(I/2) = On substituting x
f(I/2)
e-xx- i dx.
= y2 and dx = 2y dy we obtain =
1
00
o
1 e- y2 -2y dy = Y
1
00
e- y2 dy = y1f,
-00
where the last equality follows from (6.5). It now follows from Proposition 6.2.1 that 2k+" 1f2-
2k+l 2k-l ... .! IJr 2 2 2Y "
2k+ 11fk (2k
+ 1)(2k -
1)··· (3)(1)
6 Invariant measures on Grassmannians
66
2k+11f k2kk! (2k + 1)! 22k+11f k k! (2k + 1)!
o Proposition 6.2.2 leads to easy computations of [n]. Once again there are two formulas, depending on the parity of n. Proposition 6.2.3 Let k be a positive integer. Then 21f(2k - 1)!
[2k]
21fk (2k - 1) k '
= 4k(k _ 1)!(k - 1)! = 4k
and [2k+1]=
4k(k!? k (2k)-1 (2k)! =4 k
Proof Computing directly from the definition (6.1) and Proposition 6.2.2 we get [2k]
2kw2k 2W2k-1 (2k - 1)! 1 2k ~ 2 - 1fk-1(k - 1)! k1fk
21fk (2k 4k k
1)
.
Similarly, [2k+ 1]
(2k + 1)W2k+1 2W2k (2k + 1)2 2k+11f kk! k! (2k + 1)! 21fk 4kk!k! (2k)! .
o The first few values of W n , [n], and [n]! run as follows:
6.2 Computing the flag coefficients !l
71"2
r(~+1)
~ 2W n _l
n!w n 2n
n
Wn
[n]
[n]!
0
1
1
1
1
2
n
2
7r
""3
7r
71"2
371"
371"2
""2
5
871"2
""6
7
1671"3 105 71"4 24 3271"4 945 71"5 120
10
4""
8
27r 2
1571"
1571"3 -8-
16
67r 3
"3
71"3
(6.6)
4"
15
6
9
"2
2
4
8
1 71"
"2
471"
3
1 71"
67
16 "5
10571"4
3571"
16
32
35
128
247r4
31571" 256
94571"5
32
Proposition 6.2.3 leads to a number of interesting relations among the numbers [n]. For example, one can easily show that, for k > 0, [2k][2k
+ 1] = k7r,
and [2k _ 1][2k]
=
(2k - 1)7r, 2
from which it follows that [n+ 2]
~
n+ 1 n
for all n > O. Using either of Propositions 6.2.2 and 6.2.3 one may easily compute the values of the flag coefficients. Once again the formulas vary according to the parity of the parameters.
Proposition 6.2.4 Let m and k be positive integers. Then [2m] 2k [ 2m] 2k+1
=
=
(2m) (m) -1 2k k
7r (2m)! 4 m k!m!(m-k-1)!
an
2m7r (
=
4m
d 2m - 1
)
k,m,m-k-1 '
68
6 Invariant measures on Grassmannians
while [2rr;:1] =4 k (7)C:)-1
and
[~7: 11] = 4m- k (mr: k) C~~ kk)) -1 Proof Combining (6.4) with the formulas of Proposition 6.2.2, we obtain
[~7] = (~7) W2k~::-2k = (~7):~ k~~::_~! = (~7) (7)-1 With a little more effort, we have
(2~: 1) W2k+1~::-2k-1
(2m - 2k - I)! ( 2m ) 1fm (2k + I)! 2k + 1 m! 22k+11f kk! 22m-2k-11fm-k-1(m - k - I)! 2m ) _1f_ (2k + 1)!(2m - 2k - I)! 2k + 1 m!22m k!(m - k - I)! 1f (2m)! 4m k!m!(m - k - I)!' (
Computing once again,
1) 1)
W2m+ 1 ( 2m + 2k W2kW2m-2kH 2m + 22m+11fmm! k! (2m - 2k + I)! ( 2k (2m + I)! 1fk 22m - 2k+11fm-k(m - k)! k! 4km!k! (2k)!(m - k)! k! 4k(7) Ckk)-1 The final formula now follows easily from the dual symmetry of the flag coefficients:
1] =
[2m + 2k+l
[2m+l] =4m- k ( m ) (2(m-k))-1, 2(m-k) m-k m-k
by the previous computation.
o
6.2 Computing the flag coefficients
69
By means of Proposition 6.2.4 one may easily generate the following Pascal triangle for the flag coefficients [~], as n = 1, ... ,8. 1 1 8
1 1571"
1 16
1 1
3571"
32
5"
16
7
2 371"
1 "3
""4
"2
1 1
2 371"
3
4
""4
4 1571"
5
6
1 71"
1
5
!l
8
8
1 8
"3
6
10571"
35
10571"
32
3
32
1 1571"
16
7
1 16
5"
1 3571"
32
1
(6.7) Certain patterns are immediately noticeable. For example, the table (6.7) suggests that
[;]=n-1 for all n ::::: 2. This indeed follows easily from Proposition 6.2.4. It is also not difficult to verify that
[n~l ]= 2-
n-l 2- ,
for all odd positive integers n. If n is even, the value of the middle flag coefficient is given by one of two formulas, depending on the congruence class of n modulo 4. Computations similar to those preceding show that if n == 0 mod 4 then
If n
== 2 mod 4 then (n) ( ~ ) (n + 2)71' ( n ) (n + 2)71' [n] ~ = ~ n4 2 2n+2 = ~,n42, n!2 2n+2'
Recall that, for fixed n, the classical binomial coefficient (~) is maximized at k = n/2 if n is even, or k = (n - 1)/2 if n is odd. In the next section we will show that flag coefficients are also maximized 'in the middle,' as the table (6.7) suggests.
6 Invariant measures on Grassmannians
70
6.3 Properties of the flag coefficients
The flag coefficients satisfy a number of properties analogous to those of the binomial coefficients. For example,
In analogy to Pascal's triangle relations we have the following. Proposition 6.3.1 For 1 ~ k Wk-lWn-k Wn-l
[n - 1] + k -
~
n -1,
WkWn-k-l
1
[n - 1] =
Wn-l
WkWn-k
[n]
Wn
k
k
(6.8)
Proof We prove (6.8) by direct computation: Wk-lWn-k Wn-l
[n - 1] + k -
1
WkWn-k-l
[n - 1]
Wn-l
k
D
In the discrete case, Pascal's triangle for binomial coefficients is much cleaner:
(n-1) + (n -1) (n) =
k-1
k
k
.
A similar expression exists for the flag coefficients, with a relation of
inequality. The inequality will follow from a more elementary additive inequality for the numbers [nJ. Recall from elementary calculus that the volume of the unit n-ball may be computed by integrating over its (n - 1)-dimensional slices; that is,
(I
Wn
The substitutions x
J
2
= 2 o wn - l ( 1 - X )
n-l
2
dx.
= sin () and dx = cos () d() yield
6.3 Properties of the flag coefficients
71
In other words,
[n] = nWn = n {:!f;; cos n 0 dO. 2Wn - l Jo This equation leads in turn to the following proposition.
(6.9)
Proposition 6.3.2 For all positive integers m and n,
[n]
+ [m] > [n+m].
Proof If 0::; e::; 1r/2, then 0::; cose ::; 1. From (6.9) we then obtain n
[n]+[m]
l:!f;; cosn e dO + m l:!f;; cosm 0 dO
> n l:!f;; cos n +m e dO + m l:!f;; cosn +m e de (n
+ m) l:!f;; cosn +m 0 de =
[n + m]. D
Proposition 6.3.2 and Theorem 6.3.3 illustrate a fundamental difference between the combinatorics of finite sets and the measure structure of this vector space analogue. Theorem 6.3.3 (The Pascal inequality for flag coefficients) For 1 ::; k ::; n - 1,
1] + [n -k 1] > [n]k .
[nk-l
Proof For 1 ::; k ::; n - 1, we have
[n - I]! [k - 1]![n - k]!
+
[n - I]! [k]![n - k -I]!
[n - 1]!([k] + [n - k]) [k]![n-k]! By Proposition 6.3.2, [k]
1]
[n k -1
+
+ [n -
[n k
1]
k] > [k + (n - k)] = [n], so that [n - 1]![n]
[n]!
> [k]![n - k]! = [k]![n - k]!· D
72
6 Invariant measures on Grassmannians
Recall that the real numbers [nJ playa role analogous to that of the positive integers n in the discrete case. The following proposition demonstrates in part the expedience of our choice of normalization for the measure Tn; that is, for the value of [nJ. Proposition 6.3.4 The map n 1---+ [nJ is an increasing function.
Proof From Proposition 6.2.1 we have
= nJ1iT(~) = nJ1iT(~) =vnr(~). 2 r (nt ) 2 ~r (~) r (~)
[nJ= nWn 2W n-1 Define a function
f on the positive real numbers by f(t) = vn
r (t + ~) r(t)
Since fG) = [nJ, it is sufficient to show that of t. Recall that [4, p. 15J
r(t) = lim k->oo
.
f is an increasing function
ktk!
t(t + 1)··· (t + k)
This implies that
f(t)
( vnlim k->oo
( r
kt+(1/2) k' 1
1
r= l' 1m k-+oo
) 1
(t + "2)(t +"2 + 1)··· (t + "2 + k)
ktk'.
k~~ t(t + 1)··· (t + k)
y7r
.
X
)-1
t(t + 1)··· (t + k)y'k 1 1 1 (t + "2) (t + "2 + 1) ... (t + "2 + k)
(6.10)
Since the function t 1 --=1--t+~ 2t+l
is increasing with respect to t > 0, so is the product on the right-hand side of (6.10). It follows that f is increasing for t > 0, and we conclude that [nJ is an increasing function of the positive integers. D From Proposition 6.3.4 we deduce the following useful property for the generalized factorial [nJ!: Proposition 6.3.5 For 1 :::; k :::; I:::; n/2,
[kJ![n - kJ! 2:: [IJ![n - IJ!
6.4 A continuous analogue of Sperner's theorem
73
Proof To begin, note that
n o <- k <- l <2 -
l< - n - k -< n.
It follows from Proposition 6.3.4 that
[n - k]··. [n -l + 1] 2: [l]··· [k + 1]. (Note that there are l - k factors on each side of this inequality.) Multiplying on both sides, we obtain
[n - k]![k]! 2: [n - l]![l]! D
The flag coefficients satisfy the following property, in evident analogy to the classical binomial coefficients. Proposition 6.3.6 For 1 :s; k
:s; n,
Once again (n/2) denotes the greatest integer less than or equal to n/2. Proof Since
[~] = [k]![~~ k]! = [n: k]' it is sufficient to consider the case in which k < (n/2). On applying Proposition 6.3.5 to the case l = (n/2), we find that
[n - k]![k]! 2: [(n/2)]![n - (n/2)]!, so that
[n]! [n]! [n - k]![k]! :s; [(n/2)]![n - (n/2)]!' D
6.4 A continuous analogue of Sperner's theorem Define a measure Vn on Mod(n) by taking the direct sum of the measures vI:. That is, for any measurable subset A ~ Mod(n), define n
vn(A) =
L v'k(A n Gr(n, k)). k=O
The measure Vn satisfies the following analogue of the classical L.Y.M. inequality (3.1).
74
6 Invariant measures on Grassmannians
Theorem 6.4.1 (The continuous L.Y.M. inequality) Let A Mod(n) be an antichain. For 0::; k ::; n let
Ak
~
= An Gr(n, k),
so that
is a disjoint union. Then (6.11)
Proof For each 0 ::; k ::; n, the measure of flags meeting Ak is given by
by the definition (6.3) of most one point, we have
vI:.
Since every flag in Flag(n) meets A in at
n
n
k=O
k=O
L v'k(Ak) [k]![n - k]! = L 4>n(Flag(Ak)) = 4>n(Flag(A)) ::::; [n]!
It follows that
o We now have the necessary tools to prove a continuous analogue of Theorem 3.1.1.
Theorem 6.4.2 (The continuous Sperner theorem) Let A be an antichain in Mod( n). Then
Evidently equality is attained in Theorem 6.4.2 when A = Gr(n, (n/2)). Proof We reason in analogy to the proof of Theorem 3.1.1. For 0 ::; k ::; n let Ak = AnGr(n, k). Combining (6.11) and Proposition 6.3.6 we obtain
t
t
vl:(A k) < v'k(A k) < 1 k=O [(n/2J - k=O [~J -,
6.4 A continuous analogue of Sperner's theorem
75
so that
o In analogy to the discrete lattice P(S), a subset F ~ Mod(n) is called an r-family if chains in F contain no more than r elements. Given an r-family F in Mod(n) let Fk = FnGr(n, k). Since every flag in Mod(n) meets F in at most r elements, we have n
L
From (6.3) we then obtain the following generalization of (6.11):
~ VJ::(Fk)
~---
_.
(6.12)
This inequality leads in turn to a continuous analogue of Sperner's theorem for r-families (Theorem 3.1.2). Theorem 6.4.3 Let F be an r-family in Mod(n). Then
Proof As in the proof of the discrete case (Theorem 3.1.2), relabel the flag coefficients Co, Cl, ..• ,Cn in descending order, so that Co :2:: Cl :2:: ... :2:: en; then relabel the numerators VJ::(Fk) in (6.12) by xo, Xl,···, Xn so that each Xk is the numerator of that term of (6.12) having Ck as denominator. The inequality (6.12) now becomes
It then follows from Lemma 3.1.3 that
Xo
+ Xl + ... + Xn
::;
Co
+ Cl + ... + Cr-l·
In other words,
o
6 Invariant measures on Grassmannians
76
We now consider a continuous analogue of a question first posed by Sperner (see also Theorem 3.2.7). Given A <:;:; Gr(n, k), let [A]z denote the collection
[A]z = {W
E
Gr(n, l) : W
<:;:;
V for some V
E
A}.
Here we assume that 0 ::; l ::; k ::; n. Sperner's question, recast in the language of subspaces, asks for a lower bound on the measure of [A]z, given only the measure IIn (A). The continuous L.Y.M. inequality gives us such a bound. Theorem 6.4.4 For A <:;:; Gr(n, k), IIn
[k]![n- k]! ([A]z);::: [l]![n -l]! lin (A).
Proof Let Bz = Gr(n, l) - [A]z. If WE Bz then W cannot be contained in any V E A. In other words, the set A U Bz is an antichain. It then follows from (6.11) that
Meanwhile,
so that lin (A)
[~J +
1_
IIn
([A]z) < 1
[7J
_.
It follows that IIn
n] [n] -1 [k]![n - k]! ([A]z);::: [ l k IIn (A) = [l]![n -l]! lin (A).
o For example, in the case l IIn
=
k - 1, we have
[k] ([A]k-1) ;::: [ k ] lin (A). n- +1
Theorem 6.4.4 can also be expressed in the language of simplicial complexes. As for the discrete lattice P(S), define A <:;:; Mod(n) to be a simplicial complex, if whenever V E A and W <:;:; V we have W E A. (Such a set A is also called an order ideal of Mod(n).) In other words,
6.5 A continuous analogue of Meshalkin's theorem
77
if V E A, then all subspaces of V are also elements of A. Once again the set of maximal elements of a simplicial complex is an antichain, and a simplicial complex having exactly one maximal element is called a simplex (or principal order ideal). The elements of dimension k in a simplicial complex A are called the k-faces of A. Theorem 6.4.4 can now be thought of as giving a lower bound for the measure of the set of I-faces of A, given the measure of the set of k-faces of A.
6.5 A continuous analogue of Meshalkin's theorem We now turn to the continuous (vector space) analogues of the r-decompositions and s-systems on the discrete lattice P(S); see also Section 3.1. A map 8: {I, ... ,r} ---+ Mod(n) is called an r-decomposition ofRn if (i) 8( i) ~ 8(j) for i =I- j, and (ii) 8(1) EEl··· EEl 8(r) = Rn Denote by Dec(n,r) the set of all r-decompositions of Rn. Note that, for each 8 E Dec(n, r), dim 8(1)
+ ... + dim 8(r) = n.
Given positive integers aI, a2, ... , a r such that al + ... + a r = n, we shall denote by M ult (n; aI, ... , a r ) the set of all r-decompositions 8 such that dim8(i) = ai for i = 1, ... ,r. In other words, Mult(n;al, ... ,ar ) is the set of all (ordered) decompositions of Rn into direct sums of subspaces having dimensions al, ... , a r . Evidently the set Dec(n, r) can be expressed as the finite disjoint union
Dec(n,r)
=
An s-system of order r is a subset a
~
Dec(n, r) such that the set
{8(i):8Ea}
(6.13)
is an antichain in Mod(n), for each 1 :::; i :::; r. An obvious example of an s-system of order r is Mult( n; aI, ... , a r ) for some admissible selection of al, ... , a r . If 8, ( E Mult(n; al, ... , a r ) then 8(i) and ((i) both have dimension ai, so that either 8(i) = ((i) or the two subspaces are incomparable in the subset partial ordering on Mod(n). This holds for i = 1, ... , r, and so the antichain condition on (6.13) is satisfied.
6 Invariant measures on Grassmannians
78
Other disguised examples with which we have already worked are the s-systems of order 2. Let A be an antichain in Mod(n). For each V E A we can express R n as the direct sum V EEl V J.., so that the pair (V, V J.. ) is a 2-decomposition in Dec(n, 2). Moreover, the set
{VJ..: V E A} is also an antichain in Mod(n), so that the set (T
= {(V, V J..) : V
E A}
is an s-system of order 2. Thus the notion of s-system is again a generalization of the notion of an antichain. Similarly, the Grassmannian Gr(n, k) can be viewed as Mu1t(n; k, n - k) through the bijection V 1-+ (V, VJ..). In analogy to the construction of the measure v'k on Gr(n, k), define invariant measures on the sets Mult( n; aI, ... , a r ) as follows. For 8 E Mu1t(n;al,.'" ar) define a flag (xo, Xl,'" ,Xn) E Flag(n) to be compatible with 8 if (i)
(ii)
X al
= 8(1), and
Xal+.+a)Xal+.+ai_l
= 8(i),
for i 2: 2.
Here the quotient x al +.+ai / x al +.+ai-l denotes the orthogonal complement of the vector space x al +.+ai-l inside the larger space x al +.+ai' Let Flag(A) be the set of all flags (XO, Xl, ... , Xn) compatible with some 8 E A for A ~ Mult( n; aI, ... , a r ). Define
(6.14) To justify this normalization combinatorially, note that, for each 8 E A, there are exactly [al]![a2]!'" far]! flags compatible with 8; that is, to choose a flag compatible with 8 one must choose a frame for each of the vector spaces 8i , of which there are [ail! choices for each i. The measure v;;l, ... ,a r is evidently invariant under rotations, and we have
These values, called multiflag coefficients, are continuous analogues of the multinomial coefficients of the discrete lattice in Chapter 3. Define a measure vn;r on Dec( n, r) by taking the direct sum of the
6.5 A continuous analogue of Meshalkin's theorem measures v::l, ... ,a r define
79
That is, for any measurable subset A ~ Dec(n, r), (6.15)
Just as the continuous Spemer Theorem 6.4.2 gives the maximum possible measure for an antichain A in Mod(n), a generalization of this theorem gives the maximum possible measure for an s-system in Dec( n, r). En route to such a generalization we prove a multinomial version of the continuous L.Y.M. inequality. (Compare with Theorem 3.1.4.)
Theorem 6.5.1 (The continuous multinomial L.Y.M. inequality) Let u ~ Dec( n, r) be an s-system. For al + ... + a r = n denote
so that U=
u
(J'al, ... ,ar'
is a disjoint union. Then (6.16)
Proof For al + .. ·+ar = n the measure of flags compatible with ual, ... ,a r is given by A. 'f'n
(Flag(ual,···,ar ))
n = v al, ... ,a r (u al, ... ,ar )[al]"" •
[a r· ]'
by the definition (6.14) of v::l, ... ,a r ' Suppose that a flag (Xo, Xl, ... , Xn) is compatible with both ,,(,8 E u. Then "((1) = x al and 8(1) = Xbll where al = dim"((l) and bl = dim8(lt Since (Xo, Xl,"" Xn) is a flag, we have x al ~ Xb l or vice versa. However, u is an s-system, so that either "((1) = 8(1) or the two spaces are incomparable. Therefore "((1) = 8(1) and al = bl . Continuing, we have "((2) = Xal+a2/Xal and 8(2) = Xbl+b2/Xal (since al = bI). A similar argument then implies that "((2) = 8(2) and a2 = b2 • Continuing in this manner we conclude that "((i) = 8(i) for each 1 ::; i ::; r, so that "( = 8. In other words, every flag in Flag(n) is compatible with at most one r-decomposition 8 E u. It follows that
80
6 Invariant measures on Grassmannians
:::; [n]!, so that
D
The multifiag coefficients also satisfy the following property, in analogy to the classical multinomial coefficients. Proposition 6.5.2 Let r :::; n be positive integers, and suppose that n = rq + b, where q :::: 0 is an integer and 0 :::; b :::; r - I is the integer remainder. For al + ... + a r = n,
Proof Let al, ... ,ar be positive integers such that al + ... + a r = n. Without loss of generality, suppose that al < (nlr). Then ai > (nlr) for some i > 1. Again without loss of generality, suppose that
Then
a2 -
al :::: 2, so that
It then follows from Proposition 6.3.5 that
Replace al with al + I and a2 with a2 - 1. Note that the identity al + ... + a r = n is preserved. This process is repeated until ai :::: (nlr) for alII:::; i :::; r; that is, until ai = (nlr) + I for I :::; i :::; band ai = (nlr) for b + I :::; i :::; r, where b is the integer remainder upon division of n by
r. Since each iteration of this procedure decreases the value of the product [al]!··· [ar]!, it follows that
[al]!··· [ar]! :::: ([(nlr)]!r- b ([(nlr)
+ I]!)b,
81
6.6 Helly's theorem for subspaces for all al
+ ... + ar
=
n. Therefore,
[n]! [n]! [aI]!···[a r ]!::; ([(nlr)]!r- b ([(nlr)+l]!)b' for all al
+ ... + ar
= n.
D
We are now able to prove a continuous analogue to Meshalkin's Theorem 3.1.5, a multinomial generalization of the continuous Sperner Theorem 6.4.2. Theorem 6.5.3 (The continuous Meshalkin theorem) Let 17 be an s-system in Dec(n,r). Then
Vn -r(I7) < [ ,
n
],
,(nlr),···, (nlr) , ,(nlr) + 1"", (nlr) + 1"
-
",
where n
'V'
V
r-b
b
== b mod r.
