A probabilistic take on isoperimetric-type inequalities

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1 A probabilistic take on isoperimetric-type inequalities Grigoris Paouris Peter Pivovarov May 2, 211 Abstract We extend a theorem of Groemer s on the expected volume of a random polytope in a convex body. The extension involves various ways of generating random convex sets. We also treat the case of absolutely continuous probability measures rather than convex bodies. As an application, we obtain a new proof of a recent result of Lutwak, Yang and Zhang on the volume of Orlicz-centroid bodies. Keywords: isoperimetric inequalities, rearrangements, random convex sets 1 Introduction The Euclidean ball is the extremal case in a host of isoperimetric problems in convex geometry. If K n denotes the class of convex bodies in R n, then various functionals Φ : K n R + are minimized (or maximized) on the Euclidean ball. A result of this type, and the main motivation for the present article, involves the functional 1 Φ(K) := vol vol n (K) N n (conv {x 1,..., x N }) dx 1... dx N (K K n ). K K Thus Φ(K) gives the expected volume of the convex hull of independent random points sampled in K. In [11], Groemer proved that Φ(K) Φ(B n 2 ), where B n 2 is the Euclidean ball; equality holds if and only if K is an ellipsoid. Similar results hold for various functionals Φ involving the volume of random sets associated with K (e.g., [7], [2], [22], [5], [13], [1], [9, Chapter 9], [6]). We extend Groemer s theorem, and a number of related results, in two directions. Firstly, we work in the class P [n] of all probability measures on R n that are absolutely continuous The first-named author is supported by the US National Science Foundation, grant DMS The second-named author holds a Postdoctoral Fellowship award from the Natural Sciences and Engineering Research Council of Canada. 1

2 with respect to Lebesgue measure. Whereas Steiner symmetrization is typically used in K n, we make use of rearrangement inequalities; especially those related to the well-known theorem of Brascamp, Lieb and Luttinger [3]. The second difference is that we adopt an operator-theoretic viewpoint by considering random matrices applied to various convex sets. This is a natural, well-studied approach in the Local Theory of Banach spaces (see, e.g., [19] and the references therein). In our context, if N n and x 1,..., x N are independent random points with x i distributed according to µ i P [n], we treat the n N random matrix [x 1... x N ] as a linear operator from R N to R n ; applying [x 1... x N ] to a convex body C R N produces a random convex set in R n, i.e., { N } [x 1... x N ]C = c i x i : (c i ) C. We seek the minimum of the expected volume of the latter set, subject to a uniform upper bound on the densities of the µ i s. Even in the class P [n], the Euclidean ball plays a special role. Theorem 1.1. Let N n and µ 1,..., µ N P [n] ; denote the density of µ i by f i. Let C be a convex body in R N and set N F C (f 1,..., f N ) =... vol n ([x 1... x N ]C) f i (x i )dx N... dx 1. (1) R n R n If f i 1 for i = 1,..., N, then F C (f 1,..., f N ) F C (1 Dn,..., 1 Dn ), where D n R n is the Euclidean ball of volume one. If K R n is a convex body with vol n (K) = 1, we can take f i = 1 K. By choosing C R N suitably, we recover various known inequalities. If C = conv {±e 1,..., ±e N }, then [x 1... x N ]C = conv {±x 1,..., ±x N }, which corresponds to the symmetric analogue of Groemer s result mentioned above. For another example, take C = [ 1, 1] N. In this case, { N } [x 1... x N ]C = α i x i : α i 1 for i = 1,..., N, which is just the zonotope (i.e., Minkowski sum of line segments) generated by the line segments [ x i, x i ] = {αx i : α 1}. Thus Theorem 1.1 also recovers a result due to 2