Proof We reason in analogy to the proof of Meshalkin's Theorem 3.1.5. For al + ... +ar = n let l7a1 , ... ,a r = 17 nMult(n; aI, ... ,ar ). By combining (6.16) and Proposition 6.5.2 we obtain
L al
n v al,···,ar (17al,···,ar )
+·+ar=n [(n/r), .. ,(n/r),(n/r)+l, .. ,(n/r)+l]
so that n val, ... ,a r (17al,···,a r )
al+ .. ·+ar=n
<
[(nlr), ... , (nlr) , (nlr)
+
1"", (nlr) + 1]' D
6.6 Helly's theorem for subspaces Using the language of simplicial complexes in Mod(n) introduced in Section 6.4, it may be possible to apply the Euler characteristic techniques of Section 5.3 to prove the following Helly-type theorem for subspaces of R n. However, this theorem also follows easily from elementary linear algebra.
82
6 Invariant measures on Grassmannians
Theorem 6.6.1 (Helly's theorem for Mod(n)) Let F be a family of nonzero subspaces ofRn. Suppose that, for any subset G ~ F such that IGI :S n (that is, every subset of cardinality at most n of F), dim
(n x)
> O.
(n x)
> O.
xEG
Then dim
xEF
In other words, if every n elements of F contain a common line through the origin, then there is at least one line £, contained in all o( the subspaces in F. Proof To begin, suppose that F is a finite family of subspaces. For this case, the proof is by induction on the size IFI of the family F. If IFI :S n then the theorem holds trivially. Suppose that the theorem holds for the case IF I = m for some m 2: n. We then consider the case of IF I = m + l. Write F = {XI,X2,'" ,Xm+I}, and denote (6.17) Since the theorem is true for families of size m, each Yi has positive dimension. That is, for each i E {1, ... , m + 1} there exists a nonzero vector Vi E Yi. Since m 2: n, the collection {VI, V2, ... , V m + d must be linearly dependent. Without loss of generality, we may then assume that Vm+1
= CIVI
+ ... + CmVm '
where not all the coefficients Ci are zero. However, (6.17) implies that Vi E Xm+1 for all i E {1, ... ,m}. It follows that Vm+1 E Xm+1 as well. Since Vm+1 E Ym+l, we have m+1 Vm+1
E
n
Xj'
j=1
This completes the induction step and the proof of the finite case. To prove the general case, consider the set x of all lines through the origin contained in a given (nonzero) subspace x; that is
x = {£, E Gr(n, 1) : £, ~ x}.
83
6.7 Notes For all nonzero x E Mod(n), the set space Gr(n, 1). Define
F = {x:
x is a closed subset of the compact x
E F},
and define G similarly for subfamilies G of F. Suppose G is any finite subfamily of subspaces in F. Since F satisfies the intersection condition of Theorem 6.6.1, so does the subfamily G. Since G is finite, it follows from the previous argument that dim
(n x)
>
o.
xEG
oi
In other words, the family G clo~d sets has a non-empty intersection, for every finite subfamily G of F. It follows from the compactness of Gr(n, 1) that the family F of closed sets has a non-empty intersection. Equivalently, the intersection of all subspaces in F has positive D dimension.
6.7 Notes In this chapter we have assumed the existence and uniqueness of Haar (orthogonal invariant) measures on the spaces Gr(n, i) and Flag(n), which are homogeneous spaces of the orthogonal group O(n). For a complete discussion of Haar measures on Lie groups and their homogeneous spaces, see [76]. See also [97, p. 151]. The measures Tn, ¢n, V n , and vn;r may also be defined from the perspective of the action of 0 (n) on the domains of these measures. Choosing a normalization for Tn determines a normalization for the O(n)invariant measure on the group O(n) itself. The actions of O(n) on the spaces Flag(n), Mod(n), and Dec(n, r) then induce precisely the invariant measures ¢n, V n , and vn;r on these spaces. We give a brief sketch of this approach, leaving the details to the reader (see also [76]). To begin, recall that the columns of an orthogonal n x n matrix A determine a unique orthogonal frame for Rn. Conversely, given an orthogonal frame in Rn, we can construct an orthogonal matrix by choosing a unit vector from each line in the frame. Since each line in an orthogonal frame contains two unit vectors, there is a 2n-to-1 correspondence between the group 0 (n) and the set offrames for R n, and consequently between O(n) and Flag(n). Once again setting [n] = Tn (Gr(n, 1)), we can construct an O(n)-invariant measure ¢n on the group O(n) itself,
84
6 Invariant measures on Grassmannians
such that
¢n(O(n))
=
2[n]¢n-l(O(n - 1)),
so that
Recall that the group O(n) acts transitively on Gr(n, k), for 0 .::; k .::; n. Moreover, the stabilizer Stab(V) of a subspace V E Gr(n, k) under this action is isomorphic to the product group O(k) x O(n - k). This follows from an argument similar to the 'combinatorial' argument on frames given in Section 6.1. It follows from elementary Lie group theory that there exists a diffeomorphism between the space Gr( n, k) and the quotient space (no longer a group)
O(n) O(k) x O(n - k)" This diffeomorphism induces an invariant measure that
vi:
on Gr(n, k), such
[n]! [k]![n- k]!' as in our original approach. A similar construction works for the measures vn;al, ... ,ar on the spaces Mult(n;al, ... ,ar ), and again this construction via group actions agrees with that of Section 6.5. Theorems 6.4.2, 6.4.3 and 6.5.3, which give continuous analogues of Sperner's and Meshalkin's theorems, are due to Klain and Rota [55]. Many results in the lattice theory of Mod(n) hold independently of the normalization of the measure Tn; that is, the value of [n]. For example, in [24], Fisk developed a similar construction, in which the total measure of Gr(n, 1) is taken to be nwn/2, the measure suggested by the usual twoto-one quotient map from the unit sphere in Rn. However, this normalization does not permit an extension of the lattice analogy sufficient to admit analogues of Sperner's and Meshalkin's theorems, since the value of nwn /2 is not an increasing function of n. Moreover, our choice (6.1) of normalization [n] in Section 6.1 is compatible with Cauchy's surface area formula in the sense described by the equation (5.11). In subsequent sections we shall see that (6.1) is in fact the unique choice of normalization that agrees with a fundamental normalization for the rigid motion invariant measures on R n and its subspaces, namely, the intrinsic volumes (see also [55]).
6.7 Notes
85
The bound given in Theorem 6.4.4 is probably not the best. In the discrete case, Katona [48] and Kruskal [57] independently obtained a much stronger result (see Section 3.4), which unlike Theorem 6.4.4 did not depend on the size of the ambient set (or, in the language of subspaces, the dimension n of the ambient vector space). Perhaps a continuous analogue to the Katona-Kruskal theorem is waiting in the wings.
7 The intrinsic volumes for polyconvex sets
In this chapter we shift our attention from the Grassmannian (k-planes passing through the origin in Rn) to the affine Grassmannian (all kplanes in R n ), in order to extend the intrinsic volumes of Section 4.2 from the lattice of parallelotopes to the larger lattice of polyconvex sets. A crucial tool in this generalization will be the Euler characteristic of Section 5.2, which will serve as a device for testing whether a compact convex set intersects a given k-plane. Section 7.4 concludes the chapter with a preliminary version of the mean projection formula, which provides a fundamental connection between the intrinsic volumes and the lattice of subspaces. An improved version of the mean projection formula will appear in Section 9.4.
7.1 The affine Grassmannian We next consider the partially ordered set Aff(n) of all linear varieties in Rn, whether through the origin or not. The partially ordered set Aff(n) is not a well-behaved lattice. The group En of Euclidean motions acts naturally on Aff(n). Denote by Graff(n, k) the subset of Aff(n) consisting of all elements of rank k; that is, all linear varieties of dimension k. The minimal element of the partially ordered set Aff(n) is the empty set 0 (unlike in Mod(n), where the minimal element was the zero subspace
{o}). We shall prove the existence of a measure oAk on Graff( n, k) that is invariant under the Euclidean group En. To this end, we parametrize Graff(n, k) as follows. Given V E Graff(n, k), let VJ.. be the maximal linear subspace of R n orthogonal to V and containing the origin. There is a unique maximal linear subspace or(V) orthogonal to V J.. and containing the origin. The subspace or(V) is of dimension k; we shall say
7.2 The intrinsic volumes and Hadwiger's formula
87
that V and or(V) are parallel. The set V n V J.. is a point in V which we denote by p(V). Thus, to every V E Graff( n, k) there corresponds a pair (or(V), p), where or(V) E Gr(n, k) and p E or(V)J.. ~ Rn. (Note that or(V)J.. = VJ...) Conversely, given any pair (Vo,p), where Vo E Gr(n,k), and p E VoJ.., there is a unique linear variety V E Graff( n, k) such that or(V) = Vo and V n VoJ.. = p. For V E Graff( n, k) and pERn, denote by V + p the translation of the linear variety V by the vector p. If f is a real-valued measurable function on Graff(n, k), let 1: Gr(n, k) x Rn ~ R be given by the equation
/(Vo,p) = f(Vo + p), and define
J
f dAI: =
1 1
Gr(n,k) vo-L
f(Vo,p) dp dvk'(vo)'
where dp denotes the ordinary Lebesgue measure on vl ~ Rn-k. We thereby define a measure AI: on Graff( n, k) that is invariant under the group En. To see this, suppose that gv E En corresponds to translation by the vector v. Then the composition of functions
f
0
gv(Vo,p) = f
0
gv(Vo
+ p) =
f(Vo
+ P + v) =
f(Vo
+ P + vIV-L), o
where vlv-L denotes the orthogonal projection of the vector v onto the o subspace vl. We then have
1 1
Gr(n,k) Vo-L
r
r
JGr(n,k) JVo-L
f(Vo
+ P + vlv-L) dp dvk'(vo) 0
f(Vo
+ p) dp dvk'(Vo)
J
fdAI:,
by the translation invariance of the measure dp. A similar argument shows that AI: is invariant under rotations (and reflections), and consequently is rigid motion invariant.
7.2 The intrinsic volumes and Hadwiger's formula We now consider the relationship between the intrinsic volumes, as defined in Chapter 4, and the invariant measure AI: defined on Graff( n, k). Once again, fix an orthogonal coordinate system in Rn, and consider
88
7 The intrinsic volumes for poly convex sets
the lattice Par(n) of finite unions of parallelotopes with sides parallel to this fixed coordinate system. For A eRn, denote by Graff( A; k) the set of all V E Graff( n, k) such that A n V =1= 0. The relationship between Ak and the intrinsic volumes is given by the following theorem. Theorem 7.2.1 For all parallelotopes P in Par(n), P,n-k(P)
= C kAk(Graff(P; k))
(7.1)
where the constants C k depend only on nand k. Note that the equation (7.1) is asserted only for the case in which P is a parallelotope and not for arbitrary elements of Par( n) (i.e. not for all finite unions of parallelotopes). Theorem 7.2.1 immediately suggests the following question: what are the values of the constants C~? In Section 9.3 we will prove the amazing fact that C~ = 1, for all n 20 and all 0 :s; k :s; n. However, much can be accomplished in the meantime. In the course of proving Theorem 7.2.1, we shall need the following lemma. Lemma 7.2.2 Let A and B be compact convex sets in Rn such that A U B is convex. Let V be a linear variety of positive dimension such that V n A =1= 0 and V n B =1= 0. Then V nAn B =1= 0.
Proof The set V n (A U B) = (V n A) U (V n B) is convex, since it is the intersection of two convex sets. If V nAn B is empty, then choose points a E (V n A) - (V n B) and bE (V n B) - (V n A). Let 1 denote the straight line segment with endpoints at a and b. Clearly 1 ~ V. Since Au B is convex, we have 1 ~ Au B, so that 1 = (1 n A) U (1 n B). Since 1 is connected and both 1 n A and 1 n B are closed, it follows that 1 n (A n B)
=
(1 n A) n (1 n B) =1=
contradicting the assumption that V nAn B
0,
= 0.
o
Lemma 7.2.2 can be re-stated as follows. If A and B are compact convex sets such that A U B is convex, then for every k > 0 we have Graff(A n B; k)
= Graff(A; k) n
Graff(B; k),
so that Ak(Graff(A U B; k))
= Ak(Graff(A; k)) + Ak(Graff(B; k)) -
Ak(Graff(A n B; k)). (7.2)
7.2 The intrinsic volumes and Hadwiger's formula
89
We now turn to the proof of the theorem. Proof of Theorem 7.2.1 To begin, define a function rJ on parallelotopes by
rJ(P) = Ak(Graff(P; k)).
(7.3)
It follows from (7.2) and Theorem 4.1.3 that rJ has a unique extension to a valuation on all of Par(n) (although the equation (7.3) will not hold for arbitrary finite unions of parallelotopes). Since Ak is invariant, so is the valuation rJ. Now suppose that P is a parallelotope, and let fp be the indicator function of Graff(P; k) in Graff(n, k). In the language of the preceding section we have
Ak(Graff(P; k))
=
J
fp dAk
=
JJ
!p(Vo,p) dpdvk(Vo),
(7.4)
where p ranges over Vl. For a fixed Vo E Gr(n, k), we have
!p(Vo,p) = fp(Vo + p) = 1 if and only if (Vo + p) n P -I- 0 (otherwise the function takes the value zero); that is, if and only if p E PlVl, where PIVoJ.. denotes the orthogonal projection of P onto the subspace Vl. In other words, the function !p(Vo,p) is the indicator function of PIVoJ... For a > 0 we now have
Ak(Graff(aP; k))
so that
Ak(Graff(aP; k)) = ( voln-k(aPlVl) dvk'(vo), JGr(n,k)
(7.5)
where voln-k denotes the (n - k)-dimensional volume in the (n - k)dimensional space VoJ... Since (n-k)-dimensional volume is homogeneous of degree n - k, we can continue:
rJ(aP)
Ak(Graff(aP; k)) a n- k ( voln-k(PlVl) dvk'(vo) JGr(n,k)
l
an-kj IplV-LdPdvk'(vo) Gr(n,k) Vo-L 0
90
7 The intrinsic volumes lor polyconvex sets a n- k Xk(Graff(P; k)) an-krJ(P).
It follows that rJ is homogeneous of degree n - k on parallelotopes, and consequently on all of Par(n). Since the (n - k)-volume is continuous on compact convex sets in vl, it also follows from (7.5) that rJ is a continuous valuation. Therefore, by Corollary 4.2.6, there exists a constant 'Y'k E R such that
rJ(P) = 'YkJ-ln-k(P)
for all P E Par(n). It is clear from (7.5) that rJ(P) > 0 if P has a non-empty interior in Rn. It follows that 'Y'k =1= O. Setting Ck = Ih'k, we have J-ln-k(P) = C'k Ak(Graff(P; k))
for all parallelotopes P E Par( n).
D
Theorem 7.2.1 relates the intrinsic volumes, as defined in Chapter 4 in terms of product valuations, to the invariant measure Ak on Graff(n, k). So far our main result asserts that the intrinsic volume J-ln-k, which for parallelotopes is evaluated as a symmetric function of the side lengths, also evaluates the 'measure' of the set of k-dimensional planes meeting a given parallelotope. This interpretation extends to convex sets. The obvious idea for the further extension of J-ln-k to all polyconvex sets is to use Groemer's extension Theorem 5.1.1; that is, to define a continuous set function J-l~-k on a compact convex set K by J-l~_k(K)
= CkAk(Graff(K;k)).
(7.6)
It then follows from (7.2) and (7.5) that J-l~-k is a continuous valuation on compact convex sets. Theorem 5.1.1 then asserts that J-l~-k has a unique extension to all polyconvex sets in Rn. Note, however, that the equation (7.6) is valid only if K is convex. Alternatively, we can give a more constructive definition for J-l~-k on Polycon( n). It turns out that we can let the Euler characteristic do most of the work. To motivate this method, recall the definition of J-ln-k on Par(n) as a product valuation. Let I be any simple function on Par(n). We have shown that
J
I dJ-ln-k
=L 0-
JJ... J
I(Xl, X2,··., xn) dJ-l&(Xo-(l)) dJ-l&(xo-(2))
7.2 The intrinsic volumes and Hadwiger's formula
where the sum ranges over all permutations The integral
(J'
91
of the set {I, 2, ... ,n}.
JJ... J
f(xI, X2,···, xn) dfL6(XI)'" dfL6(x k)
(7.7)
can be rewritten as follows. Let V be the linear variety of dimension k parallel to the linear subspace spanned by Xl, X2, ... , Xk, and passing through the point 0, ... ,0, Xk+l, ... , Xn. Let A E Polycon(n) , and let f = h· Then the integral (7.7) simplifies to
JJ... J
hfv dfL6(XI)'" dfL6(x k)
= fLo(A n V).
To indicate the dependence of V on the coordinates Xk+l, Xk+2,···, Xn , let us write V = V(Xk+l,'" ,xn ). We then have
J
fA dfLn-k
=L
JJ... J
fLo (A n V(Xo-(k+l) , ... ,Xo-(n)))
0-
This formula suggests replacing the summation over the symmetric group of permutations by integration over the space Gr(n, k) relative to the invariant measure v'k. Specifically, let V (Vo; p) be the linear variety of dimension k parallel to the linear subspace Vo, and intersecting the linear subspace vl at the point p E Rn. We then obtain
JJ
fLo (A n V(Vo;p))
dfL~=~(P) dv'k(vo).
This can be simplified further, since the integration is practically carried out over the invariant measure on Graff(n, k). For all A E Polycon(n), define
ji~_k(A) = C'k
J
fLo (A n V) dAk(V)·
(7.8)
This formula is known as Hadwiger's formula. The integral ranges over all V E Graff(n, k). It is easily verified that fL~-k' as extended by (7.8), is a convex-continuous invariant valuation on Polycon(n). Note that fLo(K n V) = fGraff(K;k) (V) if K is a compact convex set. Hadwiger's formula (7.8) then implies that ji~_k(K)
= C'kAk(Graff(K;k)).
Thus, the valuation ji~-k defined by (7.8) agrees with the alternative extension (7.6) as well as with Theorem 7.2.1. However, the equation (7.8)
92
7 The intrinsic volumes for polyconvex sets
defines intrinsic volumes on every polyconvex set A, giving the desired extension explicitly. Recall that the intrinsic volume /tk was called 'intrinsic,' because /tk(P) remains the same for a parallelotope P, even if P is rigidly embedded within a higher dimensional space. In other words, the valuations /tk on Par(n) are normalized independently of n (see Theorem 4.2.2). We shall see later that this property carries over to the extension of /tk given by (7.8), so that our use of the term 'intrinsic' remains justified (see Theorem 8.4.1). One way to see this is to let K E Kn, embed Rn into Rn+l, and compare /t!:;(K) with /t~+l(K) as given by (7.8). The value of /t~+l is given as the measure of all (n - k + 1)-dimensional planes in Rn+l that meet K. Since a generic (n - k + l)-plane in Rn+l meets Rn in a plane of dimension n - k, the measure of all (n - k + l)-planes meeting K in Rn+l is proportional to the number of (n - k)-planes meeting K in Rn, where the proportionality constant is independent of K, and is given by the measure (in the appropriate sense) of (n - k + 1)planes in Rn+l containing a given (fixed) (n - k)-plane. If we call this proportionality constant a~,n+l, we have
for all K E Polycon(n). Meanwhile, Theorem 4.2.2 asserts /t~+l(P) = /t!:;(P) for all P E Par(n). It follows that a~,n+l = 1, and the intrinsic volumes are universally normalized. The problem in this argument is the claim that the measure of all (n - k + l)-planes meeting K in Rn+l is proportional to the number of (n-k)-planes meeting K in Rn. This statement, although true, requires additional measure-theoretic detail in order to be made precise. Instead, we defer a rigorous proof of the universal normalization to Section 8.4. For the interim, and in order to avoid ambiguity, we will continue to use the notation /t!:; for the kth intrinsic volume on Polycon( n) until we are able to prove the normalization property rigorously in Theorem 8.4.1. We will make an exception for /to, which is clearly normalized independently of the ambient space Rn, since /to(K) = 1 for all non-empty compact convex sets K. We are now in a position to generalize the results of Section 1.2. Let K and L be compact convex sets in Rn. Suppose that L has dimension n, and that K ~ L. Then Graff(K; k) ~ Graff(L; k), and the conditional probability that a linear variety of dimension k shall meet K, given that
7.3 An Euler relation for the intrinsic volumes
93
it meets L, is given by the ratio Ak(Graff(K; k)) Ak(Graff(L; k)) .
From (7.6) and (7.8) we now deduce the following.
Theorem 7.2.3 (Sylvester's theorem) Let K ~ L be compact convex sets. Suppose that L is of dimension n. The conditional probability that a linear variety of dimension k shall meet K, given that it meets L, is given by J.l~_k(K) J.l~_k(L)
. D
Note that this probability can be computed relative to any invariant measure on Graff(n, k). The normalizing constants Ck are irrelevant, since they cancel in the ratio above.
7.3 An Euler relation for the intrinsic volumes The Euler characteristic formula of Theorem 5.2.4 generalizes to intrinsic volumes. To this end, we recall from Theorem 5.2.3 that the Euler characteristic for the interior of a convex polytope P of dimension n is given by J.lo(int P)
=
(_l)n.
Let P be a compact convex polytope. For almost all V E Graff( n, n k) (with respect to the measure A~_k)' we have relint(P) n V =I- 0 whenever P n V =I- 0. Hadwiger's formula (7.8) then implies that
J.lk (relint(P)) =
Ck
= Ck
r
J.lo(relint(P) n V)
r
(_l)dim(relint(p)nv) J.lo(P n V)
}vEGraff(n,n-k)
dA~_k(V) dA~_k(V).
}vEGraff(n,n-k)
Note that if relint(P) n V =Idim(relint(P) n V)
0, then
= dim(P n V) = dimP + dim V -
dim(P U V),
where dim(PUV) denotes the dimension of the smallest plane containing
94
7 The intrinsic volumes for poly convex sets
PUV. If J1k(relint(P)) =I- 0, then dimP 2': k. Therefore, dim(PUV) = n for almost all V E Graff( n, n - k). It follows that dim(relint(P) n V)
= dimP + (n - k) - n = dimP - k,
and we obtain the formula
J1k(relint(P))
=
(-l)dimP-k J1k (P)
(7.9)
for all convex polytopes P. From Hadwiger's formula (7.8) we now derive Euler relations for the intrinsic volumes of an arbitrary polytope P. As in Section 5.2, we define a system of faces of P to be a family F of compact convex polytopes such that if Q,Q' E F and Q =I- Q' then (relint(Q)) n (relint(Q')) = 0, and such that
U(relint(Q))
=
P.
QEF
Under these conditions, the formula (7.9) yields at once
J1k(P)
=
2: (_l)dim
Q -k J1k(Q)·
QEF
7.4 The mean projection formula We conclude this section with an alternative interpretation of Hadwiger's formula (7.8) for compact convex sets.
Theorem 7.4.1 (The mean projection formula) For all K E Kn,
J1k(K)
= C;;:-k
r
JGr(n,k)
J1Z(KlVo) dvk'(vo).
Recall that J1Z denotes the k-dimensional volume on each k-dimensional subspace of Rn. Theorem 7.4.1 states that the kth intrinsic volume J1k(K) is proportional to the mean of the k-volumes of the orthogonal projections of K onto all k-dimensional subspaces of Rn. Proof For K E Kn, we have C;;:_k.A~_k(Graff(K;
C;;:-k
r
r
JGr(n,n-k) JVl
n - k))
fK(Vo,P) dp dV;;:_k(Vo)
7.5 Notes
95
Recall that lK(Vo,p) = 1 if Kn(Vo+p) -I- 0 and is zero otherwise. Since K n (Vo + p) -I- 0 if and only if p lies inside the projection KlVl, we have lK(Vo,p) = IKlVl(p). Therefore, f.1~(K)
C~_kA~_k(Graff(K;
r C~_k r C~_k r C~_k
JGr(n,n-k)
JGr(n,n-k)
JGr(n,k)
1
n - k))
vl
IKlVo(p) dp
dV~_k(Vo)
f.1~(KlVl) dV~_k(Vo)
f.1~(KlVo) dvk'(Vo),
where the last equality follows from the fact that the orthogonal duality between Gr(n, k) and Gr(n, n - k) preserves measure. D Note that the preceding argument also yields
A~_k(Graff(K; n -
k))
=
r
JGr(n,k)
f.1~(KlVo) dvk'(Vo)
for all polyconvex sets K. However, since (7.6) defines f.1~ for convex sets only, Theorem 7.4.1 need not hold for all K E Polycon(n).
7.5 Notes A brief survey of Euler relations for valuations on Polycon(n) appeared in [72, p. 218-220]. See also [37, 39, 79, 80]. For a general reference on the theory of valuations on convex bodies, see [71, 72] and [85]. For additional related integral geometric formulas and applications, see also [82] and [99]. The theory of intrinsic volumes can be extended to partially ordered sets. Let P be a finite partially ordered set, with a unique minimal element o. An order ideal of P is a subset A of P such that, if x E A and y ::; x, then YEA. The set of all order ideals of P is denoted by L(P). It is closed under unions and intersections; that is, it is a distributive lattice. (It can be shown that every finite distributive lattice can be represented in the form L(P) for some partially ordered set P). We shall consider valuations on L(P). If x E P, denote by x the order ideal consisting of all yEP such that y ::; x. A valuation f.1 on P will be called invariant if f.1(A) = f.1(B) whenever the order ideals A and Bare isomorphic as partially ordered sets. It can be shown that every valuation f.1 on L(P) extends uniquely to the Boolean algebra of all subsets of P, and that a valuation is uniquely
96
7 The intrinsic volumes for polyconvex sets
determined by assigning the value f.1(x) for each x E P. We typically consider only valuations such that f.1(a) = O. There are always at least two invariant valuations on P, namely, the size;
a(A)
= IAI- 1, A =1= {a},
and the Euler characteristic, which is uniquely determined by setting
f.10(x) = 1 if x
=1=
{a}.
It can be shown (see [78, 79, 80]) that the Euler characteristic is closely related to the Mobius function of P; that is, the integer-valued function on P uniquely defined by the conditions
f.1(a,8) Lf.1(8,q)
=
= 1
0, p > 8.
q~p
One finds that
f.1o(A)
= -
L
f.1(8,p).
pEA,p#8
Thus, one obtains a generalization of the expression for the Euler characteristic in terms of the 'number of faces'. Several special cases have already been studied in detail. Other invariant valuations are defined as follows. Let ex be an isomorphism class of 'simplices' X. For every A E £(P), let f.1a be the number of x of class ex contained in A. We conjecture that the f.1a and f.1o span the vector space of all invariant valuations. The identities holding among the f.1a are particularly interesting to determine. Actually, the theory of invariant valuations on partially ordered sets should proceed along general lines. There is a more general notion of invariant measure that deals with arbitrary segments [x, y] of P, rather than just segments x = [0, x]. Such valuations can be multiplied in a way that resembles multiplication in the incidence algebra of P (see [78, 79, 92]). There is more than one gap in the analogy between the discrete and the continuous case. On the one hand, the ease with which the theory of invariant valuations can be carried over to arbitrary partially ordered sets suggests the possibility of generalization of the intrinsic volumes to a more general space than R n. On the other hand, the analogy between £(8) (or P(8)) and Mod(n) is deficient, since we do not consider order
7.5 Notes
97
ideals in Mod(n) (or Aff(n)) as our basic building blocks, but rather the polyconvex sets. The difficulty here is that of singling out a sufficiently ample class of order ideals in Mod(n) that can be taken as simplices (so as to replace convex sets). Convexity on Grassmannians is at present too unwieldy a notion, and somehow the Schubert cell structure of the Grassmannian must be brought to bear on the problem (see, for example,
[74]).
8 A characterization theorem for volume
In this chapter we state and prove a fundamental theorem of geometric probability, namely the characterization of volume on polyconvex sets as a continuous rigid motion invariant simple valuation Polycon( n). The characterization of volume will lead in turn to a straightforward characterization for all of the intrinsic volumes (see Section 9.1). In Section 8.4 we verify the universal normalization of the intrinsic volumes. In Section 8.5 we investigate a connection between volume and random motions, leading to an analogue of the Buffon needle problem, in which evenly spaced lines are replaced by a discrete additive subgroup of Rn. The results of Section 8.5 will be used in Section 9.6 to solve the Buffon needle problem in a still more general form.
8.1 Simple valuations on polyconvex sets In this section we state and prove a characterization theorem for volume, leading to a generalization of Theorem 4.2.4 to Polycon(n). Recall that Theorem 4.2.4 characterized volume in two ways, one involving continuity and the other involving monotonicity. It turns out to be much easier to generalize Theorem 4.2.4 in the monotonic case than it is in the continuous case. Theorem 8.1.1 Suppose that J-L is a monotone translation invariant simple valuation on Polycon( n). Then there exists C E R such that J-L(K) = CJ-Ln(K), for all K E Polycon(n).
Proof We shall assume without loss of generality that J-L is an increasing valuation, for, if J-L is decreasing, then the valuation -J-L is increasing.
8.1 Simple valuations on polyconvex sets
99
Consider first the restriction of f.-l to Par(n). By Theorem 4.2.4, there exists c E R such that f.-l(P) = Cf.-ln(P), for all P E Par(n). Next, consider K E Kn. Recall from the definition of volume in elementary calculus that f.-ln(K) is given by the supremum of f.-ln(P) over all P E Par( n) such that P ~ K. Since f.-l is increasing, we have f.-l(P) ~ f.-l(K) for all P ~ K. In other words, Cf.-ln(P) ~ f.-l(K) for all P ~ K. It follows that cf.-ln(K) ~ f.-l(K). Recall again that f.-ln (K) is also given by the infimum of f.-ln (P) over all P E Par(n) such that K ~ P. It then similarly follows that f.-l(K) ~ cf.-ln(K). Hence, f.-l(K) = cf.-ln(K), for all K E Kn, and therefore also for 0 all K E Polycon(n). A characterization for volume as a continuous valuation, free of monotonicity conditions, requires considerably more work. We begin with some preliminary definitions. Recall from Section 5.1 that a non-empty compact convex set K E Kn is determined uniquely by its support function hK : sn-l - > R, defined by h K ( u) = maxxEK{ X· u}, where· denotes the standard inner product on Rn. Recall also that, if vERn and v denotes the line segment with endpoints v and -v, then hv(u) = lu, vi, for all u E sn-l. For K E Kn, denote by -K the set {x : -x E K}; that is, the reflection of K through the origin. If K = - K we say that K is centered or symmetric about the origin. A set K is centered or symmetric if some translate of K is centered about the origin. Denote by K~ the set of all centered compact convex sets in Rn. Recall that for compact convex sets K and L the Minkowski sum K + L is defined by
K
+L =
{x
+y :x
E
K and y
E
L},
and that hK +L = hK + h L . A zonotope is a finite Minkowski sum of straight line segments. A convex body Y is called a zonoid if Y can be approximated in Kn by a convergent sequence of zonotopes [85, p. 183J. We shall need the following useful fact concerning zonoids and smooth convex bodies. A complete discussion of this result and its proof may be found in [26, 34, 85J. Proposition 8.1.2 Let K E K~, and suppose that the support function hK is COO. Then there exist zonoids Y 1 , Y 2 such that
8 A characterization theorem for volume
100
Proof For 9 E coo(sn-l), the cosine transform of g, denoted Cg, is given by the equation Cg(u)
=
r
}Sn-l
Iu· vlg(v) dv.
The transform C is a bijective linear operator on the space of all even Coo functions on sn-l. This fact is a consequence of Schur's lemma (or the Funke-Hecke theorem) for spherical harmonics [85, pp. 182-189] (see also [26, 34]). Since the function hK : sn-l -----t R is Coo and even, there exists an even Coo function 9 : sn-l -----t R such that hK = Cg; that is,
hK(U) Let g+(v)
=
r
}Sn-l
Iu· vlg(v) dv.
= max{g(v),O}, and let g-(v) = max{-g(v),O}. Then
hK(U)+
r
}Sn-l
Iu,vlg-(v)dv=
r
}Sn-l
It is easy to check that the functions hY1
Iu,vlg+(v)dv.
(8.1)
= Cg+ and hY2 = Cg- each
satisfy the properties of a support function of a centered convex body, which we denote by Yl and Y2 respectively. The equation (8.1) is then equivalent to the statement that K + Y 2 = Y 1 . Moreover, since the Riemann sums converging to the integrals in (8.1) are linear combinations of support functions of line segments (Le. support functions of zonotopes), it follows that Y 1 and Y 2 are zonoids. 0 Let SO(n) denote the special orthogonal group; that is, the set of all rotations of R n. Let B = {el' ... , en} denote the standard basis for R n , and denote by SO(n, B) the set of all rotations in SO(n) that fix at least n - 2 elements of the basis B. Proposition 8.1.3 Suppose that 1> E SO(n). Then there exists a finite collection 1>1,1>2, ... , 1>m E SO(n, B) such that 1> = 1>11>2'" 1>m-
Proof The proposition holds trivially in dimension n = 2, since SO(2, B) = SO(2). Suppose that n 2: 3 and that the proposition holds for dimension n - 1. Let 1> E SO(n), and suppose that 1> tj SO(n, B). Let v = 1>en , and assume without loss of generality that v =I=- en. Let v' denote the unit normalization of the orthogonal projection of v onto Span{ el, ... , en-I} = Rn-l. There exists 'l/J E SO(n) such that 'l/Je n = en and 'l/Jv' = en-I. As v lies within Span{v', en}, it follows that 'l/Jv lies within Span{ en-I, en}.
8.1 Simple valuations on polyconvex sets
101
Let ( be the rotation that fixes e1, ... ,en-2 and rotates 'lj;v to en. Then ( E SO(n, B), and ('lj;¢e n = ('lj;v = en· Let 7] = ('lj;¢. Since 'lj; and 7] both fix en, it follows from the induction assumption on SO(n - 1) that there exist 'lj;1,' .. 'lj;i, 7]1, ... 7]j E SO(n, B) such that 'lj; = 'lj;1 ... 'lj;i and 7] = 7]1 ... 7]j. Thus,
¢ = 'lj;-lC l 7] = 'lj;:;1 ... 'lj;llC l 7]l ... 7]j. D
A valuation J-L on Kn is said to be simple if J-L vanishes on sets of dimension less than n. Theorem 8.1.4 Suppose that J-L is a continuous translation invariant
simple valuation on Kn. Suppose also that J-L([O,l]n) = 0, and that J-L(K) = J-L(-K), for all K E Kn. Then J-L(K) = 0, for all K E Kn. Here [O,l]n denotes the n-fold Cartesian product of the closed unit interval [0, 1] with itself; that is, a unit n-cube. Proof If n = 1 then the result follows readily, since a compact convex subset of R is merely a closed line segment. Since J-L is simple and vanishes on the closed line segment [0,1]' it must vanish on all closed line segments of rational length. It then follows from continuity that J-L vanishes on all closed line segments. For n > 1, assume that Theorem 8.1.4 holds for valuations on Kn-1. Since J-L is translation invariant and simple, the fact that J-L( [0, l]n) = implies that J-L([O, l/k]n) = for all integers k > 0. Therefore, J-L(C) = for every box C of rational dimensions, with sides parallel to the coordinate axes. This follows from the fact that such a box can be built up out of cubes of the form [0, l/k]n for some k > 0. The continuity of J-L then implies that J-L( C) = for every box C of positive real dimensions, with sides parallel to the coordinate axes. Next, suppose that D is a box with sides parallel to a different set of orthogonal axes. If n = 2 then it is easy to see that D can be cut into a finite number of pieces, translations of which can be pasted to form a box C with sides parallel to the original coordinate axes (see Figure 8.1). Since J-L is simple and translation invariant, it follows that J-L(D) = J-L(C) = 0. If n > 2, then for all rotations (E SO(n,B), a box with sides parallel to the basis (B can be cut, translated, and re-pasted into a box parallel to B, using precisely the operations followed in the case n = 2. This works because the rotation ( fixes at least n - 2 of the original coordinate axes. More generally, for 'lj; E SO(n), Proposition 8.1.3 states that 'lj; is a finite product of elements of SO(n, B).
°
°
°
°
102
8 A characterization theorem for volume
________ 1
Fig. 8.1. Re-orient a frame without use of rotations.
Therefore, a box with sides parallel to the basis 'If;E can be cut, translated, and re-pasted into a box parallel to E, using a finite iteration of operations of the type used in the case n = 2. It follows that, if D is a box with sides parallel to any orthogonal frame in Rn, then D can be transformed into a box C with sides parallel to the original coordinate axes, by means of cutting, pasting, and translations. Therefore, we have p,(D) = p,(C) = O. Next, define a valuation 7 on Kn-l as follows. Given a compact convex subset K of R n - l , set
7(K) = p,(K x [0,1]). Note that 7([0, l]n-l) = p,([o,l]n) = O. Notice also that 7 satisfies the hypotheses of Theorem 8.1.4 in dimension n - 1. The induction hypothesis then implies that 7 = O. Since p, is simple, it follows that p,(K x [a, b]) = 0, for any convex body K ~ R n - l and any rational numbers a and b, with a :::; b. The continuity of p, then implies that p,(K x [a, b]) = 0 for all a, bE R. Said differently, p, is zero on any right cylinder with a convex base. Let Xl, ... ,Xn be the coordinates on R n. We can represent R n-l by the hyperplane Xn = O. The right cylinders for which we have shown p, to be zero have tops and bottoms that are congruent and that lie directly above and below each other. In other words, the edges connecting the top face to the bottom face are orthogonal to the hyperplane Xn = O. This process can be applied to right cylinders with base in any (n-1)dimensional subspace of Rn. Since p, = 0 on boxes in every orientation, it follows (from the preceding argument) that p, = 0 on right cylinders of every orientation. Suppose that M is a prism, or slanting cylinder, for which the top and bottom faces are congruent and both parallel to the hyperplane Xn =
8.1 Simple valuations on polyconvex sets
103
Fig. 8.2. Turn a prism into a right cylinder.
0, but whose cylindrical boundary is no longer orthogonal to Xn = 0, meeting it instead at some constant angle. See Figure 8.2. Cut M into two pieces, MI and M 2 , separated by a hyperplane that is orthogonal to the cylindrical boundary of the prism. Rearrange the pieces Mi and re-paste them together along the original (and congruent) top and bottom faces. We are then left with a right cylinder C whose surrounding boundary is orthogonal to the new top and bottom faces. Since f.1 remains constant under this operation, it follows that
(Actually, such a cutting and rearrangement is possible only if the diameter of the top/bottom of M is sufficiently small compared with the height and angle of the cylindrical boundary; i.e. provided that M is not too 'fat'. If the base of M is too large, however, we can subdivide M into 'skinny' prisms by subdividing the top/bottom of M into convex bodies of sufficiently small diameter and considering separately each prism formed by taking the convex hull of the (disjoint) union of a piece of the bottom of M with its corresponding congruent piece of the top of
M.) Now let P be a convex polytope having facets PI, ... , Pm, and corresponding outward unit normal vectors UI, ... , U m . Let v ERn, and let v denote the straight line segment connecting the point v to the origin o. Without loss of generality, let us assume that PI, ... ,Pj are exactly those facets of P such that Ui . v > 0, for each 1 ~ i ~ j. In this case,
104
8 A characterization theorem for volume
the Minkowski sum P
+ v can be expressed in the form
where each term of the above union is either disjoint from the others, or intersects another in a convex body of dimension at most n - 1. It follows that
Notice, however, that each term of the form Pi f.1(Pi + v) = 0. Hence, f.1(P
+ v) =
+ v is a
prism, so that (8.2)
f.1(P),
for all convex polytopes P and all line segments V. By induction over finite Minkowski sums of line segments, it immediately follows from (8.2) that, for all convex polytopes P and all zonotopes Z,
f.1(Z)
= 0,
and f.1(P
+ Z) = f.1(P).
The continuity of f.1 then implies that f.1(Y) = 0, and f.1(K
+ Y)
=
for all K E K n and all zonoids Y. Next, suppose that K E K~ has a Coo support function h K Proposition 8.1.2, there exist zonoids Y1 and Y2 such that K + Y2 In this case, (8.3) implies that f.1(K) = f.1(K
+ 1'2) =
(8.3)
f.1(K),
.
=
By Y1 .
f.1(Y1 ) = 0.
Since any centered convex body K can be approximated by a sequence Ki of Coo centered convex bodies, it follows (by continuity) that f.1 is zero on all of K~. Now let .6. be an n-dimensional simplex, with one vertex at the origin. Let Ul, ... ,Un denote the other vertices of .6., and let P be the parallelotope spanned by the vectors Ul,"" Un. Let v = Ul + ... + Un. Let 6 be the hyperplane passing through the points Ul, ... , Un, and let 6 be the hyperplane passing through the points v - Ul, .•. ,v - Un. Finally, denote by P* the set of all points of P lying between the hyperplanes 6 and 6. We can now write P
= .6. U P* U (-.6. + v),
105
8.1 Simple valuations on polyconvex sets
where each term of the union intersects another in dimension at most n - 1. Since P and P* are centered, we have
0= p,(P)
=
p,(I:::.)
+ p,(P*) + p,( -I:::. + v) =
p,(I:::.)
+ p,( -1:::.).
In other words, p,(I:::.) = -p,( -1:::.). Meanwhile, we are given that p,(I:::.) = p,( -1:::.). Therefore p,(I:::.) = 0, for any simplex 1:::.. Let P be a convex polytope in Rn. The polytope P can be expressed as a finite union of simplices
such that the intersection I:::. i n I:::. j has dimension less than n, for all i =I- j. It follows that m
p,(P)
=
"Lp,(l:::. i )
=
O.
i=l
Since the set of all convex polytopes is dense in Kn, the continuity of p, then implies that p,(K) = 0 for all K E Kn. D Theorem 8.1.4 is equivalent to the following theorem. Theorem 8.1.5 (The volume characterization theorem) Suppose that p, is a continuous translation invariant simple valuation on Kn. Then there exists c E R such that p,(K) + p,(-K) = cp,n(K), for all KEKn. Note that Theorem 8.1.5 implies that p,(K) = (c/2)P,n(K) for all centered convex bodies K E K~. Proof of equivalence Suppose that p, is a continuous translation invariant simple valuation on Kn. For K E Kn, define
v(K)
= p,(K) + p,( -K) - 2p,([0, l]n)P,n(K).
Then v satisfies the hypotheses of Theorem 8.1.4, so that v(K) all K E Kn. Therefore,
p,(K)
+ p,( -
K)
=
= 0 for
cp,n(K),
where c = 2p,([0,1]n). Hence, Theorem 8.1.4 implies Theorem 8.1.5. The reverse implication is obvious. D
106
8 A characterization theorem for volume
8.2 Even and odd valuations A valuation f.1 on Kn is said to be even if f.1(-K) = f.1(K)
for all K E Kn. If f.1(-K) = -f.1(K)
for all K E Kn then f.1 is said to be odd. Every valuation f.1 on Kn has a decomposition f.1 =
where
f.1even
f.1odd
+ f.1odd,
is the even valuation defined by
=
f.1even(K)
and
f.1even
~(f.1(K)
+ f.1( -K))
is the odd valuation defined by f.1odd(K)
=
~(f.1(K) - f.1(-K)).
Theorem 8.1.5 implies that the volume f.1n is the only continuous, translation invariant, even, simple valuation on Kn, up to a constant factor. More generally, for any continuous, translation invariant, simple valuation f.1, there exists C E R such that f.1(K)
= cf.1n(K) + f.1odd(K)
(8.4)
for all K. A natural question at this point is that of whether the property of evenness is necessary to characterize volume. Do there exist any nontrivial continuous, translation invariant, odd, simple valuations? The answer turns out to be yes, even in the case of dimension 2. For example, let .6. denote the equilateral triangle in R 2 of unit side length, centered at the origin and with a side parallel to the x-axis. Define a valuation 'T/ on K2 by the equation (8.5) It is clear that 'T/ is a translation invariant continuous function of K. To see that 'T/ is a valuation, we require the following proposition.
Proposition 8.2.1 Suppose K, L, M E Kn such that K U L is convex. Then
(K U L)
+M
=
(K + M) U (L + M),
(8.6)
8.2 Even and odd valuations
107
and
(KnL) +M = (K +M) n (L+M).
(8.7)
Proof The equation (8.6) is obvious, even when K U L is not convex. To prove (8.7), suppose that x E (K n L) + M. Then x = y + m, where y E KnL and m EM. Evidently x E K +M and x E L+M, so that
(K n L) + M
~
(K + M) n (L + M).
Next, suppose that x E (K +M)n(L+M). Then x = a+ml = b+m2, where a E K, bEL, and ml,m2 E M. Since K U L is convex, with K and L each compact, the line segment with endpoints at a and b must contain a point y E K n L, which we can express in the form y = (1 - t)a + tb for some 0 :s:; t :s:; 1. Let m = (1 - t)ml + tm2. Since M is convex, mE M, and y + mE (K n L) + M. Meanwhile,
=
y+ m
(1 - t)a + tb + (1 - t)ml + tm2
=
(1 - t)x + tx
= x,
so that x E (K n L) + M, and
(K n L) + M
~
(K + M) n (L + M). o
This completes the proof.