3 Bourgain, Meyer, Milman and Pajor [2]. For the class K n, a general framework for proofs of results of this type is discussed in [6]; the underlying principle goes back to a result of Shephard [24]. In addition to the extension to P [n], a new insight provided by Theorem 1.1 is that rather than applying a particular method to a given functional, it applies to many functionals at once; one need only select C R N. Furthermore, one is not limited to choosing a single C R N. By taking a sequence of convex bodies C N R N for N = n, n + 1,... and applying a simple limiting argument, we get additional applications. We obtain a family of isoperimetric inequalities, not necessarily involving random sets. For instance, we retrieve, and extend to the class P [n], the following theorem of Lutwak, Yang and Zhang [18] (here we deal only with the symmetric case; cf. Remark 5.5). Theorem 1.2. Let ψ : [, ) [, ) be a Young function, i.e., convex, strictly increasing with ψ() =. Let µ P [n]. Define the Orlicz-centroid body Z ψ (µ) of µ corresponding to ψ by its support function h(z ψ (f), y) = inf { λ > : ψ R n If f denotes the density of µ and if f 1, then ( x, y vol n (Z ψ (µ)) vol n (Z ψ (λ Dn )), where λ Dn is the restriction of Lebesgue measure to D n. λ ) } dµ(x) 1. Despite the fact that the latter theorem involves non-random sets, our proof shows that it can be seen as a Law of Large Numbers, which is the probabilistic take referred to in the title. In the present paper, we do not consider equality cases in Theorems 1.1 and 1.2. When µ = 1 K and K R n is a convex body (with the origin in its interior) equality holds in Theorem 1.2 if and only if K is a centered ellipsoid [18]. The paper is organized as follows. In Section 2, we collect definitions and basic facts about rearrangements and give an overview of inequalities related to [3]. In Section 3, we isolate a condition (which we call Groemer s Convexity Condition (GCC)) under which one can conclude a minimization result such as Theorem 1.1. In the presence of (GCC), rearrangement inequalities allow us to pass to densities that are rotationally invariant; moving then to the Euclidean ball is done in 3.1. In Section 4, we verify that the particular integrand in F C (f 1,..., f N ) satisfies (GCC). Section 5 concludes with applications; in particular, the proof of Theorem Preliminaries on rearrangements of functions Let A be a Borel subset of R n with finite Lebesgue measure. The symmetric rearrangement A of A is the open ball with centre at the origin, whose volume is equal to the 3

4 measure of A. Since we choose A to be open, χ A is lower semicontinuous. The symmetric decreasing rearrangement of χ A is defined by χ A = χ A. We consider Borel measurable functions f : R n R + which satisfy the following condition: for every t >, the set {x R n : f(x) > t} has finite Lebesgue measure. In this case, we say that f vanishes at infinity. For such f, the symmetric decreasing rearrangement f is defined by f (x) = χ {f>t}(x)dt = χ {f>t} (x)dt. Let f : R n R + be a measurable function vanishing at infinity. For θ S n 1, we fix a coordinate system such that e 1 := θ. The Steiner symmetrization f ( θ) of f with respect to θ is defined as follows: for x 2,..., x n R, we set h(t) = f(t, x 2,..., x n ) and define f (t, x 2,..., x n θ) := h (t). (2) We refer the reader to the book [16] or the introductory notes [4] for further background material on rearrangements of functions. 2.1 Brascamp, Lieb & Luttinger and consequences In this section we give an overview of results related to the Brascamp, Lieb & Luttinger rearrangement inequality [3, Theorem 1.2] (for functions of one variable). The main consequence which we use here was observed by M. Christ [7, Theorem 4.2]. We prefer to explicitly state the ingredients used in the proof to point out connections to pertinent results in the literature. Theorem 2.1 ([3]). Let f 1,..., f M : R R + be non-negative measurable functions. Let u 1,..., u M R n. Then R n M f i ( x, u i )dx R n M fi ( x, u i )dx (3) Corollary 2.2. Let K be a symmetric convex set in R n. Suppose that f 1,..., f n are nonnegative measurable functions defined on R. Then n n f i (x i )dx fi (x i )dx. K K 4