M
Now suppose that K, L E K} such that K U L is convex. If we set = .6. then Proposition 8.2.1 implies that
J-l2((K U L) +.6.) + J-l2((K n L) +.6.) =
J-l2((K +.6.) U (L + .6.)) + J-l2((K +.6.) n (L + .6.))
=
J-l2(K +.6.) + J-l2(L + .6.),
and similarly if.6. is replaced with -.6.. Consequently, we have
77(K U L)
+ 77(K n L) = 77(K) + 77(L).
Moreover, since J-l2 is even,
77(-K)
= J-l2(-K +.6.) - J-l2(-K + -J-l2(K +.6.) -77(K),
so that 77 is an odd valuation on K2.
(-.6.))
= J-l2(K +
(-.6.))
8 A characterization theorem for volume
108
Fig. 8.3. The Minkowski sum 6
+ (-6).
It immediately follows that 'f} is simple! To see this, suppose that K is a symmetric convex body; i.e., K = -K. Then we have
'f}(K) = 'f}( -K) = -'f}(K), so that 'f}(K) = O. However, all points and line segments are symmetric convex bodies, so that 'f} must vanish in dimensions 0 and 1. Finally, we check that 'f} -I- O. The fact that 'f}(f:l.) -I- 0 follows easily from the Brunn-Minkowski inequality in the plane [85, p. 309]. However, this fact can also be seen by a simple and direct computation. It can be seen in Figure 8.3 that the Minkowski sum f:l. + (-f:l.) is a regular hexagon of unit edge length, having area
whereas
Hence,
'f}(f:l.) = /12(f:l. + f:l.) - /12(f:l.
/3 -I- o. + (-f:l.)) = -2/12(f:l.) = -2"
Shortly after the discovery of Theorem 8.1.5, which characterizes the continuous translation invariant even simple valuations, Schneider used an analogous approach to characterize the continuous translation invariant odd simple valuations. This led to the following theorem.
Theorem 8.2.2 (Schneider's characterization theorem) Suppose that /1 is a continuous translation invariant odd simple valuation on Kn. Then there exists a continuous odd function 9 : sn-l - > R and a measure S K on sn-l such that
/1(K) for all K E Kn.
=
r
lSn-l
g(u) dSK
o
8.3 The volume theorem
109
Here sn-l denotes the unit sphere in Rn. The measure SK is sometimes called the Aleksandrov-Fenchel-Jessen measure associated with K. Since any continuous translation invariant simple valuation f..t can be expressed as a sum f..t = f..teven + f..todd of even and odd valuations, Theorems 8.1.5 and 8.2.2 combine to improve the characterization (8.4). In other words, a continuous translation invariant simple valuation f..t must have the form f..t(K)
where c E Rand 9 :
= cf..tn(K) +
sn-l ----t
r
}Sn-l
g(u) dSK
,
R is a continuous odd function.
8.3 The volume theorem In order to generalize Theorem 4.2.4 to the lattice Polycon(n), we require the following proposition relating rotation invariance to invariance under reflections. Proposition 8.3.1 (Sah) Let .6. be an n-dimensional simplex. There exist polytopes PI, ... , Pm such that
where each term of this union intersects another in dimension at most n -1, and where each of the polytopes Pi is symmetric under a reflection across a hyperplane. Proof Let xo, ... , Xn be the vertices of .6., and let .6. i be the facet of .6. opposite to Xi. Let Z be the center of the inscribed sphere of .6., and let Zi be the foot of the perpendicular from Z to the facet .6. i . For all i < j, let Ai,j denote the convex hull of Z, Zi, Zj, and the face .6. i n .6. j . Then
.6. =
U
Ai,j,
O~i<j~n
where the distinct terms Ai,j of this union intersect in at most dimension n-1. It is also evident that each Ai,j is symmetric under reflection across the n - 1 hyperplane determined by the point Z and the face .6. i n .6. j • Now relabel the polytopes Ai,j by a linear ordering PI' ... ' Pm, where m = ~n(n + 1). This gives
.6.
= PI U··· U Pm,
where the polytopes Pi satisfy the desired conditions.
o
110
8 A characterization theorem for volume
We now generalize Theorem 4.2.4 to the lattice Polycon(n).
Theorem 8.3.2 (The volume theorem for Polycon( n)) Suppose that f1 is a continuous rigid motion invariant simple valuation on Kn (or Polycon(n)). Then there exists c E R such that f1(K) = Cf1n(K), for all K E Kn (or Polycon(n)). Recall from Groemer's extension Theorem 5.1.1 that a convex-continuous valuation is well defined on Polycon( n) if and only if it is well defined on the generating set Kn. Proof Since f1 is translation invariant (as well as rotation invariant) and simple, Theorem 8.1.5 implies the existence of a E R such that f1(K) + f1( -K) = af1n(K), for all K E Kn. Let .6. be a simplex in R n. Then we have (8.8) If the dimension n of the ambient Euclidean space is even, then .6. differs from -.6. by a rotation, so that
Meanwhile, if n is odd, from Proposition 8.3.1 there exist polytopes PI,"" Pm such that
where each term of this union intersects another in dimension at most n -1, and where each of the polytopes Pi is symmetric under a reflection across a hyperplane. It follows that each Pi differs from - Pi by a proper rigid motion (i.e. by a rotation followed by a translation), so that f1( -Pi) = f1(Pi). Therefore, m
m
i=l
i=l
(8.9) Taken together, (8.8) and (8.9) imply that f1(.6.) = (a/2)f1n(.6.) for any simplex .6.. Let c = a/2, and suppose that P is a convex polytope in Rn. The polytope P can be expressed as a finite union of simplices
8.4 The normalization of the intrinsic volumes such that the intersection i =1= j. It follows that
M(P)
~i
n
~j
111
has dimension less than n, for all
+ ... + M(~7n) CMn(~l) + ... + CMn(~7n)
M(~l)
CMn(P). Since the set of all convex polytopes is dense in Kn, the continuity of M then implies that M(K) = CMn(K) for all K E Kn. This concludes the proof of Theorem 8.3.2. 0
8.4 The normalization of the intrinsic volumes Recall from Theorem 4.2.2 that if P is a parallelotope in Rl, and if we consider Rl ~ Rn for some n > l, then Ml(p) = M'k(P), for all k ~ o. We will now show that this universal normalization holds when P is replaced by any polyconvex set. Theorem 8.4.1 (The universal normalization theorem) The valuations Mi on Polycon(n) are normalized independently of the dimension n.
Proof Theorem 8.4.1 clearly holds for i = 0, since Mo(K) = Mo(K) = 1 for all non-empty compact convex subsets of R n for all dimensions n. Moreover, it is clear that M~ restricts to Ml for all l < k, since both valuations are identically zero on polyconvex sets of dimension l < k. Let n > k, and suppose that M~-l restricts to Ml for all k ~ l ~ n - 1.
(8.10)
We need to show that the condition (8.10) holds for M'k as well. Since M'k vanishes in dimension less than k, the restriction of M'k to a k-dimensional plane in Rn is a continuous, invariant, simple valuation on Polycon(k). By the Volume Theorem 8.3.2 there exists C E R such that M'k(K) = CM~(K) for all K E Polycon(k). Since M'k(P) = M~(P) for all parallelotopes P E Par(k) (by Theorem 4.2.2), it follows that C = 1 and M'k = M~ on Polycon(k). If k = n - 1 then we have achieved our goal. If k < n - 1, suppose that
M'k restricts to Ml for some k ~ l < n - 1.
(8.11)
8 A characterization theorem for volume
112
To complete the induction step and prove the theorem, we need to show that the condition (8.11) holds for Ikk and 11~+1 as well. Denote by v the restriction of 11k to Polycon( l + 1). Then v restricts to 11~ on Polycon(l) by the condition (8.11), whereas 11~+1 restricts to 11~ on Polycon(l) by the condition (8.10). It follows that v - 11~+1 vanishes on Polycon(l), so that v - 11~+1 is a continuous invariant simple valuation on Polycon(l + 1). By Theorem 8.3.2 there exists c E R such that v - 11~+1 = on Polycon(l + 1). Since v - 11~+1 vanishes on all of Par(l + 1) by Theorem 4.2.2, we have c = 0 and v = 11~+ I. The theorem now follows by double induction, first on l and then on n. 0
cl1lti
Henceforth we are justified in omitting reference to ambient spaces when discussing the intrinsic volumes, and so we shall simplify the notation 11k to 11k. The induction technique used in the proof of Theorem 8.4.1 actually leads to a much more powerful result, with which we shall begin Chapter 9.
8.5 Lattice points and volume Let B = {VI, ... , v n } be a basis for R n, and let £ denote the collection of points in R n consisting of all vectors having integer coordinates with respect to B; that is,
£={alvl+···+anvn
al, ... ,anEZ}.
Here Z denotes the set of integers. The set £ forms a discrete subgroup of R n with respect to vector addition. Discrete subgroups of R n are traditionally called lattices, and the elements of a lattice £ are called lattice points. Lattices in R n are not to be confused with the distributive lattices of sets treated in the preceding chapters ~ although the two notions are related, this relation will not be relevant in the present context. For the remainder of Chapter 8, the term lattice will refer to discrete additive subgroups of R n. In this section we ask the following question. If a convex body K is moved in R n by a random Euclidean motion, what is the expected number of lattice points to be found in K? In other words, what is the expectation of the random variable IgK n £1, where 9 is a random Euclidean motion? Using Theorem 8.1.1 we will show that this expected number is proportional to the volume of K. Let C denote the parallelotope C
= {alvl + ... +anvn
0:::; al,.' .,an :::; I}.
8.5 Lattice points and volume
113
The parallelotope G is called a fundamental domain for the lattice 'c. If A is a matrix whose ith column is given by the coordinates of the vector Vi (with respect to the standard orthonormal basis for R n), then the volume of G is given by J-tn(G) = Idet AI. Proposition 8.5.1 For any x E R n and any positive integer k, the translate kG + x of kG contains at least k n points of ,C and at most (k + l)n points of ,c. Proof For x = XlVI + ... + XnVn the set (kG + x) n,C consists of all vectors V = YI VI + ... + Yn Vn such that Xi ::; Yi ::; k + Xi. There are either k or k + 1 possible integer values for Yi within this interval, for each i E {I, ... , n}. Therefore kG + x contains between k n and (k + l)n points of ,c. 0
We consider next a simplified version of our original question. Suppose that a polyconvex set K is moved in Rn by a random translation x, resulting in a new body K + x. What is the expected number of lattice points in K + x? Theorem 8.5.2 Let K E Polycon(n), and let XK denote the number of points in the set (K + x) n,C, for a random translation x. Then the expectation E(XK) is given by E(X ) K
=
J-tn(K) J-tn(G) .
Proof To begin, note that for x E Rn the number of lattice points in K + x is given by J-to (( K + x) n £), where J-to is the Euler characteristic. In order to compute the expectation E(XK) we must average over all x E R n the function J-to(gK n ,c). This makes no sense, however, since such an integral would diverge. Nonetheless, because ,C is symmetric under the set of translations by points of 'c, it is sufficient to average over the set of translations by vectors x E G. Thus we have
so that E(XK) is a translation invariant monotonically increasing valuation in the parameter K. Evidently E(XK ) = 0 if K has dimension less than n, so the valuation E(XK) is also simple. It then follows from
114
8 A characterization theorem for volume
Theorem 8.1.1 that there exists
Ct
E
R such that (8.12)
for all K E Polycon(n). To compute the constant Ct, consider K k, Proposition 8.5.1 asserts that
= C.
For all positive integers
It then follows that
From (8.12) we then obtain
for all positive integers k. It follows that
Ct
=
1/J-tn (C).
o
We are now able to answer the question of how many lattice points are expected inside gK, for a random Euclidean motion g. Theorem 8.5.3 Let K E Polycon(n), and let X K denote the number of points in the set gK n £, for a random Euclidean motion g. Then the expectation E(XK) is given by E(X ) = J-tn(K) K J-tn(C)·
Proof To begin, note that, for 9 E En, the number of lattice points in gK is given by J-to(gK n C), where J-to is the Euler characteristic. In order to compute the expectation E(XK) we must average the function J-to(gKn£) over all motions g. Again this appears to make no sense, since the required integral would diverge. However, because £ is symmetric under the set of translations by points of £, it is sufficient to average over the set of motions En consisting of any rotation or reflection ¢ E O(n) followed by a translation by a vector x E C. Hence, we have E(XK)
=
k
J-to(gKn£)dg
En
r r J-to((¢K+x)n£)dxd¢,
JO(n) Jc
8.6 Remarks on Hilbert's third problem
115
where the integrals are taken with respect to the unique invariant probability measures on C and O(n). Since
1
cJto((¢K +x) nC)dx
=
Jtn(¢K) Jtn(C)
=
Jtn(K) Jtn(C)
by Theorem 8.5.2, it follows that
o
as well.
8.6 Remarks on Hilbert's third problem Let P be a polytope in R n. A dissection of P is an expression of P as a union
of polytopes PI, ... , Pm such that each intersection Pi nPj has dimension less than n for all i f j. Two polytopes P and Q in R n are said to be scissors congruent if there exist dissections of P and Q into a finite set of polytopes PI, P2 , ... , Pm and Ql , Q2, ... , Q m respectively, such that each Pi is congruent to Q i by some rigid motion of Rn; that is, such that for each i we have ~ = giQi for some rigid motion gi of Rn. It is clear that, if P and Q are scissors congruent, then they have the same volume: Jtn(P) = Jtn(Q). At the Paris International Congress of Mathematicians in 1900, David Hilbert posed the converse question (see [44]): if two polytopes P and Q in Rn have the same volume, are they necessarily scissors congruent? This question, along with subsequent variations, has come to be known as Hilbert's third problem. It can easily be shown that the answer is 'yes' for polytopes in the plane; that is, if n = 2 (see [81, p. 5]). For dimensions n ~ 3, however, a negative answer was given by Dehn in [19] only a year after Hilbert had posed the problem (see also [8, 81]). Dehn discovered a functional 1jJ on the set of polytopes in R n (for n ~ 3), known as the Dehn invariant, such that 1jJ(P) = 1jJ(Q) whenever P and Q are scissors congruent. He then exhibited two polytopes P and Q having the same volume such that 1jJ(P) f 1jJ(Q). In modern parlance the term 'Dehn invariant' actually refers to a family of scissors congruence invariant functionals that distinguish between
116
8 A characterization theorem for volume
some polytopes ofthe same volume. An example in R3 is constructed as follows. Let a = (arccos(I/3))/1f, and note that a is irrational (see, for example, [8, p. 102]). Recall again that R is a vector space of infinite dimension over the field Q of rational numbers. Let f : R -----t R be a Q-linear functional on R such that f(l) = 0 and f(a) = 1. (Such a function is possible because a is irrational.) For a compact convex polyhedron P in R3, let M I , ... , Mm denote the edges of P, and let (h, ... ,()m denote the outer dihedral angles between the facets adjacent to each corresponding edge. Define
With some effort one verifies that 'IjJ is invariant under scissors congruence. Let .6. denote the regular tetrahedron in R3 having unit volume. Since the dihedral angle between any two adjacent facets of .6. is given by arccos(I/3), we have 'IjJ(.6.) > O. Meanwhile, the unit cube C in R3 has outer dihedral angles 31f/2, so that 'IjJ( C) = O. It follows that C and .6. are not scissors congruent, in spite of having the same volume. A more general form of Dehn invariant is given in terms of tensor products (see also [81, p. 2]). Suppose n 2: 3, and let P be a compact convex polytope in Rn having (n - 2)-dimensional faces M I , ... , Mm. Let ()I,"" ()m denote the angles between the facets adjacent to each corresponding (n - 2)-face. Denote by R/Z the normalized circle group, and let R @Z R/Z denote the tensor product of the Abelian groups R and R/Z. Define a functional W : pn -----t R @Z R/Z by
The functional W is a Dehn invariant for all n 2: 3. Hilbert's original question has many variations. Let G be any group of rigid motions of Rn, for example, the group of translations, or some finite group of symmetries. We say that two polytopes P and Q are scissors congruent with respect to G if there exist dissections of P and Q into a finite set of polytopes PI, P2, ... , Pm and QI, Q2, ... , Qm respectively, so that each Pi is congruent to Qi by some motion in G; that is, so that for each i we have Pi = giQi for some gi E G. Once again it is clear that, if P and Q are scissors congruent with respect to G, then J-tn(P) = J-tn(Q). Once again one may pose the converse question. The answer remains in the negative for n 2: 3, but, for the case n = 2 (the Euclidean plane),
8.7 Notes
117
the answer is more interesting. In particular, suppose that G = T 2 , the set of all translations of the plane. In this case Hilbert's question has once again a negative answer, even in the plane! This follows from our construction (8.5) of the valuation TJ in Section 8.2. Recall that TJ was defined by
where K E K2 and ~ is the equilateral triangle of unit side length, centered at the origin. The valuation TJ is simple and translation invariant. It follows that, if P and Q are polygons in R2 that are scissors congruent with respect to T2 , then TJ(P) = TJ(Q). However, recall also that TJ(K) = 0 whenever K is a centered convex body (such as a rectangle), while TJ(~) -I- O. Let P = ~, and let Q be a square with the same area as P. Since TJ(P) -I- TJ(Q), the polygons P and Q cannot be scissors congruent with respect to the translation group T 2 . In view of Theorem 8.1.5, one might suspect that the answer to Hilbert's third problem is once again 'Yes' in dimension 2, provided that we allow the group G to contain both the translations of the plane and the reflection through the origin (corresponding to scalar multiplication by -1). Indeed, this was shown to be true by Hadwiger and Glur in [40]. A detailed treatment of the results of Hadwiger and Glur may also be found in [8, pp. 69-92] and [81]. The remarks in this section give only a hint of the beauty and complexity to be found in the study of polytopes, dissections, and congruences. For a more thorough treatment of this theory, see [8, 71, 81].
8.7 Notes Theorem 8.3.2 is due to Hadwiger, who gave a long and difficult proof in [39]. The more general Theorem 8.1.5 is due to Klain [50]. Schneider's characterization theorem appeared in [86]. Proposition 8.3.1 is due to Sah [81, pp. 16-17]. For a discussion of spherical harmonics and other background for Proposition 8.1.2, see [85, p. 184] and [33,34]. Theorem 8.1.5 leads in turn to a connection between continuous even valuations on compact convex sets and continuous functions on Grassmannians. This connection can be described in part by using generating distributions for symmetric compact convex sets. For a detailed discussion, see [54]. The question of how to characterize volume as a valuation on defined only on polytopes remains open (see [72]).
9 Hadwiger's characterization theorem
In Section 9.1 we use the volume Theorem 8.3.2 to complete our characterization of invariant valuations on polyconvex sets with one of the most beautiful and important theorems in geometric probability, Hadwiger's characterization theorem. We then use Hadwiger's theorem to derive simple proofs of numerous results in integral geometry and geometric probability. In Sections 9.3 and 9.4 we give cleaner re-statements of the intrinsic volume formulas of Chapter 6. Sections 9.2, 9.7, and 9.8 deal with the computation of intrinsic volumes in special cases. In Section 9.6 the mean projection formula of Section 9.4 is combined with the results of Section 8.5 to yield a generalization of the Buffon needle problem to spaces and planes of arbitrary finite dimension.
9.1 A proof of Hadwiger's characterization theorem The following result generalizes Theorem 4.2.5 to the lattice Polycon(n). As in the case for parallelotopes, this theorem is, in fact, equivalent to the associated volume Theorem 8.3.2.
Theorem 9.1.1 (Hadwiger's characterization theorem) The valuations Jto, JtI, ... , Jtn form a basis for the vector space of all convexcontinuous rigid motion invariant valuations defined on polyconvex sets in Rn.
Proof Let Jt be a convex-continuous invariant valuation on Polycon(n). Let H be a hyperplane in R n; that is, a linear variety of dimension n -1. The restriction of Jt to H is an invariant valuation on H. Proceeding by
9.1 A proof of Hadwiger's characterization theorem
119
induction, we may assume that n-l
jt(A)
=
L
Cijti(A),
i=O
for every polyconvex A
H. Thus, the valuation
<:;;;
n-l
jt -
L
Cijti
i=O
vanishes on all lower dimensional polyconvex sets of R n. This follows from the invariance of the valuations jti and from the fact that any lower dimensional polyconvex set is contained in the image of some rigid motion of the hyperplane H. By the volume Theorem 8.3.2, n-l
jt -
L
Cijti = Cnjtn,
i=O
where jtn is the volume on words,
Rn,
and where
Cn
is a real constant. In other
n
jt =
L
Cijti·
i=O
D
A valuation jt on Polycon(n) is said to be homogeneous of degree k > 0 if
for all K E Polycon(n) and all a
~
O.
Corollary 9.1.2 Let
jt be a convex-continuous rigid motion invariant valuation defined on Polycon(n) that is also homogeneous of degree k, for some 0 :s: k :s: n. Then there exists C E R such that jt(K) = cjtk(K) for all K E Polycon(n).
Proof By Theorem 9.1.1 there exist
Cl,""
n
jt
=L i=O
If P
=
[0, 1]n then, for a > 0,
Cijti·
Cn
E
R such that
120
9 Hadwiger's characterization theorem
Meanwhile,
Therefore,
Ci
= 0 if i =I-
k, and /-l
o
= Ck/-lk.
9.2 The intrinsic volumes of the unit ball As an application of Theorem 9.1.1, we compute the intrinsic volumes /-li(Bn) of the unit ball in Rn. We require the following facts about Minkowski sums. Let K, L be compact convex sets, and let a > O. Recall that the Minkowski sum K + aL is defined by
K
+ aL = {x + ay : x
E
K and y
E
L}.
Proposition 9.2.1 For K E Kn and any unit vector u ERn,
Here u denotes the straight line segment connecting the point u to the origin o. Proof Let L = K + EU. The volume /-In of L can be computed by integrating over the hyperplane u~ the length of each linear slice of L; that is,
where ex denotes the straight line parallel to u through the point x E u~. Since /-ll(L n ex) = /-ll(K n ex) + E for all x E Klu~, while L n ex = 0 if x ~ Klu~, we have
11-
/-ll (L
r
iKlu1-
/-In(K)
n ex) dx
/-ll(Knex)+Edx
+ E/-ln-l(Klu~). o
9.2 The intrinsic volumes of the unit ball
Let that
for
en denote the n-dimensional unit cube.
°: ;
121
Recall from Theorem 4.2.1
i ::; n.
Proposition 9.2.2 For
E ~
0,
Proof Let Ul, U2, ... , Un denote the standard orthonormal basis for Rn. Let Ui denote the line segment with endpoints at the origin 0 and the point Ui. From Proposition 9.2.1 we have
for all n ~ l. Suppose that the equation (9.1) holds for lower dimensions, and that
for some 1 ::; k < n. Then
+ EUI + ... + EUk+l) = J-ln(Bn + EUI + ... + EUk) + EJ-ln-l(Bn + EUI + ... + EUklut+l)
J-ln(Bn
=
L k
(k) i
.+
Wn-i E'
EJ-ln-l (Bn - l
+ EUI + ... + EUk)
,=0 =
t,
G)Wn-i Ei
+
t,
G)Wn-l-iEi+ l
9 Hadwiger's characterization theorem
122 Since
the equation (9.1) follows by induction to the step k
+ 1 = n.
Theorem 9.2.3 (Steiner's formula) For K E Kn and
E
0
2: 0,
n
J-ln(K
+ EEn) = LJ-li(K)wn_i En - i .
(9.2)
i=O
Proof Let 'fJ(K) = J-ln(K + En), for K E Kn. It follows from Proposition 8.2.1 that 'fJ is a continuous invariant valuation. Theorem 9.1.1 then implies the existence of constants co, ... ,Cn E R, such that n
'fJ(K) = L CiJ-li(K), i=O
for all K E Kn. Therefore, for
E
EnJ-ln
> 0, we have
(~K + En) = ~CiJ-li(K)~ En
n
L CiJ-li(K)E n- i .
(9.3)
i=O
Setting K
=
en and comparing equations (9.1) and
(9.3) we find that
o We are now able to compute the intrinsic volumes J-li(En).
Theorem 9.2.4 (The intrinsic volumes of the unit ball) For 0 S is n,
Proof Setting K
= En
in (9.2), we obtain
n
L J-li(En)wn_iEn-i i=O
L n
(
n) i
n
i
WnE - ,
.=0
for all E > O. The proposition then follows from a comparison of the coefficients of each En-i. 0
123
9.3 Crofton's formula
9.3 Crofton's formula We are now ready to compute the constants C k from Theorem 7.2.1. Specifically, we have the following remarkable result. Theorem 9.3.1 For 0:::; k:::; nand K E Kn,
J-ln-k(K)
=
Ak(Graff(K; k)).
(9.4)
In other words, C k = 1 for all n, k 2: O. We shall see that our judicious choice of normalization (6.1) for [n] leads in turn to this expression of the equation (9.4), free from normalizing constant factors. Once again, note that the equation (9.4) is asserted only for the case in which K is convex and not for arbitrary polyconvex sets. Proof Recall from Section 7.2 that
Ck Ak(Graff(K; k))
=
J-ln-k(K)
for all compact convex sets K. For the case K
=
CkAk(Graff(Bn;k)) = J-ln-k(Bn) = (n: k)
B n , we have
~:
= Wn-k [~l
Meanwhile,
r
r
r
rI
JGr(n,k) JVl JGr(n,k) JVl
J-lo(Bn n (Vo Bn _ k
+ p)) dpdvk(Vo)
dpdvk(Vo)
r
Wn-k dVk(Vo) JGr(n,k) wn-kvk(Gr(n, k)) Wn-k
[~l
Hence, we have
o
so that Ck = 1.
Theorem 9.3.1 enables us in turn to rewrite Hadwiger's formula (7.8):
J-ln-k(K)
=
r
JGraff(n,k)
J-lo(K n V) dAk(V)'
(9.5)
124
9 Hadwiger's characterization theorem
Unlike (9.4), the formula (9.5) is valid for all polyconvex sets K, although it reduces to (9.4) in the event that K is actually convex. In a similar vein, we obtain the following relation between intrinsic volumes of different degree, thereby generalizing Theorem 9.3.1.
Theorem 9.3.2 (Crofton's formula) For 0 Polycon(n),
!
/1j(K n V)
:s:
i,j
:s:
nand K
[i; j] /1i+j(K).
d).~_JV) =
E
(9.6)
Graff( n, n-i)
Note that, if j
= 0 and K
E
Kn, then the identity (9.6) reduces to (9.4).
Proof If i + j > n then both sides of (9.6) are zero. Suppose that + j :s: n. For K E Kn, define
i
ry( K)
!
=
/1j (K n V)
d).~-i (V).
Graff( n, n-i)
On applying (9.5) we then obtain
ry(K) =
!
!
/10 (K n V n W) d).~=i_ j (W) d).~_i (V),
Graff(n,n-i) Graff(V,n-i-j)
where Graff(V, n - i - j) denotes the space of linear varieties W ~ V having dimension n - i - j. Since each W ~ V, we can rewrite the previous integral as
r
r
/1o(K n W)
JGraff(n,n-i) JGraff(V,n-i-j)
= [oEGr(n,n-i)
[l
/1o(K n (Wo
!woEGr(VO,n-i-j)
d).~=Lj(W) d).~_i(V)
!Wlnvo
+ p + q)) dq dl/;:-=:_j (Wo) dpdl/~_i(VO)
= kr(n,n-i) kr(VO,n-i-j) [l !Wlnvo /1o(K n (Wo
+ p + q)) dqdpdl/;:-=Lj(Wo) dl/~_i(VO)
= kr(n,n-i) kr(VO,n-i-j) [ltB(Wlnvo) /1o(K n (Wo + v)) dvdv;:-=:_j(Wo) dl/~_i(VO)
9.4 The mean projection formula revisited =
krCn,n-i) krcvO,n-i-j) JwoJ.. J-lo(K n (wo
=
125
{
+ v)) dv dll~=Lj(Wo) dll~_i(VO)
{
(
}GrCn,n-i) }Grcvo,n-i-j) }wt
= {
(
}GrCn,n-i) }Grcvo,n-i-j)
IK1wJ..
dvdll~=:_j(WO) dll~_i(VO)
0
J-li+j(KIWl)
dll~=Lj(Wo) dll~_i(VO),
since dim(Wl) = i + j. Because J-li+j is a continuous valuation, homogeneous of degree i + j, the linearity of the integrals above implies that 'f/ is a continuous valuation on Kn, homogeneous of degree i + j. It then follows from the invariance of J-li+j and of the measures lI~=Lj and lI~_i that 'f/ is invariant. By Corollary 9.1.2, there exists c E R such that 'f/ = cJ-li+j' To compute the constant c, set K = En, the unit ball in Rn, and note that
J J J-li+j (En Iwl ) dll~=L j (Wo) dll~_i (Vo) GrCn,n-i) GrCvo,n-i-j) n .J [ n ~ i .J Wi+j , [n-z n-z-] while
cJ-li+j (En) =
C
[i : j JWi+j,
by Theorem 9.2.4. Hence,
o 9.4 The mean projection formula revisited The conclusion of Theorem 9.3.1 that C k = 1 for all 0 :::; k :::; n also allows us to re-state Theorem 7.4.1, thereby generalizing Cauchy's formula 5.5.2. Theorem 9.4.1 (The mean projection formula) For 0 :::; k < n and K E Kn,
J-lk(K) = ( J-lk(KlVo) dllk(Vo). }GrCn,k)
(9.7)
o
126
9 Hadwiger's characterization theorem
In other words, the kth intrinsic volume /-lk(K) of a compact convex subset K of any dimension in R n is equal to the integral of the k-volumes of the projections of K onto all k-dimensional subspaces of Rn. Recall that the intrinsic volumes J-tk are normalized; that is, /-lk(L) of an l-dimensional convex body L is the same regardless of the dimension n 2: l of the ambient space R n. The absence of any additional normalizing factor in (9.7) demonstrates once again the importance of making the correct choice of normalization for the Grassmannian measures v'k. The mean projection formula can also be expressed in probabilistic terms, using random variables. For a compact convex set K in R n let Xk(K) denote the k-volume of a projection of K onto a randomly chosen k-dimensional subspace V E Gr(n, k). The expectation E(Xk(K)) is computed by averaging over all projections; that is, by integrating over all subspaces with respect to the Haar probability measure on Gr(n, k), to wit, E(Xk(K))
=
r
/-lk(KIV) dV,
lGr(n,k)
where
r
dV
= 1.
lGr(n,k)
Integrating instead with respect to the measure v'k, we have E(Xk(K)) = [n k ] -1
r
/-lk(KIV) dv'k.
lGr(n,k)
Thus, we obtain the following version of Theorem 9.4.1. Corollary 9.4.2 For 0
:s: k :s: nand K
E JCn,
D
Hadwiger's Theorem 9.1.1 yields a simple proof of a yet more general form of Theorem 9.4.1. Theorem 9.4.3 (Kubota's theorem) For 0 :s: k
:s: l :s: nand K
E JCn,
9.4 The mean projection formula revisited
127
Proof Define a valuation 'f/ on Kn by
r
'f/(K) =
/1k(KIV) dvi(V).
lGr(n,l)
Evidently 'f/ is continuous, invariant, and homogeneous of degree k. It then follows from Corollary 9.1.2 that there exists c E R such that 'f/ = C/1k· To compute the constant c we consider the case K = En:
so that c
[l]! [k]![n - k]! [n]! [k]![l - k]! [n]! [l]![n -l]! [n-k]! [n-k] [n - l]![l - k]! - 1 - k .
o Kubota's theorem can also be viewed combinatorially. For 0 :::; k :::; 1 and K E KI, the mean projection formula (9.7) states that
J
/1k(K) =
/1k(KIW) dvk(W).
Gr(l,k)
Consequently,
J
/1k(KIV) dvi(V)
Gr(n,l)
J J
=
Gr(n,l) Gr(V,k)
Since each W S;; V, we have (KIV)IW
J Gr(n,l)
/1k((KIV)IW) dvk(W) dvi(V)·
/1k(KIV) dvi(V) =
= KIW,
J J
so that
/1k(KIW) dVk(W) dvi(v)·
Gr(n,l) Gr(V,k)
(9.8) How does this last integral in (9.8) compare with the ordinary mean projection formula (9.7)? The difference is that in (9.8) we are 'counting' a k-subspace W E Gr(n, k) once for each l-subspace V E Gr(n, l)
128
9 Hadwiger's characterization theorem
containing W. Ideally, our combinatorial intuition would suggest that counting Z-subspaces that contain W E Gr(n, k) is the same as counting the (Z- k)-subspaces of the quotient space RnjW ~ Rn-k, of which there are
k] [nZ- k . From this observation it would then follow that
r
r
JGr(n,l) JGr(V,k)
= [nz ~kk]
!1k(KIW) dVk(W) dvz(V)
r
JGr(n,k)
!1k(KIW) dvk'(W),
which is precisely what Kubota's theorem 9.4.3 asserts.
9.5 Mean cross-sectional volume
A k-dimensional cross-section of a compact convex set K is a subset of the form KnV, where V E Graff(K; k). Let Yk(K) denote the k-volume of a randomly chosen k-dimensional cross-section of K. The mean kcross-sectional volume of K is given by the expectation E(Yk(K)), which is computed by averaging over all V E Graff(K; k). In other words, E(Yk(K))
=
An(G k ra
~(K-, k)) JGraff(n,k) r !1k(K n V) dAk(V)'
By Theorem 9.3.1 we have Ak(Graff(K; k)) = !1n-k(K), so that
r r
JGraff(n,k)
!1k(K n V) dAk(V)
f
JGr(n,k) Jv.L
!1k(K n (V
+ x)) dx dvk'(v).
Recall from elementary calculus that
r
Jv.L
!1k(K
n (V + x)) dx
= !1n(K),
for each V E Gr(n, k). It follows that
r
JGr(n,k) !1n(K)
!1n(K) dvk'(v)
[~l
129
9.6 The Buffon needle problem revisited
The mean k-cross-sectional volume of K is then given by
E(Yk(K))
=
Mn(K) [n]. Mn-k(K) k
A one-dimensional cross-section is sometimes called a chord. Recall that Mn-l(K) = (1j2)S(K), where S(K) is the surface area of K. The expected length of a random chord in K is now given by
E(Yl(K)) = Mn(K) [n] = 2[nl Mn (K). Mn-l(K) 1 S(K) In particular, the expected length of a random chord in the unit n-ball B is
9.6 The Buffon needle problem revisited We are now ready to generalize the Buffon needle problem to n-dimensional spaces. Suppose that V E Gr( n, k) and that {Vl, ... ,vn-d is a basis for V J... Let V denote the collection V = {V
+ alVl + ... + an-kVn-k : al, ... , an-k
E
Z}
of k- planes in R n. If a compact convex set K is moved randomly in R n , what is the expected number of intersections of K with the collection V? In other words, if we denote by X K the number of connected components of gK n V for a random Euclidean motion g, what is E(XK)? Let 12 denote the lattice in V J.. given by
and let C denote a fundamental domain of C. Once again the concept of random motion is meaningful, since the symmetry of the collection V allows that we consider only motions involving translations by vectors in the fundamental domain C. Therefore, the collection of motions under consideration forms a compact Lie group, on which there exists a unique Haar probability measure. Consider the set ¢K + x, where ¢ is a rotation and x is a vector. For a fixed ¢ E O( n), the number of intersections of ¢K + x with V is equal to the number of elements of (¢K + x) IV J.. n C. Hence, the expected number of intersections of ¢K + x with V is equal to
Mn-k((¢K + x)IVJ..) Mn-k(C)
Mn-k( (¢K) IV J..) Mn-k(C)
(9.9)
130
9 Hadwiger's characterization theorem
by Theorem 8.5.2. Therefore, the expected number of intersections of K +x with V over all x and all is equal to the expected value of (9.9) over all orthogonal transformations . We now apply Corollary 9.4.2 to obtain E(XK)
= [ n ] -1 !1n-k(K). n- k
!1n-k(C)
As an example, consider the case in which K is a needle of length L in R 2 and V is a collection of lines evenly spaced by a distance d. In this case £ is a one-dimensional lattice of points on the line V J.., evenly spaced by a distance d. Since
[2]
-1
=
1
(2)
-1 WIW1
1
W2
= ~, 7f
it follows that E(X )
= ~!11(needle) = 2L
K
d
7f
7fd'
which agrees with Buffon's original solution.
9.7 Intrinsic volumes on products Next we use Hadwiger's characterization Theorem 9.1.1 to examine how the intrinsic volumes !1i evaluate on orthogonal Cartesian products. We have the following generalization of Proposition 4.2.3.
Theorem 9.7.1 Let 0 ::; k ::; n, and suppose that K ~ Rk and L ~ Rn-k are polyconvex sets. Then !1i(K x L)
=
L
!1r(K)!1s(L).
(9.10)
r+s=i
Proof Evidently the set function !1i(K x L) is a continuous valuation in each of the variables K and L when the other is held fixed. Note that each motion E Ek of Rk is the restriction of a motion E En that restricts to the identity on the complementary space Rn-k. Therefore,
In other words, the set function !1i(K x L) is a continuous invariant valuation in each variable. By an iteration of Theorem 9.1.1 in each
131
9.7 Intrinsic volumes on products
variable, there exist constants Crs E R such that k n-k
Jti(K x L) =
LL
crsJtr(K)Jts(L)
r=Os=O
for all K E Kk and L E Kn-k. Let C m denote the unit m-dimensional cube. For a, f3 2': 0, k n-k
L L crsJtr(Ck)Jts(Cn-k)ar f3s r=O
s=O
Meanwhile,
by Proposition 4.2.3. Therefore, for 0 ~ r ~ k and 0 ~ s have Crs = 1 if r + s = i and Crs = 0 otherwise. Hence, k n-k
Jti(K x L)
=L
L
r=Os=O
crsJtr(K)Jts(L)
=
L
~
n - k, we
Jtr(K)Jts(L).
r+s=i D
Corollary 9.7.2 Suppose that Jt is a convex-continuous invariant valuation on Polycon( n) such that
(9.11) for all K ~ Rk and L ~ Rn-k, where 0 ~ k ~ n. Then either Jt there exists C E R such that
=0
or
(9.12) Conversely, if Jt is a valuation of the form (9.12) then Jt also satisfies the multiplicative rule (9.11). Proof By Hadwiger's Theorem 9.1.1, there exist constants that Jt = coJto
+ clJtl + c2Jt2 + ... + cnJtn·
Ci
E R such
132
9 Hadwiger's characterization theorem
For k E {O, 1, ... ,n} denote by C k the unit k-cube in a, /3 2: 0, the condition (9.11) implies that
Rk.
Then, for all
k n-k L L crCsJ-tr(Ck)J-ts(Cn-k)ar/3s. r=Os=O Meanwhile, by Theorem 9.7.1, n
LCiJ-ti(aCk x /3Cn- k) i=O n
L J-tr(Ck)J-ts(Cn_k)a r/3s r+s=i k n-k L L Cr+sJ-tr (Ck)J-ts (Cn_k)a r /3s. r=Os=O L
Ci
i=O
Therefore, cr+s = CrC s for all 0 ::; r, s ::; n. In particular, Co = Co+o = c6, so that Co = 0 or Co = 1. If Co = 0 then Cr = cr+o = CrCo = 0, so that J-t = O. If Co = 1 then relabel C = Cl. For r > 0, we then have Cr = cHH .. +l = cl = c r , from which (9.12) follows. The converse follows from Theorem 9.7.1 by means of a similar argument. 0 We conclude this chapter with another example of a convex-continuous invariant valuation. Choose a real-valued non-negative continuous function f(t) of a non-negative variable t, such that f(t) is decreasing sufficiently fast to 0 as t -+ 00. For any n, and for any non-empty compact convex set K in R n, set
J-t(f; K)
=
r
JRn
f(d(p, K)) dp
where d(p, K) is the distance from the point p to the set K, and where dp denotes the ordinary n-dimensional Lebesgue measure. We show that J-t satisfies the inclusion--exclusion principle; that is,
J-t(f; Kl U ... U Km)
=
L J-t(f; K i ) - L J-t(f; Ki n K j ) i<j
+ ...
(9.13)
9.7 Intrinsic volumes on products
133
whenever K 1 ,K2 , ... , Km and Kl U··· UKm are convex. We begin with a direct proof of the case m = 2. Let p ERn. Since Kl and K2 are compact, there exist unique points ql E Kl and q2 E K2 such that d(p, qi) = d(p, Ki). We assume that Kl U K2 is convex, so that Kl n K2 of- 0. Evidently, d(p, Kl U K 2) = min{d(p,K1 ),d(p,K2)}, and d(p,K 1 n K 2) 2: max{d(p,K1 ),d(p,K2)}. Since Kl U K2 is convex, the line segment I connecting ql and q2 must intersect Kl nK2 at some point q'. Either d(p, q') :::; d(p, qd or d(p, q') :::; d(p, q2). Therefore,
n K 2) :::; d(p, q') :::; max{ d(p, K 1 ), d(p, K 2)}
d(p, Kl
as well, so that d(p, K 1 nK2)
=
max{d(p, K 1 ), d(p, K 2)}. We then obtain
+ f(d(p, Kl n K 2)) = f(min{ d(p, K 1 ), d(p, K 2 )}) + f(max{ d(p, Kd, d(p, K 2)}) = f(d(p,Kd) + f(d(p,K2 )),
f(d(p, Kl U K 2))
for all p ERn. Hence,
Since f is a continuous function, it follows easily from the definition of the Hausdorff topology that /-l is a continuous function of K. The general case of (9.13) then follows from Theorems 5.1.1 and 2.2.1. Thus we can extend /-lU; K) to an invariant valuation defined on all polyconvex sets K. The expression of /-lU; K) as a linear combination of the intrinsic volumes (given by Theorem 9.1.1) exhibits the moments
mjU) of the function
=
1=
x j f(x) dx
f and has the form n
/-lU; K) = f(O)/-ln(K)
+L
Cimi-l (f)/-ln-i(K),
i=l
where the constants Ci are independent of the function f. To understand this, and to compute the Ci, consider the case of the n-dimensional ball aB of radius a, centered at the origin. From the definition, we have /-lU; aB)
=
(
JRn
f(d(p, aB)) dp
9 Hadwiger's characterization theorem
134
1B f(O) dp + JRn-aB f(lpl - a) dp f(O)P,n(aB)
+
1 1
00
uESn-l
f(r - a)r n- 1 drdu
'"
f(O)P,n(aB)
+ nWn
f(O)P,n(aB)
+ nwn
1 1
f(O)P,n(aB)
+ nWn
I: ~ 1) 1
00
f(r - a)r n- 1 dr
00
f(y)(y
00
(n
j=O
f(O)P,n(aB)
+ a)n-l dy
J
f(y)yja n- 1 - j dy
0
.
~(n-1) + nWn ~ i _ 1 an-'mi_l(J)
1)
nWnmi-l (J) () () ( ) ~ (n f 0 P,n aB + ~ i _ 1 P,n-i(B) P,n-i aB . Once again Wn denotes the volume of the unit ball B, so that the surface area of the unit sphere sn-l is given by nWn- It follows that Ci =
(n
-1) _n_w-:-:::n::-;P,n-i(B)
=
i-1
(n -l)nwn ( n )-l_Wi i-1 n-i Wn
=
iWi,
so that n
p,(J; K)
=
f(O)P,n(K)
+ 2: iWimi-l(J)P,n-i(K).
(9.14)
i=l
A notable special case is obtained by setting f(x) = e-7rX 2 • The resulting valuation is called the Wills functional and is denoted by W(K).
Theorem 9.7.3 If K ~ V and L ~
vJ.. for some linear variety V,
W(K x L) = W(K)W(L).
Proof The moment
mi-l
of f(x)
=
e- 7rX2 is given by
then
9.8 Computing the intrinsic volumes
On substituting u
=
mi-I
= 7rX 2 , we
1
00
1
27ri/2
0
135
have
i - I -u U 2 e
du
=
1
27ri/2
(i)
r "2 =
1 iWi·
On substituting into (9.14), we have n
fLU; K) = fLn(K)
+L
n
L fLn-i(K).
iWimi-1 U)fLn-i(K) =
i=1
(9.15)
i=O
It then follows from Corollary 9.7.2 that W(K x L) = W(K)W(L).
o From Theorem 9.7.3 we obtain the following characterization for the Wills functional W. Once again let Cn denote the unit n-cube in Rn.