5 The corollary can be proved by approximating K by intersections of slabs of the form m K m = {x R n : x, u i 1} for suitable u 1,..., u m R n. In this case, 1 Km = m 1 [ 1,1](, u i ) and one can apply (3) with M = m + n. For an extension of Corollary 2.2 to certain cases when K is non-convex, see [8]; see [22] for the case when f i is the indicator of a compact subset of R; related results appear in [1]. We say that F : R N R is quasi-concave if for all s the set {x : F (x) > s} is convex. Similarly, F : R N R is quasi-convex if for all s the set {x : F (x) < s} is convex. An immediate consequence of Corollary 2.2 is the following. Corollary 2.3. Let F : R N R + be an even quasi-concave function and g i be real nonnegative integrable functions. Then F (t)g 1 (t 1 ) g N (t N )dt R N F (t)g1(t 1 ) gn(t N )dt. R N If F : R N R + is even and quasi-convex then F (t)g 1 (t 1 ) g N (t N )dt F (t)g1(t 1 ) gn(t N )dt. R N R N Proof. For s, let K(s) := {x : F (x) > s}. Then K(s) is symmetric and convex. Using the layer-cake representation (cf. [16, Theorem 1.13]), Fubini s Theorem, and Proposition 2.1, we have F (t)g 1 (t 1 ) g N (t N )dt = g 1 (t 1 ) g N (t N )dtds R N K(s) g1(t 1 ) gn(t N )dtds K(s) = F (t)g 1 (t 1 ) g N (t N )dt. R N For the second assertion, one can use the fact that 1 {F s} + 1 {F >s} = 1. 3 Groemer s Convexity Condition For a function F : N R n R, set F F (f 1,..., f N ) :=... R n F (x 1,... x N )f 1 (x 1 )... f N (x N )dx 1,..., dx N R n 5

6 In this section we isolate a condition on F from which one can conclude a minimization result such as Theorem 1.1. We will say that F : N R n R + satisfies Groemer s Convexity Condition, or simply (GCC) in short, if for every z R n and for every y 1..., y N z the function F Y : R N R + defined by is even and convex. F Y (t) = F (y 1 + t 1 z,..., y N + t N z) Proposition 3.1. Let F : N R n R + be a function that satisfies (GCC). Let f 1,..., f n be non-negative integrable functions defined on R n and let θ S n 1. Then F F (f 1,..., f N ) F F (f 1 ( θ),..., f N( θ)), where f ( θ) is the Steiner symmetrization of f about θ (cf. (2)). Proof. Using the notation for the Steiner symmetrization of f with respect to θ, F F (f 1,..., f N ) is equal to =... R n 1 =... R n 1... R n 1 R n 1 R n 1 R n 1 R... F (y 1 + t 1 e 1,... y N + t N e 1 ) R N f i (y i + t i e 1 )dt 1... dt N dy 1... dy N F Y (t 1,, t N )h 1 (t 1 )... h N (t N )dt 1... dt N dy 1... dy N R N F Y (t 1,, t N )h 1(t 1 )... h N (t N )dt 1... dt N dy 1... dy N, R N which is simply equal to F F (f1 ( θ),..., fn ( θ)) (cf. Corollary 2.3). Successive symmetrizations with respect to n 1 dimensional subspaces yield the symmetric rearrangement fi for each f i, i N. In particular, we will make use of the following result, proved in [3]. Proposition 3.2. Let f : R n R + be a measurable function with compact support. Then there exists a sequence of functions f n, where f = f and f n+1 = f n ( θ) for some θ S n 1, such that lim n f n f L1 =. By a standard approximation argument, we obtain the following proposition. Proposition 3.3. Suppose F : N R n R + satisfies (GCC) and f 1,..., f n are nonnegative integrable functions on R n. Then F F (f 1,..., f N ) F F (f 1,..., f N). (4) 6

7 Remark 3.4. (1) As the proof shows, in Proposition 3.3, one can replace the (GCC) assumption on F by the following: for almost all θ S n 1 and almost all choices of y 1,..., y N θ, the sets {F Y s} are centrally symmetric and convex. (2) If F satisfies the quasi-concave analogue of (1) above, i.e., if for almost all θ S n 1 and almost all choices of y 1,..., y N θ, the level sets {F Y > s} are centrally symmetric and convex, then F F (f 1,..., f N ) F F (f1,..., fn). The latter inequality was observed by M. Christ [7, Theorem 4.2]; such functions F are referred to there as Steiner convex. 3.1 From rotational invariance to the ball Let f 1,..., f N be bounded integrable functions with f(x)dx = 1. We will say that f R n is rotationally invariant if f(x) = f(y) whenever x 2 = y 2. As in the introduction, let P [n] be the class of probability measures on R n that are absolutely continuous with respect to Lebesgue measure; let RP [n] P [n] be the subclass consisting of rotationally invariant measures. The previous proposition shows that if F satisfies (GCC), then inf F F (f 1,..., f N ) = inf F F (f 1,..., f N ), P [n] RP [n] where the f i s are the densities of measures in P [n] and RP [n], respectively. The remainder of this section is devoted to studying the quantity inf F F (f 1,..., f N ) RP [n] under the additional assumption that f i 1, for 1 i N. The following lemma is standard; the proof given for completeness. Lemma 3.5. Let f : R + [, 1] be a measurable function and assume that A := f(t)t n 1 dt <. Let g = 1 [,(na) 1/n ]. Then for any increasing function φ : R + R, Proof. Note that φ(t)f(t)t n 1 dt f(t)t n 1 dt = φ(t)g(t)t n 1 dt. g(t)t n 1 dt. 7