Corollary 9.1.4 Suppose that fL is a convex-continuous invariant valuation on Polycon( n) such that fL(K x L)
= fL(K)fL(L),
for all K ~ Rk and L ~ Rn-k, where 0 :::; k :::; n. If fL(C I ) = 2 then fL(K) = W(K) for all K E Polycon(n). Proof Since fL(Cd
i-
0, Corollary 9.7.2 implies the existence of some
c E R such that n
fL
= fLo + CfLI + C2fL2 + ... + CnfLn =
L CifLi. i=O
Hence, n
2
= fL(CI ) = L
cifLi(Cd
= fLo(Cd + CfLI(CI ) = 1 + c,
i=O
so that
C
= 1.
It then follows from (9.15) that fL
= W.
o
9.8 Computing the intrinsic volumes The actual computation of the intrinsic volumes of a polyconvex set is difficult in general. It is difficult even for (ordinary) volume, and also for the surface area, which is given by 2fLn-l. It can be difficult even in low dimensions: consider the perimeter of an ellipse, for example. However, we are already able to compute the intrinsic volumes of a large class of compact convex sets. Theorem 4.2.1 gives a formula for the kth intrinsic volume of an orthogonal parallelotope in Rn, and
136
9 Hadwiger's characterization theorem
Theorem 9.2.4 gives the intrinsic volumes of the unit ball. Theorem 9.7.1 gives a formula for computing the intrinsic volume of an orthogonal Cartesian product of two polyconvex sets whose intrinsic volumes are already known. In this section we work with two additional classes of polyconvex sets. First, we compute the intrinsic volumes of an arbitrary (possibly nonorthogonal) parallelotope. We then comment on the intrinsic volumes of an arbitrary convex polytope and give an explicit formula for the first intrinsic volume f.-LI of a convex polytope in R3 in terms of the lengths of its edges and the angles between the outer normals to its adjacent facets. In Section 4.2 we proved Theorem 4.2.4, which characterized volume on Par(n). Recall that Par(n) denoted the lattice of finite unions of orthogonal parallelotopes having edges parallel to the coordinate axes of Rn. A similar theorem holds for more general parallelotopes. Let VI, ... , Vn be a basis for Rn, and let Par( VI, ... , v n ) denote the lattice of finite unions of parallelotopes with edges parallel to the vectors Vi. The characterization theorem for volume in Par(n), Theorem 4.2.4, generalizes immediately to the lattice Par (VI, ... , v n ). Theorem 9.8.1 (The volume theorem for Par(VI, ... , v n ) Let f.-L be a translation invariant simple valuation defined on Par( VI, ... , vn ), and suppose that f.-L is either continuous or monotone. Then there exists C E R such that f.-L(P) = Cf.-Ln(P) for all P E Par(vI, ... , v n ); that is, f.-L is equal to the volume, up to a constant factor.
Proof For each i, let Vi denote the line segment having endpoints at Vi and the origin o. The proof of Theorem 9.S.1 is the same as the proof of Theorem 4.2.4, provided that we replace the unit cube [0,1]n with the 'unit parallelotope' C =VI
+ ... +vn
of Par(vI, ... , v n ). It then follows that f.-L
= Cf.-Ln, where C = f.-L(C)/ f.-Ln(C),
o Theorem 9.S.1leads in turn to a formula for the kth intrinsic volume of an arbitrary parallelotope. Note that if P E Par(vI, ... , v n ) is a parallelotope, then P is a translate of a parallelotope of the form
for some al, ... , an ;::: O.
9.8 Computing the intrinsic volumes Theorem 9.8.2 For
all
1 :::; k :::;
n and all al,"" an
137
2: 0,
The value of each term /-tk (ail ViI + ... + aik Vik) is easily computed using elementary linear algebra. If A is a k x n matrix whose jth row is given by the coordinates of the vector aij Vij' for j = 1, ... , k, then /-tk(ai l ViI
+ ... + aik Vik)
=
Jdet (AAT).
See, for example, [94, p. 234]. Proof of Theorem 9.8.2 Define
and extend Vk to a valuation on all ofPar(vl, ... , v n ) with Theorem 4.1.3. Let'TJ = /-tk-Vk. We will prove that 'TJ(P) = 0, for all P E Par(vI, ... , vn ). Since both /-tk and Vk vanish in dimensions less than k, it follows that 'TJ vanishes in dimensions less than k as well. If we restrict 'TJ to the k-plane Vil, ... ,ik spanned by the (independent) vectors Vip •.• ,Vik' then 'TJ becomes a continuous translation invariant simple valuation on Par(vi l , ... , Vik)' It follows from Theorem 9.8.1 (in dimension k) that there exists c E R such that 'TJ( P) = C/-tk (P) for all P E Par( ViI' ... , Vik)' However,
by the definitions of 'TJ and Vk. Therefore C = 0, and 'TJ vanishes in dimension k. In other words, if we restrict 'TJ to a k + I-plane Vil, ... ,ik+l spanned by the vectors ViI' ... , Vik+l' then we have once again a continuous translation invariant simple valuation, this time on a lattice of parallelotopes in a space of dimension k + 1. Theorem 9.8.1 again applies, and there exists C E R such that 'TJ( P) = C/-tk+ I (P) for all P E Par( ViI' ... , Vik+J. However, 'TJ is homogeneous of degree k, whereas /-tk+l is homogeneous of degree k + 1. Hence, we have c = 0, and 'TJ vanishes in dimension k + 1. Continuing this argument in each higher dimension we conclude that 'TJ(P) = 0 for P of any dimension in Par(vI, ... , v n ). 0
138
9 Hadwiger's characterization theorem
Computing the intrinsic volumes of arbitrary polytopes is more difficult. Recall from Steiner's formula 9.2.3 that, for all r ~ 0, n
/-tn(P
+ rB)
=
L /-ti(P)wn_irn-i,
(9.16)
i=O
where B denotes the unit ball in R n. Let us consider the convex set P + r B more carefully. If x E P + r B, then there exists a unique point Xp E P such that
Ix-yl,
Ix-xpl S;
for all YEP. This follows from the fact that P is compact and convex. If x E P, then evidently x = x p. If x ~ P, then x p lies on the boundary OP of P. Moreover, if x ~ P and y E OP, then y = Xp if and only if x - y.l H, where H is a support plane of P and yEP n H. Let Pi(r) denote the set of all x E P + rB such that Xp lies in the relative interior of an i-face of P. So, for example, Pn(r) = int(P). Evidently n
P+rB= UPi(r),
(9.17)
i=O
a disjoint union. Denote by Fi(P) the set of all i-dimensional faces of P. For each proper face Q of P denote by M (Q, r) the set M(Q,r) = {y+O'v : 0::; 8::; r},
where y E relint( Q) and v is any outward unit normal to oP at the point y. See Figure 9.l. It turns out that, for 0 ::; i < n, Pi(r)
=
U
M(Q, r).
QEFi(P)
To see this, suppose that x E Pi(r). Then Xp E relint(Q) for some i-face Q of P. If x E relint(Q), then x E M(Q, r). If x ~ relint(Q), then x of- Xp. Let v = (x - xp)/Ix - xpl. Then v .1 H for some support plane H of P at x p, and x = x p + O'v for 0 < 8 = Ix - x pi::; r. In other words, x E M(Q, r). Conversely, suppose that x = y + 8v E M (Q, r) for some i-face Q. If 8 = 0 then x E relint(Q). Otherwise, we have x - y .1 H for some support plane H at the point y E relint(Q). It follows that y = Xp and x E Pi(r).
9.8 Computing the intrinsic volumes
139
M({x},r)
Fig. 9.1. The Decomposition of P
+ rB.
Given a subset A of Rn, the affine hull of A is the intersection of all planes in Rn (not necessarily through the origin) containing A. Let A~ denote the set of vectors in Rn orthogonal to the affine hull of A. If Q is an i-face then Q~ has dimension n - i, so that f-tn(M(Q, r)) = rn-if-tn(M( Q, 1)) It follows that f-tn(Pi(r)) = r n- i f-tn(Pi (l)). From (9.17) we then obtain n
f-tn(P + rB) = Lf-tn(Pi(l))r n- i ,
(9.18)
i=O
for all r
> O. By comparing (9.16) and (9.18) we find that f-ti(P)
=
f-t n(Pi (l)). Wn-i
(9.19)
For example, consider a convex polyhedron P in R3 having edges Each edge Zi is formed by the meeting of two facets Qil and Qi2 of P, having outward unit normals Uil and Ui2 respectively. Therefore, the volume of M(Zi' 1) is given by
z'l, ... ,ZTn'
f-t3 (
M(-. 1)) _ Z"
-
f-tl
(_.)arccos(uil 'Ui,) _ f-tl(Zi)(}i 2
Z,
-
2
'
where (}i is the angle between the outer normals to the facets Qil and 7f, it follows that
Qi 2 • Since W3-1 = W2 =
1 Tn
f-tl(P) =
'2 Lf-t1(Zi)(}i. 7f i=l
In other words, the first intrinsic volume of a polyhedron P in R3 is given
140
9 Hadwiger's characterization theorem
by 1/(21f) multiplied by the sum over all edges of P of the product ofthe length of each edge and the corresponding angle between the normals to the facets adjacent to that edge. Suppose that P is an orthogonal parallelotope in R3 having dimensions al x a2 x a3. In this case P has 12 edges, four of length ai for each i = 1,2,3, while the angles (h are all equal to 1f /2. We then obtain 1
/-tl(P)
=
21f
3
L 4ai~ = al + a2 + a3, ,=1
as asserted by Theorem 4.2.1. Similarly, it is not difficult to show that if Ts is a regular tetrahedron in R 3 of edge length s, then
formulas for /-tl (under a slightly different normalization) evaluated on a variety of common solids in R3 are described by Hadwiger in [38, pp. 36-37J. 9.9 Notes
For the original proof of Hadwiger's characterization Theorem 9.1.1, see [39J. In addition to providing straightforward verification of many results in integral geometry, Theorem 9.1.1 also provides a connection between rigid motion invariant set functions and symmetric polynomials [13J. The normalization Ak of the invariant measures on Graff(n, k) which we have used in the text is natural for Roo; it leads to a good definition of the intrinsic volumes in Hilbert space. The invariant measure under O(n) in Gr(n, k), corresponding to the measure Ak on Graff(n, k) under En, is the Gauss-Wiener measure (as can be easily verified). The article by Schanuel [83J provides an interesting and accessible perspective on Steiner's formula and the first intrinsic volume, or generalized length, /-tl, for compact polyhedra in R3. See also [38, pp. 36-37J for extensive computations of /-tl in R3. Steiner's formula (Theorem 9.2.3) is a special case of the polynomial formula for mixed volumes. For K l , K 2, ... , Km E Kn, and real numbers AI, A2' ... ' Am > 0, the Minkowski linear combination K = AlKl + A2K2 + ... + AmKm is the convex body consisting of the set
The basis of the theory of mixed volumes is the polylinearization of
141
9.9 Notes
volume with respect to Minkowski linear combinations. If Kl' ... ' Km E IC n and AI, ... , Am > 0, then the volume /-tn is a homogeneous polynomial in the positive variables AI, ... , Am; that is, m
/-tn(AlKl
+ ... + AmKm) =
L
V(Ki1 , ... , KiJAil ... Ai n , (9.20)
where each symmetric coefficient V(Kil' .. . , KiJ depends only on the bodies K i1 , ... , Kin. Given Kl, ... ,Kn E IC n , the coefficient V(Kl, ... ,Kn) is called the mixed volume of the convex bodies Kl' ... ' Kn. It is well known, but not trivial, that the mixed volume V(Kl' ... ' Kn) is a non-negative continuous function in n variables on the set IC n , symmetric in the variables K i , and monotonic with respect to the subset partial ordering on IC n . Moreover, the mixed volume V(Kl' ... ' Kn) is a valuation in each of its parameters K i , provided that the other parameters are held fixed. The mixed volumes also satisfy several useful inequalities, such as the Brunn-Minkowski and Aleksandrov-Fenchel inequalities. The intrinsic volumes /-ti studied in the present work are normalizations of the mixed volume of a convex body K with the unit ball Bn. Specifically,
/-ti(K)
=
(~) _l_V(K, ... , K'fJn'.~.' B n), Z Wn-i '---v----' i
n-i
for all K E IC n . For a thorough treatment of the theory of mixed volumes, see [85]. Theorem 9.3.2 is originally due to Crofton [14, 15, 16] (see also [85, p. 235]). Crofton's formula generalizes the ideas of Barbier presented in Chapter 1. Theorem 9.4.3 was proved by Kubota in [58]; see also [85, p. 295]. Numerous variations of the formulas of Crofton and Kubota can also be found in [1, 2, 82, 99]. Wills first defined the functional W(K) in the context of lattice point enumeration [101]. For a discussion of the Wills functional and its applications, see also [85, p. 302] and [96]. There exist many important functionals on IC n that are invariant with respect to rigid motions, but that elude Hadwiger's characterization. One striking example, pointed out by Hadwiger in [38, p. 44], is the affine surface area, denoted n. Although originally defined only for polytopes and smooth convex bodies [85, p. 419], the affine surface area was later extended to an affine invariant functional on all of IC n (see, for example, [20,59,60,61,66, 89, 100]). Extended affine surface area was
142
9 Hadwiger's characterization theorem
shown to be a valuation by Schlitt [88]. It turns out, however, that affine surface area cannot be expressed as a linear combination of intrinsic volumes, the reason being that it is not continuous, but only upper semicontinuous, on Kn. Instead, the affine surface area satisfies a very strong invariance property: If K E Kn, then n(¢K) = n(K), for all affine transformations ¢; that is, for all translations and linear transformations having unit determinant. The affine surface area also satisfies a number of useful inequalities [67]. The question of how to characterize the affine surface area as a valuation possessing certain properties (in the style of Hadwiger's characterization theorem) remains open. Another interesting open problem of convex set theory is the problem of classification of invariant set functions that are convex-continuous and of polynomial type. For example, a bivaluation J.l(K, L), defined for polyconvex sets K and L, is a real-valued set function that is a valuation in either variable, provided that the other variable is held fixed. In a similar vein one defines a trivaluation and more generally an n-valuation. If J.l is an n-valuation, set (for K convex)
v(K) = J.l(K, K, ... ,K). The set function v is called a homogeneous polynomial set function of degree n. A finite sum of homogeneous polynomial set functions is called a set function of polynomial type. It would be interesting to give an 'intrinsic' definition of set functions of polynomial type. It is tempting to conjecture that the classification of invariant set functions of polynomial type is related to symmetric functions. For example, if f(XI,"" xn) is any homogeneous symmetric polynomial, and if one sets v(P) = f(XI,'" ,x n ) for any parallelotope with side lengths Xl, ... , X n , one can extend v to all polyconvex sets by using the expression of f in terms of elementary symmetric functions and then replacing each occurrence of an elementary symmetric function by the corresponding intrinsic volume. Does one get all invariant set functions of polynomial type in this way? (See [13].) Hadwiger's characterization Theorem 9.1.1 and Theorem 9.3.1 together suggest yet another striking interpretation in the context of simplicial complexes. Recall that elements of the distributive lattice L(8) of Chapter 3 consist of finite unions of simplices, a simplex being a collection of subsets of a finite set 8 which is closed under the taking of subsets; i.e. an order ideal with respect to subset inclusion. The distributive lattice Polycon(n) admits a partial analogy to this notion. Loosely speaking, we can think of Polycon(n) as a collection of
9.9 Notes
143
simplicial complexes, such that the simplices are the compact convex sets in R n, and the faces of such a simplex (a convex set) are its intersections with lower dimensional planes; i.e. the k-faces of an n-dimensional body K would be the sets K n V for V E Graff(n, n - k). Note, however, the codimensional duality. More precisely, define a filter F of a partially ordered set P to be a subset of P such that if x E F and y 2: x then y E F. In other words, a filter of P is an order ideal of the dual P* of P (in which all of the order relations in P are reversed). In dual analogy to order ideals, the set of minimal elements of a filter F is an antichain in P, and we can view the filters of P as simplicial complexes, such that a simplex is now defined to be a filter with a unique minimal element. As with order ideals, the collection of all filters in P forms a distributive lattice. Now consider the case P = Aff(n), ordered by subset inclusion. For V E Aff( n), denote by V the filter consisting of all linear varieties in Rn containing V; that is, V = {W E Aff(n) : V ~ W}. One may view V as a simplex in the distributive lattice of filters in Aff(n). Note that, although this lattice is also graded from 0 to n, this grading is dual to that of Aff(n). In other words, the k-faces of V are the (n - k)dimensional linear varieties in V. If K E Polycon(n) and VnK =I=- 0, then WnK =I=- 0 for all W E V. This motivates us to define k = {V E Aff(n) : V n K =I=- 0}. Evidently, k is a filter in Aff(n). Consequently one may view k as a simplicial complex in the lattice of filters of Aff( n), and it is then natural to ask how the lattice Polycon(n) relates to the lattice of filters of Aff(n). (Is it a quotient lattice?) Viewed in this light, the intrinsic volumes J.tk become the polyconvex analogues of the face enumerators (also denoted J.tk) of Section 3.2. This analogy is buttressed by the formula (9.4), which states that the intrinsic volume J.tk is, in some sense, an enumerator (measure) of the 'k-faces' of a 'simplex' (convex set) Kin Polycon(n). In a similar spirit, one may view the discrete basis Theorem 3.2.4 and Hadwiger's characterization Theorem 9.1.1 as entirely analogous classifications of invariant valuations on the discrete and polyconvex lattices, respectively. The analogy is pressed further by the existence of Helly-type theorems (Theorem 3.3.1 and Theorem 5.3.3) in both contexts. Such remarkable similarities suggest the possibility of additional ties between the simplicial and order structures of these two lattices. For example, do there exist polyconvex analogues of the binomial coeffi.-
144
9 Hadwiger's characterization theorem
cients, and if so, how are these analogues related to the flag coefficients of Chapter 6? Similar observations may be made concerning the lattice Mod(n). Although the discrete basis Theorem 3.2.4, Theorem 4.2.5 and Hadwiger's characterization Theorem 9.1.1 characterize all suitably invariant valuations on the lattices L(8), Par(n), and Polycon(n) respectively, there remains the question of how to characterize invariant valuations on simplices in Mod(n), as defined in Section 6.4. Indeed the success of the lattice analogy for Mod(n) in Chapter 6 suggests the possibility of a comprehensive theory of simplicial complexes in Mod(n) that has yet to be developed. The extension of the intrinsic volumes from elementary symmetric functions on parallelotope edge lengths (see Theorem 4.2.1) to invariant valuations on compact convex sets motivates the following question regarding total positivity. Let A be an n x n matrix of real numbers, and denote by Aij the entry of A in the ith row and jth column. The matrix A is said to be totally positive if det A : : : 0 for all minors A of A. See, for example, [47]. For K E Kn, denote by A(K) the matrix with entries Aij(K) = J.lj-i(K) if j - i ::::: 0; otherwise let Aij(K) = O. An interesting open question is that of whether the matrix A(K) is totally positive. For example, if we let K denote an orthogonal parallelotope with dimensions al x ... x an, then the total positivity condition on the 2 x 2 minors of A(K) is equivalent to Newton's inequality (see [69, p. 106] or [41, pp. 51-55]) for the elementary symmetric functions el, ... , en of the variables ai:
For general compact convex sets K, it may be necessary to replace the intrinsic volumes by a renormalization of the J.li in order to obtain total positivity for the matrix A(K). Among related open questions, the problem of syzygies seems particularly difficult. Let AI,"" Aj be compact convex sets, and let l~~) be the indicator functions of the set Graff( A; k) on Graff( n, k). The problem is to find a basis for all linear relations
for all k ::::: 0 (so that l~ = fA). Even for k = 0, where l~k) = lA, the problem is not entirely trivial (though it has been solved fully). The
9.9 Notes
145
restricted inclusion-exclusion principle for convex sets yields some such identities, but there are probably some more special identities for each k. No doubt there exist additional lattices, for which these questions may be asked and answered. For example, many results known for the lattice Polycon(n) (such as Hadwiger's characterization theorem) remain open questions for the ostensibly simpler lattice pn of polytopes in Rn, as well as for the lattice of polyconvex subsets of the sphere sn (see also Chapter 11). For a survey of open questions in this area, see [71, 72]. Another lattice sharing many qualities with Polycon(n) is the lattice of star-shaped sets in R n . A set A ~ R n is said to be star-shaped, if o E A, and if for each line £ passing through the origin in R n, the set A n £ is a closed interval. Advances have recently been made in the development of a comprehensive, though far from complete, theory of star-shaped sets, a theory dual to the theory of convex sets developed in the present text. In the dual theory, convex bodies are replaced by star-shaped sets, and projections onto subspaces are replaced by intersections with subspaces. In [64] Lutwak introduced dual mixed volumes, in analogy to the classical mixed volumes of Minkowski (see also [65]). Dual (star-shaped set) analogues also exist for the intrinsic volumes, the mean projection formula (replaced in the dual theory by a mean intersection formula; see [65]), kinematic formulas [103], and Hadwiger's characterization theorem [51, 52]. A comprehensive introduction to the theory of star-shaped sets and geometric tomography has been presented in Gardner's book [26]. Finally, one can ask whether an analogous lattice theory is possible for the collection of finite unions of compact convex subsets of the Grassmannians Gr(n, k), for which the notion of 'convex set' is suitably defined. Indeed, one may go even further, to the study of polyconvex subsets of compact Lie groups, about which almost nothing is known at present. A characterization theorem in the spirit of Hadwiger's characterization theorem for invariant valuations on compact Lie groups would have profound consequences throughout mathematics.
10 Kinematic formulas for polyconvex sets
In this chapter we use Hadwiger's characterization Theorem 9.1.1 to generalize the kinematic formulas of Chapter 3 to the lattice of polyconvex sets. These formulas will have application to additional questions regarding random motions of polyconvex sets. In Section 10.2 we use the principal kinematic formula of Section 10.1 to give a condition under which one compact convex set in R2 must be contained in a translate of another.