8 By assumption, f 1 and hence for any s (na) 1/n, s Consequently, for any s, f(t)t n 1 dt s f(t)t n 1 dt s s g(t)t n 1 dt. g(t)t n 1 dt. Without loss of generality, we may assume that φ() =. Then φ(t)f(t)t n 1 dt = = = = t φ (s) φ (s) t φ (s)f(t)t n 1 dsdt s s f(t)t n 1 dtds g(t)t n 1 dtds φ (s)g(t)t n 1 dsdt φ(t)g(t)t n 1 dt. Lemma 3.6. Let µ RP [n] and assume that its density f : R n R + satisfies f 1. For φ S n 1 and s, set Then H(φ, s) = {x R n : x, φ s}. µ(h(φ, s)) vol n (D n H(φ, s)) Proof. Let g = 1 Dn. For each fixed θ S n 1, the function from R + to R + defined by r 1 H(φ,s) (rθ) is increasing and hence so is r 1 H(φ,s) (rθ)dσ(θ). S n 1 8

9 Using spherical coordinates and applying Lemma 3.5, we get f(x)dx = nω n H(φ,s) = 1 H(φ,s) (rθ)f(rθ)r n 1 dσ(θ)dr S n 1 nω n 1 H(φ,s) (rθ)g(rθ)r n 1 dσ(θ)dr S n 1 g(x)dx. H(φ,s) Lemma 3.7. Let ρ : R n R be a function such that for any x R n, the function from R to R defined by s ρ(sx) is convex. Let X be a symmetric random vector with values in R n. Then the function from R + to R + defined by s Eρ(sX) is an increasing function. Proof. It is sufficient to show that Eρ(aX) Eρ(X) (5) for any a 1. For such a, we can write a = b(1) + (1 b)( 1) with b 1 and use the convexity assumption ρ(ax) bρ(x) + (1 b)ρ( X), from which (5) follows on taking expectations. Lemma 3.8. If F : N R n R + satisfies (GCC) then for any x 1,..., x N R n and any 1 j N, the function from R to R defined by is convex. s F (x 1,..., sx j,..., x N ) (6) Proof. For 1 i N with i j, write x i = x i + s i x j with s i R and x i x j. In the definition of (GCC), take z = x j, y j = and y i = x i for all i j. Then the map G Y : R N R + given by G Y (t) := F (y 1 + t 1 s 1 z,..., t j z,..., y N + s N t N z) 9

10 is convex since G Y (t) = F Y (s 1 t 1,..., t j,..., s N t N ). But the restriction of G Y to the line {t R N : t j R, t i = 1 for each i j} is just the function in (6). Proposition 3.9. Let f i : R n [, 1] be rotationally invariant probability densities. Suppose F : N R n R + satisfies (GCC). Then F F (f 1,..., f N ) F F (1 Dn,..., 1 Dn ). Proof. Using spherical coordinates for each x i R n, we will write x i := r i θ i, with r i <, and θ i S n 1 for i = 1,..., N. Then F F (f 1,..., f N ) is equal to (nω n ) N... S n 1... S n 1 F (r 1 θ 1,..., r N θ N ) N f i (r i θ i )r n 1 i dσ(θ 1 )... dσ(θ N )dr 1... dr N. Fix 1 j N and suppose r 1,..., r j 1, r j+1,..., r N are fixed non-negative scalars. Suppose momentarily that θ 1,..., θ N S n 1 are fixed vectors. By Lemma 3.8, the function from R + to R + defined by r j F (r 1 θ 1,..., r j θ j,..., r N θ N ) is convex. Averaging now in θ j S n 1, Lemma 3.7 implies that the function r j F (r 1 θ 1,..., r j θ j,..., r N θ N )dσ(θ j ) S n 1 is increasing. By assumption, we have 1 = f j (x)dx R n = nω n S n 1 f j (r j θ j )r n 1 j dσ(θ j )dr j. Since f j depends only on the value of r j, we have that for any θ j S n 1, f j (r j θ j )r n 1 j dr j = (nω n ) 1. Thus we apply Lemma 3.5 with A = (nω n ) 1 to see that F (r 1 θ 1,..., r j θ j,..., r N θ N )f j (r j θ j )r n 1 j S n 1 1 dσ(θ j )dr j

11 is at least as large as 1/n ω n F (r 1 θ 1,..., r j θ j,..., r N θ N )r n 1 j dσ(θ j )dr j S n 1 Applying Fubini s theorem iteratively, we have that F F (f 1,..., f N ) is larger than or equal to (nω n ) N ωn 1/n... ωn 1/n S n 1... S n 1 F (r 1 θ 1,..., r N θ N ) which is simply F F (1 Dn,..., 1 Dn ) in spherical coordinates. We summarize the results of this section with the following theorem. N r n 1 i dσ(θ 1 )... dσ(θ N )dr 1... dr N, Theorem 3.1. Let µ 1,..., µ N P [n] ; denote the density of µ i by f i. Suppose F : N R n R + satisfies (GCC) and set N F F (f 1,..., f N ) :=... F (x 1,..., x N ) f i (x i )dx 1... dx N. (7) R n R n Then Moreover, if f i = f i 4 Verifying GCC F F (f 1,..., f N ) F F (f 1,..., f N). and f i 1 for i = 1,..., N, we also have F F (f 1,..., f N ) F F (1 Dn,..., 1 Dn ). Let C be a symmetric convex body in R N. For x 1,..., x N R n, let T (x 1,... x N ) = [x 1 x N ] be the n N matrix with columns the x i s. Throughout this section, we let F : N R n R + be the function F (x 1,..., x N ) := vol n (T (x 1,..., x N )C). (8) Note that for any S SL n, F (S(x 1 ),..., S(x N )) = F (x 1,..., x N ). (9) Indeed, for any n n matrix M, we have F (M(x 1 ),..., M(x N )) = ( vol n [M(x1) M(x N )]C ) = vol n (M[x 1 x N ])C) = det(m) F (x 1,..., x N ). Our goal is to show that F satisfies (GCC) so that we can apply Theorem

12 Proposition 4.1. Let F be as defined in (8). Let θ S n 1 and y 1,..., y N θ. Set Y := {y 1,... y N }. Let T Y (t) := [y i + t i θ] and define F Y : R N R + by Then F Y F Y (t) = vol n (T Y (t)c). is (i) even and (ii) convex. In particular, F satisfies (GCC). Proof. The proof is analogous to that of [11, Lemma 3]. Note that { N } [y 1 + t 1 θ... y N + t N θ]c = c i (y i + t i θ) : (c i ) C, while { N } [y 1 t 1 θ... y N t N θ]c = c i (y i t i θ) : (c i ) C. The latter two sets are reflections of each other about θ, hence F Y (t) = F Y ( t). For the second assertion, let us set P := P θ, the orthogonal projection onto θ. For any compact, convex set A R n, define functions f A, g A : P A R by and Then f A is concave and g A is convex. Let s, t R N and consider the functions and f A (y) := sup{λ : y + λθ A} (1) g A (y) := inf{λ : y + λθ A}. (11) f TY (s)c, g TY (s)c : P T Y (s)c R f TY (t)c, g TY (t)c : P T Y (t)c R defined as in (1) and (11). For convenience of notation, set and f s := f TY (s)c, f t := f TY (t)c, Since P is the orthogonal projection on θ, we have g s := g TY (s)c g t := g TY (t)c. P T Y (s)c = P [y i + s i θ]c = [y i ]C = P [y i + t i θ]c = P T Y (t)c. Thus setting D = P T Y (s)c = P T Y (t)c, we can define f, g : D R by f = (1/2)f s + (1/2)f t, g = (1/2)g s + (1/2)g t. 12