10.1 The principal kinematic formula Before pursuing additional applications of Theorem 9.1.1, we digress briefly to discuss the Haar measure on the group of Euclidean motions (including reflections across hyperplanes). It is easy to derive an explicit formula for such a measure by the arguments we have already employed for Grassmannians. The Haar measure on the orthogonal group O(n) is closely related to the invariant measure on the set of frames (and, therefore, the set of flags of subspaces) in R n, since orthogonal transformations are parametrized by Cartesian coordinate systems. In other words, after fixing a Cartesian coordinate system Xl, X2, ... ,Xn in R n, an arbitrary orthogonal transformation is uniquely determined by the choice of another Cartesian coordinate system UI, U2, ... ,Un with the same origin. (See also Section 6.7.) Therefore, an invariant measure on O(n) is obtained from the invariant measure on the set of frames in Mod( n) after multiplying it by 2n. However, for the purposes of this section we shall normalize the invariant measure on O(n) so that the total measure of O(n) equals one; i.e. we use the Haar probability measure on O(n). Recall that En denotes the set of all Euclidean motions of R n . Since every Euclidean motion is
147
10.1 The principal kinematic formula
a (possibly improper) rotation followed by a translation, an invariant measure is induced on En by taking the product of the Haar probability measure on O(n) with the n-dimensional Lebesgue measure. We denote this Haar measure on En by dg, where 9 E En. It turns out that Sylvester's Theorem 7.2.3 can be generalized to the case in which, in place of a linear variety, we substitute any compact convex set. Let A and K be compact convex sets. For 9 E En, denote by gK the set {g(b) : b E K}, and consider the integral /-to (A,
K) = ( /-to (A n gK) dg.
(10.1)
lEn
This integral has an evident geometric interpretation as the measure of the set of all 9 E En such that AngK =I=- 0. Alternatively one may think of (10.1) as the 'measure' of all convex sets gK in Rn congruent to K that meet A. If A and C are both compact convex sets of dimension n and if C ;2 A, then the quotient
f /-to (A n gK) dg f /-to(C n gK) dg
(10.2)
gives the probability that a 'rigid' convex set K, dropped at random on Rn so as to meet C, shall also meet A. Note that this probability is independent of the choice of normalizing constant for the Haar measure dg. The computation of the probability (10.2) can be carried out using Hadwiger's Theorem 9.1.1. Indeed, the set function /-to (A, K) is a continuous valuation in each of its variables K and A when the other is held fixed. Moreover,
/-to(g'A,K)
J J
/-to(g'AngK) dg /-to (A
=
J
/-to(Ang'-lgK) dg
n gK) dg
by the invariance of the Euler characteristic and of Haar measure. One verifies similarly that /-to (A, g' K) = /-to (A, K) for all g' E En. Finally, note that
J
/-to (A
n gK) dg
J J
/-to (g-l An
K) dg
=
J
/-to(gA n K) dg- 1
/-to(gA n K) dg,
since the Haar measure on En is invariant under the inversion map 9
1--+
148
10 Kinematic formulas for polyconvex sets
g-l. Thus, f-to(A, K) = f-to(K, A). From two applications of Hadwiger's
Theorem 9.1.1 we obtain the expansion f-to(A, K) =
n
n
i=O
j=O
LL
Cijf-ti(A)f-tj(K),
where the coefficients Cij = Cji depend only on n, and where f-ti and f-tj denote the intrinsic volumes. It turns out that most of the constants Cij are equal to zero. In fact, they can even be computed explicitly.
Theorem 10.1.1 (The principal kinematic formula) For all A, K E Polycon(n),
Proof From the argument above we know that f-to(A, K)
=
1
f-to(A
n gK)
dg
En
=
t
Cijf-ti(A)f-tj(K),
i,j=O
where the Cij are constants depending only on i, j, and n, and where Cij
= Cji'
Let En denote the unit ball in Rn, centered at the origin. For a, b 2: 0, denote by aEn and bEn the balls of radii a and b in R n, centered at the origin. Then
1
f-tO(aEn
En
r r r r
JO(n) JRn JO(n) JRn
n gbEn) dg
f-to(aE n
n (¢bEn + V)) dv d¢
f-to(aE n
n (bEn + v)) dv d¢,
since ¢En = En for any orthogonal transformation ¢. Since the total measure of O(n) is equal to one, we continue:
10.1 The principal kinematic formula
149
iRn I(a+b)Bn dv
(a + b)nwn Wn
t (~)aibn-i
.
• =0
In other words, J.lO(aBn, bBn) is a homogeneous polynomial in the nonnegative variables a and b. Meanwhile, n
L
CijJ.li(aBn)J.lj(bBn)
i,j=O n
L
Cijaibl J.li(Bn)J.lj(Bn).
i,j=O Therefore, Cij
=
0 if i
+j
=I=-
n, and
by Theorem 9.2.4.
o
In view of Theorem 10.1.1, the expression (10.2) becomes
J J.lo(A n gK) dg J J.lo(C n gK) dg giving once again the probability that a 'rigid' convex set K, dropped at random on Rn so as to meet C, shall also meet A. The principal kinematic formula can also be expressed in more probabilistic terms, using random variables. For a compact convex set K in Rn let Xi(K) again denote the i-volume of a projection of K onto a randomly chosen i-dimensional subspace V E Gr(n, i). Combining Corollary 9.4.2 with the principal kinematic formula (Theorem 10.1.1) we obtain
a formula mysteriously reminiscent of the classical binomial theorem.
150
10 Kinematic formulas for polyconvex sets
10.2 Hadwiger's containment theorem As an application of the principal kinematic formula 10.1.1, we derive Hadwiger's condition for the containment of one planar convex region by another. Theorem 10.1.1 takes the following form for polyconvex subsets of R2:
for all K, L E K2. The formula (6.4) for flag coefficients then yields
JrE2 J-Lo(K n gL) dg =
J-LO(K)J-L2(L)
+ ~J-Ll(K)J-Ll(L) + J-L2 (K)J-LO (L). K
(10.3) Now suppose that K and L are convex polygons in R2 with non-empty interiors. Denote by 8K and 8L the boundaries of K and L respectively, and let 9 be a Euclidean motion of the plane. If K n gL = 0 then 8Kng8L = 0 as well. However, if KngL -I- 0 there are two possibilities. The first possibility is that 8K n g8L -I- 0 as well, in which case 8K n g8L consists of an even number of distinct points in R2. (There is a possibility that 8K and g8L will intersect in some other way, but the set of such motions 9 is easily seen to be of measure zero in E2.) The second possibility is that 8K n g8L is still empty, which implies that either K S;;; int gL or gL S;;; int K. In other words, the motion 9 moves one of the regions K or L into the interior of the other. Suppose that neither K nor L can be moved into the interior of the other by any rigid motion of the plane. This eliminates the second possibility. In terms of the Euler characteristic, this non-containment assumption implies that (i) If J-Lo(K n gL) = 0 then J-Lo(8K n 9 8L) = 0 as well, and (ii) If J-Lo(K n gL) = 1 then J-Lo(8K n g8L) = 2k for some positive integer k, for all 9 E E2 (except a set of measure zero). In other words, if neither K nor L can be moved into the interior of the other by any rigid motion of the plane, then we have
J-Lo(8K n 9 8L) ?: 2J-Lo(K n gL).
10.2 Hadwiger's containment theorem
151
It follows that
( /-to(8K n g8L) dg
iE2
~
(
iE2
2/-to(K n gL) dg.
The principal kinematic formula (10.3) for R2 then implies that /-to (8K)/-t2(8L)
+
2
-/-tl (8K)/-tl (8L) 1f
+ /-t2(8K)/-to(8L) 4
> 2/-to(K)/-t2(L) + -/-tl (K)/-tl (L) + 2/-t2(K)/-to(L). 1f
Because K and L are convex, we have /-to(K) = /-to(L) = 1. Since K and L are polygons, 8K and 8L consist of finite unions of line segments. Therefore, /-t2(8K) = /-t2(8L) = 0, and we have 2
-/-tl(8K)/-tl(8L) 1f
~
2/-t2(L)
4
+ -/-tl (K)/-tl (L) + 2/-t2(K). 1f
(10.4)
If we denote by A(K) and P(K) the area and perimeter of a compact convex subset K of R2 (with a non-empty interior), the inequality (10.4) becomes
~P(K)P(L) ~ 2A(L) + ~ P~) P~L) + 2A(K), so that
21f(A(K) + A(L)) - P(K)P(L) :S
o.
(10.5)
Given K, L E K2 with non-empty interiors, define fl(K, L) by
fl(K, L)
=
21f(A(K)
+ A(L)) -
P(K)P(L).
(10.6)
We have shown that, if K and L are convex polygons with non-empty interiors, and if neither K nor L can be moved into the interior of the other by any rigid motion of the plane, then fl(K,L) :S O. Since the set of convex polygons is dense in the set of all compact convex subsets of R2, a simple continuity argument implies that fl(K, L) :S 0 if K, L E K2 (not necessarily polygons) such that neither contains a translate of the other in its interior. Thus, we have the following theorem.
Theorem 10.2.1 (Hadwiger's containment theorem) Let K, L E K2 with non-empty interiors. If fl(K, L) > 0 then there exists a Euclidean motion g E E2 such that either K ~ int gL or L ~ int gK.
o Which way will the containment go? If fl(K, L) > 0 then Theorem 10.2.1 implies that A(K) =I=- A(L). Therefore, the convex body of larger area will contain a rigid motion of the other.
152
10 Kinematic formulas for polyconvex sets
10.3 Higher kinematic formulas In addition to the principal kinematic formula 10.1.1 one can derive analogous kinematic formulas for the remaining intrinsic volumes J-L1, ... , J-Ln. Define a function (k on pairs of polyconvex sets by the formula (k(A, K) =
r
J-Lk(A n gK) dg
JEn
For convex A and K, one may interpret the expression
fEn J-Lk(A n gK) dg fEn J-Lo(A n gK) dg as the mean value, or the expected value, of the kth intrinsic volume of An gK, taken over all gK in Rn congruent to K that meet A. If A and C are compact convex sets of dimension n and if C ;2 A, then the quotient
f J-Lk(A n gK)dg f J-Lo(C n gK)dg
(10.7)
gives the expected value of J-Lk(A n gK) given that K meets C. For A, K convex, Theorem 9.3.1 asserts that J-Lk(A n gK) = '\~_k(Graff(A n gK; n - k))
Therefore, we have (k(A,K)
=
hn 1 '\~_k(Graff(A
J-Lk(AngK) dg
En
n gK; n - k)) dg
r r J-Lo«AngK)nV)d'\~_k(V)dg r r J-Lo«A n V) n gK) dgd'\~_k(V) JGraff(n,n-k) J En J En JGraff(n,n-k)
r
JGraff(n,n-k)
~ [7] i=O
n-k
~
t [~] i=O
-1 J-Li(A
n V)J-Ln-i(K)
d'\~_k(V)
2
-1 J-Ln-i(K)
r
JGraff(n,n-k)
J-Li(A n V)
[i +k k] [n]-l i J-Ln-i (K)J-LHk (A),
d'\~-k(V)
153
10.4 Notes
where the last equality follows from Theorem 9.3.2. Hence, we have the following generalization of Theorem 10.1.1 for the intrinsic volumes. Theorem 10.3.1 (The general kinematic formula) For 0::; k ::; n,
]-1
r
[. + k] [ lEn Jtk(A n gK) dg = n-k ~ 't k ~ for all A, K
E
Jtk+i(A)Jtn-i(K).
o
Polycon( n).
Theorem 10.3.1 and Hadwiger's characterization Theorem 9.1.1 together provide a kinematic formula for the integral
r
lEn
Jt(A n gK) dg,
for any continuous invariant valuation Jt on Polycon(n).
10.4 Notes The principal kinematic formula 10.1.1 has variously been attributed to Blaschke, Chern, Hadwiger, and Santal6 (see [82, p. 262] and [85, p. 253]). In [46], Howard gave a generalization of the kinematic formula to Riemannian homogeneous spaces. By a different approach Fu proved a generalized principal kinematic formula for invariant differential forms on homogeneous spaces [25]. Zhang [103] recently developed dual kinematic formulas for the lattice of star-shaped sets in Rn. For numerous variations and applications of kinematic formulas, see also [82, 85, 87, 98, 106]. Hadwiger's containment Theorem 10.2.1 first appeared in [35, 36]; see also [82, pp. 121-123]. A spherical version of Hadwiger's containment theorem is treated in Section 11.4; see also [82, p. 324]. The question of how to generalize Theorem 10.2.1 to higher dimensions remains open in many cases; some new ideas have recently been put forward by Zhou [77, 104, 105, 106, 107] for convex bodies in R3 and in R2n, and by Zhang [102] for Rn in general. For a generalization of Theorem 10.2.1 to the projective and hyperbolic planes, see [31]. Once again we emphasize the analogical nature of these results. In particular, the kinematic formula 3.2.5 of Chapter 3 is a discrete analogue of the general kinematic formula 10.3.1 for Polycon(n). This analogy suggests the possibility of additional kinematic formulas for the lattice Mod(n), with applications to appropriate probabilistic questions in this context.
11 Polyconvex sets in the sphere
The successful characterization of many classes of valuations on polytopes and compact convex sets in Euclidean space motivates similar questions about valuations on polytopes and convex sets in non-Euclidean spaces. Unfortunately little is yet known about valuations in a non-Euclidean context. In this chapter we consider valuations on the lattice of polyconvex sets in the unit sphere sn, with emphasis on the elementary example of continuous O(3)-invariant valuations on the unit sphere S2 in R3.
11.1 Convexity in the sphere Recall that sn denotes the set of unit vectors in Rn+l. The intersection En sn of a two-dimensional subspace E of Rn+l with the sphere is called a great circle. More generally, the set E n sn is called a great k-subsphere of sn if E is a subspace of Rn+l of dimension k + 1. Two points x, y, E sn-l are said to be antipodal if y = -x. The sphere sn inherits a Riemannian structure from the ambient space R n+ 1 , in which the shortest path (or geodesic) between two nonantipodal points x -I=- y E sn is given by the shorter arc of the unique great circle in sn passing through x and y. A great (n -l)-subsphere (J separates sn into two hemispheres, each the antipode of the other. The intersection of n + 1 distinct hemispheres having linearly independent normals is called a spherical n-simplex. The intersection of at most n hemispheres is called a lune. For u E sn denote by u-L the n-dimensional subspace of Rn+l orthogonal to u. If ~ is a simplex inside the great (n - 1)-subsphere u-L n sn, the lune through ~, denoted L( ~), consists of the union of all half circles with endpoints at u and -u and bisected by a point of ~.
11.1 Convexity in the sphere
155
A set P ~ sn is a convex spherical polytope if P can be expressed as a finite intersection of hemispheres. Denote by p(sn) the set of all convex spherical polytopes in sn. A spherical polytope is a finite union of convex spherical polytopes. More generally, a set K ~ sn will be called convex if K is contained in some hemisphere of sn and if any two points of K can be connected by a geodesic (i.e. an arc of a great circle) inside K. Alternatively, a set K lying inside a hemisphere of sn is convex if the cone 0 * K in R n+1 defined by 0*
K
=
{Au : u
E
K and
°: ; ). : ;
1}
is convex in R n+ 1 . Denote by K(sn) the set of all compact convex sets in sn. The set K(sn) is endowed with the topology induced by the Hausdorff metric on compact sets in Rn+l (see Section 4.1). In analogy to the Euclidean case, we call a set K ~ sn poly convex if K can be expressed as a finite union of compact convex sets in sn. Evidently the collection of polyconvex sets in sn forms a distributive lattice under union and intersection of sets. A function cp : p(sn) ---t R is called a valuation on p(sn) if cp(0) = 0, where 0 is the empty set, and if cp(P U Q) = cp(P)
+ cp(Q) -
cp(P n Q),
(11.1)
for all P,Q E p(sn) such that PUQ E p(sn) as well. A valuation cp on p(sn) is simple if it vanishes on spherical polytopes of dimension less than n. A valuation cp on p(sn) will be called invariant if cp(gP) = cp(P) for all orthogonal transformations (rotations and reflections) g of sn. Similarly, a function cp : K(sn) ---t R is said to be a valuation if cp(0) = and if (11.1) is satisfied for all compact convex sets K, L ~ sn such that K U L E K(sn) as well. The following is an adaptation of Groemer's extension Theorem 5.1.1 for valuations on the sphere sn.
°
Theorem 11.1.1 (Groemer's extension theorem for sn)
(i) A valuation cp defined on convex polytopes in sn admits a unique extension to a valuation on the lattice of all polytopes in sn. (ii) A continuous valuation cp on compact convex sets in sn admits a unique extension to a valuation on the lattice of polyconvex sets in sn. In each case the extension of cp to finite unions is given by iteration of the inclusion-exclusion identity (11.1). 0
156
11 Polyconvex sets in the sphere
The proof of part (i) of Theorem 11.1.1 follows the same lines as the proof of Theorem 4.1.3. Similarly, the proof of Theorem 5.1.1 (for the Euclidean case) goes through for the sphere sn without essential change, since the original proof is based not on the geometry of Rn, but rather on the algebra of indicator functions, and on the fact that a polytope is the intersection of half spaces in R n, a property which carries over analogously to spherical polytopes and hemispheres. (See Section 5.1.) In the arguments that follow, the unique extension of a valuation 'P given by Theorem 11.1.1 will allow us to consider the value of'P on all finite unions of convex spherical polytopes (or compact spherical convex sets in sn), whether or not such unions are actually convex.
11.2 A characterization for spherical area
We now turn our attention to the two-dimensional sphere S2. Important examples of continuous invariant valuations on P(S2) (and K(S2)) include spherical area, denoted 'P2, spherical length 'PI, and the Euler characteristic 'Po. The spherical length 'PI (K) of a spherical convex region K with a non-empty interior in S2 is given by one half of the perimeter of K; that is, one half of the length of the curve in S2 that forms the boundary of K. It is easy to verify that 'PI is an extension of geodesic length in S2 to a continuous invariant valuation on K(S2). To see this, note that
for all K E K(S2). The Euler characteristic 'Po (K) of a spherical convex region K is defined to be 1 if K i=- 0, while 'Po(0) = o. Theorem 11.1.1 insures that 'Po has a unique continuous and invariant extension to all finite unions of spherical convex sets in K. As in the Euclidean case, this unique extension of 'Po coincides with the Euler characteristic of algebraic topology. It can be shown that in fact every continuous invariant valuation on p(S2) (or K(S2)) is a linear combination of the valuations 'PO,'PI,'P2. To this end, we prove a characterization theorem for spherical area 'P2. Theorem 11.2.1 (The spherical area theorem) Suppose that 'P is a continuous invariant simple valuation on p(S2). Then there exists c E R such that 'P(P) = C'P2(P), for all P E P(S2). Remark: Theorem 11.2.1 actually holds under the weaker assumption
11.2 A characterization for spherical area
157
that cp is invariant under the group SO(3) of rotations of S2. This follows from an argument similar to that in the proof of Proposition 8.3.1. Before proving Theorem 11.2.1, we consider two preliminary cases. Proposition 11.2.2 Suppose that cp is a continuous invariant simple valuation on closed arcs I of the circle SI. Then there exists c E R such that cp(I) = CCPl (I), for all closed arcs I ~ SI.
Proof Let c = cp(SI) / (27r), and define 1/(1) = cp(I) - CCPl (I), for all closed arcs I ~ SI. Note that 1/ is also continuous, invariant, and simple. In addition, I/(SI) = o. It now suffices to show that 1/(1) = 0 for all I. Suppose that In is a closed arc of length 27r / n, where n is a positive integer. Since the circle SI can be tiled with n rotations of In, the invariance and simplicity of 1/ imply that nl/(In) = I/(SI) = 0, so that I/(In) = O. Since any closed arc I of length 27rm/n can be tiled with rotations of In, for any positive integers m < n, it follows that 1/ vanishes on arcs of length rational in proportion to the circle. It then follows from the continuity of 1/ that 1/(1) = 0 for all closed arcs I. 0 Proposition 11.2.3 Suppose that cp is a continuous invariant simple valuation on p(S2) such that cp(S2) = O. Then cp(~) = 0, for all spherical simplices ~ C S2.
Proof Let 0" denote a great circle in S2. For any closed arc I contained in a half-circle in 0", denote by £(1) the lune through I, and define 1/(1) = cp(£(I)). Evidently the valuation 1/ satisfies the conditions of Proposition 11.2.2. Therefore, there exists c E R such that 1/(1) = ccpl(I) for all I ~ 0". However, 1/(0") = cp(S2) = 0, so that c = O. It follows that cp vanishes on alllunes through closed arcs of 0". Since cp is invariant, it follows in turn that cp vanishes on alllunes in S2. Now suppose that ~ is a spherical 2-simplex in S2; that is, a spherical triangle; given by the intersection of hemispheres ~ = HI nH2 nH3 . See Figure 11.1. For A ~ S2 denote by AC the closure of the complement S2 - A. Note that
where - ~ denotes the orthogonal reflection of R3. Since cp is simple and invariant, we obtain
cp(Hl U H2 U H 3 ) = cp(S2) - CP((HI UH2 U H 3
~
through the origin in
n = 0-
cp( -~) = -cp(~). (11.2)
11 Polyconvex sets in the sphere
158
Fig. 11.1. An intersection of hemispheres in 8 2 •
Meanwhile, iteration of the inclusion-exclusion identity (11.1) gives 3
cp(Hl UH2UH3) =
L cp(Hi) - L cp(HinHj)+cp(HlnH2nH3). i=l
(11.3)
i<j
Note that the intersections Hi n Hj are lunes in 8 2 for i =1= j. Since cp vanishes on hemispheres and lunes, the identity (11.3) implies that (11.4) It then follows from (11.2) and (11.4) that cp(~) = O.
cp(~) =
-cp(~);
that is, 0
Remark: A spherical triangle ~ in 8 2 has angle measures ct, /3, 'Y E [0,11'] if the inner Euclidean (dihedral) angles between planar sides of the cone o * ~ in R3 are given by ct, /3, 'Y. It is well known that the spherical area of ~ is given by the spherical excess:
(11.5) One can prove (11.5) by the same inclusion-exclusion technique that was used in the proof of Proposition 11.2.3, using the fact that the lune through a great circular arc of length 8 has spherical area 28. We are now ready to prove Theorem 11.2.1. Proof of Theorem 11.2.1 Suppose that cp is a continuous invariant simple valuation on P(8 2). Let c = cp(8 2)j(411') and define v(P) = cp(p) Ccp2(P) for all P E P(8 2). Since the valuation v satisfies the conditions of Proposition 11.2.3, it follows that v(~) = 0 for all spherical simplices ~ <;;; 8 2 . For P E P(8 2 ) express P as a union of spherical simplices
P= ~l U··· U~m'
11.3 Invariant valuations on spherical polytopes where
dim(~i
n ~j) < 2 for all i
=1= j.
159
Since v is simple, it follows that
m
v(P)
= L V(~i) = 0, i=1
for all P E p(S2).
o
Since any continuous invariant simple valuation
for all K
E
K(S2).
0
11.3 Invariant valuations on spherical polytopes
Theorem 11.2.1 leads to the following characterization theorem for continuous invariant valuations. Theorem 11.3.1 Suppose that
p(S2). Then there exist Co, Cl, C2 E R such that, for all P E p(S2),
Theorem 11.3.1 can be thought of as a spherical analogue of Hadwiger's characterization Theorem 9.1.1. Since the set p(S2) is dense in K(S2), Theorem 11.3.1 also holds if p(S2) is replaced with the larger collection K(S2). Proof Suppose that
160
11 Polyconvex sets in the sphere
vanishes on all P E P(S2) of dimension less than 2; that is, v is a continuous invariant simple valuation on p(S2). Theorem 11.2.1 then implies the existence of C2 E R such that v(P) = C2!.p2(P) for all P E
p(S2).
0
Theorem 11.3.1 implies that any continuous invariant valuation on
p(S2) is determined completely by its values on a singleton {x}, a great circle 0", and the entire sphere S2. Indeed, the coefficients Ci in Theorem 11.3.1 can be computed using the following table of values:
{x}
0"
S2
1 0 0
0 21f 0
2 0 41f
!.po !.pI
!.p2
(11.6)
Note well that !.pI (S2) = O. This follows from the fact that !.pI (H) = 1f, for any closed hemisphere H, while !.p1(0") = 21f if 0" is a great circle. Denote by -H the antipode of H. We then obtain
since H n -His a great circle. One of the advantages of Hadwiger's characterization Theorem 9.1.1 for invariant valuations on Euclidean convex sets is the ease with which that theorem allows one to prove a variety of classical results in integral geometry. We now demonstrate similar advantages of the spherical area Theorem 11.2.1 and the resultant characterization Theorem 11.3.1. We begin with a spherical analogue of the mean projection formula (9.7) of Section 9.4. For K E K(S2) and u E S2, define the projection Ku of K onto the great circle u.l n S2 by
Ku
= {x
E u.l
n S2 : lux] n K -:f. 0},
where lux] denotes the (unique) half great circle through x with endpoints at u and -u. The mean spherical width sw(K) of a compact convex set Kin S2 is defined by (11. 7) The functional sw(K) is continuous on K(S2). To see this, note that, for a fixed u E S2 and K E K(S2), the interval Ku varies continuously in a neighborhood of (K, u) provided that either u E int(K) or u 1:. K. Thus the functional !.pI (Ku) is continuous at (K, u) unless u E oK. Since oK
11.3 Invariant valuations on spherical polytopes
161
has measure zero in S2, the functional ifJl (Ku) is integrable with respect to spherical Lebesgue measure, and the integral (11. 7) varies continuously with respect to K. Although we do not apply the formula (11.7) to non-convex sets, it is possible to extend the functional sw to a continuous valuation on all polyconvex sets in S2 as follows. Suppose that K c S2 is compact and convex. If u E S2 and x E u-L n ~2, then ifJo(Kn lux]) = 1 if and only if x E K u , otherwise ifJo(Kn lux]) = 0. (An exception may occur if K contains a great half-circle, in which case this exception will occur with measure zero.) Therefore, the equation (11.7) can be rewritten sw(K) =
r
iS2
ifJl(Ku ) du =
r
1,
iS2 u-Ln S2
ifJo(K n lux]) dx duo
(11.8)
Since ifJo is an invariant valuation, it follows from (11.8) that the mean spherical width sw(K) is a continuous invariant valuation of K. Moreover, the integral (11.8) is defined for all polyconvex sets in S2, giving the desired extension of sw. Proposition 11.3.2 For K E K(S2), sw(K)
= 41rifJl(K) + 21rifJ2(K).
Proof Since sw is a continuous invariant valuation on K(S2), Theorem 11.3.1 implies that
for some constants Ci E R. Since the formula (11.8) is valid (as a valuation) for all finite unions of compact convex sets, we can evaluate (11.8) to compute sw on great circles and on the full sphere S2. To compute the coefficients Ci, note that sw vanishes on singleton sets, so that Co = 0. If 0" is a great circle, then 0" n lux] consists of a single point almost always; that is, provided that u rf- 0". Thus, ifJo(O" n lux]) = 1 almost always, and we obtain
Cl ifJl (0")
= 21rCl =
r 1,u-LnS2 iS2
dx du
= 81r2,
or Cl = 41r. Finally, if K = S2 then S2 n lux] = lux]' a half-circle, so that ifJO(S2 n lux]) = 1. Since ifJl(S2) = 0, we obtain
C2ifJ2(S2) or C2
= 21r.
= 41rC2 =
r 1,
iS2 u-Ln S2
dx du
= 81r2, D
11 Polyconvex sets in the sphere
162
In a similar manner Theorem 11.3.1 can be used to obtain an easy proof of the spherical Crofton formula, which expresses the spherical perimeter (or length) of a spherical polyconvex set K as the measure of the great circles meeting K; that is,
r 4?o(K nul..) du
JS2
=
44?1(K),
(11.9)
for all polyconvex sets K in S2.
11.4 Spherical kinematic formulas In analogy to the methods of Sections 10.1 and 10.3, the characterization Theorem 11.3.1 yields an easy proof of a spherical kinematic formula for polyconvex sets in S2, which generalizes the spherical Crofton formula (11.9).
Theorem 11.4.1 (The principal kinematic formula for S2) all poly convex sets K, L ~ S2,
1
4?o(K n gL) dg
=
1
41f 4?0 (K)4?2 (L)
0(3)
For
1
+ 21f24?1(K)4?1(L)
1
+ 41f4?2(K)4?0(L) -
1
81f 2 4?2(K)4?2(L).
Here the integral is taken with respect to the invariant Haar probability measure on the orthogonal group 0(3). Proof To begin, define
4?o(K, L)
=
1
4?o(K n gL) dg.
0(3)
For fixed K, the set function 4?o(K, L) is a valuation in the variable L; in fact, it is an invariant valuation, since
4?o(K,goL)
1 1
4?o(K n ggoL) dg
0(3)
4?o(K n gL) dg,
0(3)
for each go E 0(3). By Theorem 11.3.1, the functional 4?o(K, L) can be expressed as a linear combination of the valuations 4?i' with coefficients
11.4 Spherical kinematic formulas
163
ci(K) depending on K: 2
'Po(K, L)
=
I>i(K)'Pi(L). i=O
Meanwhile, for fixed L, the set function 'Po(K, L) is a valuation in the variable K. From this it follows that each of the coefficients ci(K) is a valuation in the variable K. One way to check this fact is to insert special values for L and use the table (11.6). Since 'Po is invariant, we ~liave
r r r
'Po(K, L)
'Po(K n gL) dg
JO(3)
'PO(g-1 K n L) dg
JO(3)
'Po(gK n L) dg
= 'Po(L, K).
JO(3)
Therefore, the coefficients ci(K) are invariant valuations in the parameter K, so that 2
'Po(K, L)
=
L
Cij'Pi(K)'Pj(L),
(11.10)
i,j=O again by Theorem 11.3.1. Since 'Po(K,L) = 'Po(L,K), it is evident that Cij = Cji' One can verify this, as well as compute the values of the coefficients Cij, by evaluating 'Po(K, L) for the cases in which each of K and L is either a point (singleton) {x}, a great circle 0", or the whole sphere 8 2 , using the values of 'Pi given in the table (11.6). For example, if K = L = {x}, then (11.10) takes the form: 0= Coo. We then let K
o
= 0"
and L
= {x},
COl 'Po (0" )'PI ( {x})
so that (11.10) becomes
+ ClO'PI (0" )'Po ({x}) + Cll 'PI (0" )'PI ({ X } )
0+ 27rClO + 0 so that ClO = COl = O. In a similar manner one evaluates both sides of (11.10) for each case of K, L E {{x}, 0", 8 2 } (using the table (11.6)) to compute the values of Cij, thereby completing the proof of Theorem 11.4.1. 0
164
11 Polyconvex sets in the sphere
Using similar methods one can easily show that, for polyconvex sets K,L<:::; S2,
and
r
JO(3)
ifJ2(K n gL) dg = 41 ifJ2(K)ifJ2(L), n
thereby classifying all kinematic formulas for continuous invariant valuations on S2 (by Theorem 11.3.1). Theorem 11.4.1 and its higher dimensional generalizations have numerous applications to questions in spherical geometric probability. For example, Theorem 11.4.1 (as stated for S2) leads in turn to a spherical analogue of Hadwiger's containment Theorem 10.2.1. Theorem 11.4.2 Let K, L E K(S2) with non-empty interiors, and suppose that (11.11) Then there exists 9 E 0(3) such that either K <:::; int gL or L <:::; int gK. D
To determine the direction of containment in Theorem 11.4.2, one compares the values of ifJ2(K) and ifJ2(L). Evidently the set of larger spherical area will contain a rigid motion of the other. If we define A(K) = ifJ2(K) and P(K) = 2ifJl(K) (spherical area and perimeter respectively), then the inequality condition (11.11) becomes: 2n(A(K)
+ A(L)) -
A(K)A(L) - P(K)P(L)
> O.
Compare this with (10.5) in Section 10.2. In analogy to the proof of Theorem 10.2.1, the proof of Theorem 11.4.2 makes use ofthe spherical principal kinematic formula (Theorem 11.4.1), along with the fact that two polygonal convex curves in the sphere will almost always intersect at an even number of points. For details, see Section 10.2.
11.5 Remarks on higher dimensional spheres A natural question at this point (or even earlier) is that of how spherical volume on polytopes or convex sets in higher dimensional spheres is
11.5 Remarks on higher dimensional spheres
165
to be characterized. In order to characterize spherical area (on S2) we proved the following sequence of assertions for a given continuous invariant simple valuation cp:
cp is simple
t-t
cp
=
area on lunes (by induction on dimension) t-t cp = spherical area,
where the second implication follows from an inclusion--exclusion argument (see (11.3)). Unfortunately, this inclusion-exclusion argument fails in S3 and for s2n+1 in general. Without a characterization theorem for spherical volume in S3, the induction step for the first implication is not possible, so that the proof of Theorem 11.2.1 also fails to generalize to S4 and so on. However, we do have the following partial result. Denote by S the (invariant) spherical volume on s2n. Recall that a lune in s2n is a subset of s2n consisting of the intersection of at most 2n hemispheres. Theorem 11.5.1 Suppose that cp is a continuous invariant simple valuation on p(S2n). Ifcp(L) = S(L) for alllunes L ~ S2n, then cp(P) =
S(P) for all P E p(S2n). Proof Without loss of generality we may assume that cp(S2n) = 0, so that cp(L) = 0 for alllunes L ~ s2n. (Just substitute cp - S for cp.) It remains to show that cp(P) = 0 for all P E p(S2n). Suppose that b. is a spherical simplex in s2n given by the intersection of hemispheres b. = HI n ... n H 2n + l . Again denote by AC the closure of the complement S2n - A, for any A ~ s2n. Note that
Since cp is simple and invariant, we obtain
= 0 - cp( -b.) = -cp(b.). (11.12) Meanwhile, the inclusion-exclusion principle gives
cp (
2n+1
U Hi )
2n+1
L
CP(Hi) -
L
CP(Hil n H i2 ) + ...
i=1
(11.13)
166
11 Polyconvex sets in the sphere
Since cp vanishes on hemispheres and lunes, all of the terms on the righthand side of (11.13) vanish except for the last term: (-1)2ncp(Hl n· .. n H 2nH ) = cp(~). It then follows from (11.12) that
cp(~) = cp
ild Hi
2n+l (
)
= -cp(~),
so that cp(~) = O. Since every polytope P E p(S2n) can be expressed as a union of spherical simplices intersecting in dimension less that 2n, it follows that cp(P) = 0 for all P E p(S2n). 0 This proof of Theorem 11.5.1 fails for s2nH because the sign of the last term on the right-hand side of (11.13) is negative in this case, so that, on combining it with (11.12), we would obtain the tautological observation that
cp(~) = -cp
ild Hi
2n+l (
)
= cp(~).
For this reason a proof of such a spherical volume characterization for S2nH, if possible, will require some new ingredient besides the decomposition of simplices by inclusion-€xclusion. Indeed the question of whether spherical volume is the only continuous invariant simple valuation on p(sn) (or K(sn)) remains open for dimension n 2: 3.
11.6 Notes
Theorems 11.2.1 and 11.5.1 are both originally due to McMullen. In [70] McMullen developed the polytope algebra, which places many dissection and inclusion-exclusion techniques in an algebraic framework (see also [72]). The proofs of Theorems 11.2.1 and 11.5.1 given in Section 11.2 give a more simplified approach to these specific questions. Groemer made observations regarding Theorem 11.1.1 at the end of his paper on extensions of additive set functionals [32]. Theorem 11.4.2 and other non-Euclidean analogues of Hadwiger's containment theorem can be found in [82, p. 324]. For additional references see Section 10.4. Theorem 11.4.1 generalizes to a principal kinematic formula for spheres of arbitrary dimension; see [82, p. 321]. Similarly, the spherical Crofton formula (11.9) generalizes to higher dimensions [82, p. 316]. However, it is not known whether the characterization Theorem 11.3.1 generalizes to spheres of dimension n 2: 3.
11.6 Notes
167
Theorems 11.2.1 and 11.2.4 vastly simplify valuation characterization theorems for valuations on star-shaped sets in R3. In [51, 52] characterizations are given for continuous homogeneous valuations and continuous rotation invariant valuations on a class of star-shaped sets called Ln-stars; that is, star-shaped sets in R n with n-integrable radial nmctions (see also [26]). For the general case of star-shaped sets in R n these characterizations require that the valuations in question be defined on the entire class of Ln-stars. However, by using Theorem 11.2.4 one can prove similar characterizations for valuations defined merely on finite unions of convex cones in R 3 (of finite height) having a common apex at the origin. This is a much smaller and more manageable class of objects, which is nonetheless dense in the original and larger class of L3- stars. Clearly a higher dimensional version of Theorem 11.2.1 or Theorem 11.2.4 would have similar and important consequences in the theory of valuations on star-shaped sets in Rn.
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Index of symbols
B n ,31 CI:,123 C~, 88 En, 86 Fi(P), 138 H+,34 H-,34 lA, 7, 144 K +EBn, 31 K +L, 43, 99 KIV,55 L(B), 21 L(/l), 154 M(Q, r), 138 O(n),60 P(P(B)), 21 P(B), 13 Pier), 138 Pk(B), 13 Pal , ... ,a r (B), 18 B(K),55 BO(n), 100 BO(n,B),100 BK,109 T2,117 Tn, 43 W(K),134 Zc,4 [0, l]n, 38, 101 [n]!,62 [n], 61, 66 [x, y], 13 /l(K,L),151 ret), 63 0,141 Q,39 R2 1
Rn',30 R Q ,39
Sn-l, 56 Z,112
k,143 143 b(K, L), 31 ix, 120
V,
0, 7
x, 95 Ai, 3 Ak, 86
(n/2), 14 (nlr),20 f.L(J; K), 132 f.LQ, 22, 35, 37, 46 f.Lk, 23, 36, 91 f.Leven, 106 f.Lodd, 106 vI:, 63 vn,73 1/{}:1, ... ,a r l78 v n ;r(A),79
55, 63, 66 I8>Z,116 'it, 120 Wn,
V, 103
x,
21 8K,4
19
61
58
'ei , 156 Tn, 30 36
Iir,
~k'
90
pn, 83
62 (k, 152
Tn,
d(x, A), 31
Index of symbols ek(tl,t2, ... ,tn ),23 hK, 42, 99 o*K,155 rex), 13 u-L,
56
x V y, 6, 60 x 1\ y, 6, 60
K(Sn),155 Kn,42 K'd, 99 £,112
p(Sn), 154 V,129
Ll,~aJ, [~l ' 63
m,
78
14,69
(al ,n,aJ, 18, 19 IAI,14
sw(K), 160
175
Index
Aff(n), 86 affine Grassmannian, 86 affine hull, 138 affine surface area, 141 Ambartzumian, R. V., 5 antichain, 13 Barbier, E., xi, 5, 141 binomial, 14 bivaluation, 142 Boolean algebra, 11, 13 box, 30 Brunn-Minkowski inequality, 108 Buffon needle problem, xi, 1, 129 Caratheodory's theorem, 52 Cauchy's surface area formula, 56 centrally symmetric, 99 chain, 13 Chen, B., 29, 58 chord, 129 combinatorics, xi, 62, 63, 71, 78, 84, 127 compatible discrete flag, 19 flag of subspaces, 78 cone, 155 continuity, 31, 44 convex hull, 51 convex ring, 42 convex set centered, 99 in Rn, 42 in the plane, 3 random, 147 spherical, 155 symmetric, 107 convex-continuity, 44 cosine transform, 100 Crofton's formula, 124 Crofton, M. W., xi, 3, 5, 141
cross-section, 128 De Giorgi, E., 42
Dec(n,r),77 Dec(S,r),18 Dehn invariant, 115 discrete basis theorem, 23, 144 dissection, 115 Erdos, P., 29 Euclidean motion, 43, 146 Euler characteristic, 29, 58 discrete, 22 on parallelotopes, 37 on polyconvex sets, 46 on spherical polytopes, 156 Euler's formula discrete, 23 for intrinsic volumes, 93 for polytopes, 49 expectation, 1, 112, 126, 128, 129, 149, 152 face enumerator, 23, 29, 143 facet, 21 filter, 143 Fisk, S., 84 flag, 13 flag coefficient, 63 formulas for, 67 maximal, 73 Pascal's triangle, 69 Flag(n),61 frame, 62 Fu, J., 153 fundamental domain, 112 gamma function, 63, 72 Gardner, R., 145 generating set, 8
Index geometric probability, xi, 1, 5, 25, 92, 147
join, 6, 60
Gr(n, k), 61
Katona-Kruskal theorem, 29 kinematic formula discrete, 25 general, 153 principal, 148, 153 spherical, 162 Klain, D., 84, 117 Klee's theorem, 50 Kubota's theorem, 126, 141
Graff(2, 1), 3 Graff(n, k), 86 Graff(A; k), 88 Grassmannian,61 affine, 86 great circle, 154 Groemer's extension theorem, 44 for parallelotopes, 32 in the sphere, 155 Groemer's integral theorem, 8 Groemer, H., 12, 58 group action, 60, 83, 86 Haar measure on Grassmannians, 61 on projective space, 57 on the Euclidean motions, 146 on the orthogonal group, 83 probability, 57, 126 Hadwiger's characterization theorem, 118 for parallelotopes, 40 spherical analogue, 159 Hadwiger's containment theorem, 151 spherical analogue, 164 Hadwiger's formula, 91, 123 Hadwiger, H., xii, 58, 117 Harper, L., 29 Hausdorff distance, 31 Helly's theorem, 52 discrete, 28 for subspaces, 81 hemisphere, 154 Hilbert's third problem, 11, 115 Hirsch, W. M., 29 Hochberg, M., 29 homogeneous space, 83 Howard, R., 153 hyperplane, 30 inclusion--exclusion principle, 7 indicator function, 7, 144 integral, 8 intrinsic volume, 58, 88, 94, 130 computation, 135 discrete, 23 normalization, 111 of a convex polytope, 137 of a convex set, 90 of a parallelotope, 37, 136 of a product, 37, 130 of the unit ball, 122 invariance, 22, 31, 43, 95
L.Y.M. inequality continuous, 73, 79 discrete, 14, 19 lattice, 6, 29 distributive, 6, 21 in Rn, 112 of parallelotopes, 30 of polyconvex sets, 42 of subsets, 13 of subspaces, 60 lattice point, 112 Lie group, 83, 145 linear functional, 10, 46 Lubell, D., 29 lune, 154, 165 Lutwak's containment theorem, 54 Lutwak, E., 145 Mobius function, 96 McMullen, P., 11, 58, 166 mean chord length, 129 mean cross-sectional volume, 128 mean projection formula, 94, 125 mean spherical width, 160 meet, 6, 60 Meshalkin's theorem continuous, 81 discrete, 20 Minkowski linear combination, 140 Minkowski sum, 43, 99 mixed volumes, 140, 141 Mod(n), 60 moment, 133 monotone, 32
Mult(n;al, ... ,ar ),77 multiflag coefficient, 78 multinomial coefficient, 18 needle, xi, 1, 129 normalization theorem, 37, 111
or(V),87 order ideal, 76, 95 orthogonal group, 60, 83, 146 special, 100
177
178
Index
Par(n),30 Par(Vl, ... , v n ), 136 parallelotopes, 30 partially ordered set, 6, 21, 60, 86, 95, 143 Pascal inequality, 71 Pascal's triangle for flag coefficients, 69, 70 for binomial coefficients, 70 permutation group, 22, 24 permutation of coordinates, 30, 36, 40 Polycon( n), 42 polyconvex set, 42 polygon, 2, 117, 150 polynomial type, 142 polytope, 47 spherical, 154 quotient space, 128 r-decomposition discrete, 18 in Mod(n), 77 r-family discrete, 15 in Mod(n), 75 random variable, 1, 3, 25, 112, 126, 128, 129, 149 rank, 13 relative interior, 48 relint(P), 48 Rota, G.-C., 29, 84 s-system, 18, 77 Sah, C.-H., 109, 117 Santal6, L., 153 Schutt, C., 141 Schanuel, S., xii, 29, 140 Schneider's characterization theorem, 108, 117 Schneider, R., 11, 58 scissors congruent, 115 segment, 96 of a Boolean algebra, 13 simple function, 7 simple valuation, 101 simplex, 21, 54, 77, 142 dissection of, 109 random, 25 simplicial complex, 21, 77, 142 special orthogonal group, 100 Sperner's theorem continuous, 74 discrete, 14, 29 for r-families, 15, 75 spherical area theorem, 156 spherical convex set, 155
spherical excess, 158 spherical geometry, 154 spherical length, 156 spherical polytope, 154 stabilizer, 84 star-shaped set, 145, 167 Steiner's formula, 122, 140 support function, 42, 99 support plane, 43 surface area, 56 Sylvester's theorem, 4, 93, 147 Sylvester J.J., 3, 5 symmetric convex set, 99 symmetric function, 23, 35 syzygy ,144 tensor product 116 totally positive 144 translation, 30, 43, 101, 116 translation invariance, 36, 44 valuation, 6, 22, 30, 32, 42, 44 even, 106 homogeneous, 40 increasing, 32 monotone, 32 odd, 106 on the sphere, 155 simple, 101 volume characterization theorem, 105 volume of unit ball, 63 volume theorem for parallelotopes, 38, 136 for polyconvex sets, 110 spherical, 164 Wills functional, 134, 141 wire, 2 Zhang, G., 153 Zhou, J., 153 zonoid,99 zonotope, 99