13 Set We claim that Ĉ := {y + λθ : y D, g(y) λ f(y)}. T Y (s/2 + t/2)c Ĉ. (12) Indeed, let x T Y (s/2 + t/2)c so that for some c = (c 1,..., c N ) C, we have x = N c i (y i + (s i /2 + t i /2)θ) = y + with y := N c iy i D. Note that ( N ) y + c i s i θ = and hence Similarly, Thus g s (y) g t (y) N c i (s i /2 + t i /2) θ, N c i (y i + s i θ) T Y (s)c N c i s i f s (y). N c i t i f t (y). g(y) = (1/2)g s (y) + (1/2)g t (y) N N (1/2) c i s i + (1/2) c i t i (1/2)f s (y) + (1/2)f t (y) = f(y), which shows that x = y + N c i(s i /2 + t i /2)θ Ĉ and establishes (12). Next, observe that ) vol d (Ĉ = f(y) g(y)dy D = (1/2) f s (y) g s (y)dy + (1/2) f t (y) g t (y)dy This shows that F Y D = (1/2) vol d (T Y (s)c) + (1/2) vol d (T Y (t)c). is convex. D 13

14 Proof of Theorem 1.1. The desired inequality follows from Theorem 3.1 and Proposition Further Extensions of Theorem 1.1 Before proceeding to applications, we briefly mention two natural extensions of Theorem 1.1. Remark 4.2. In Theorem 1.1, one can replace vol n ( ) by intrinsic volumes (refer to e.g., [23] for background on intrinsic volumes) by using the argument in [13, Lemma 2.3]. We omit the details. Remark 4.3. Let g 1 : (, ) (, ) be an increasing function and g 2 : (, ) (, ) be decreasing. If F : N R n R satisfies (GCC) then g 1 F satisfies the condition in Remark 3.4 (1); similarly, g 2 F satisfies the condition in Remark 3.4 (2). Thus if f 1,..., f N are non-negative integrable functions on R n, then and F g1 F (f 1,..., f N ) F g1 F (f 1,..., f N). (13) F g2 F (f 1,..., f N ) F g2 F (f 1,..., f N). (14) For instance, if g 2 (t) = t p for p >, then (14) gives upper bounds for F F p(f 1,..., f N ) provided that one can compute the corresponding quantity in the rotationally invariant case. This is possible in several cases but beyond our present scope. 5 Applications In this section we prove a corollary of Theorem 1.1 and use it to derive various isoperimetric inequalities. As in the introduction, let K n denote the collection of all convex bodies in R n. Denote the Hausdorff metric by δ H, i.e., for K 1, K 2 K n, δ H (K 1, K 2 ) := inf{δ > : K 1 K 2 + δb n 2, K 2 K 1 + δb n 2 }. We assume that µ 1, µ 2,... are probability measures in P [n] ; denote the density of µ i by f i. Let X 1, X 2,... be independent random vectors distributed according to densities f 1, f 2,... respectively. Let X1, X2,... be independent random vectors distributed according to f1, f2,... respectively. For each N n, let T N = T N (X 1,..., X N ) : R N R n be the operator represented by the n N matrix Similarly, for each N n, let T sym N T N = [X 1 X N ]. : R N R n be the operator with matrix T sym N = [X 1 X N]. 14

15 For notational reasons, it is convenient to assume that all random vectors X i and Xi are defined on an underlying probability space (Ω, Σ, P) and E denotes expectation with respect to P. Corollary 5.1. Suppose that (C N ) N=n is a sequence of convex bodies with C N R N. Let T N and T sym N be the linear operators defined above. Let M L 1 (Ω, Σ, P). Assume that vol n (T N C N ) M (a.s.) and vol n (T sym N C N) M (a.s.). Suppose that C and C are (random) convex bodies in R n defined by the following and C := lim N T NC N (a.s.) C := lim T sym N N C N (a.s.), where the convergence is in the Hausdorff metric. Then E vol n (C) E vol n (C ). Proof. We use the following three facts: (1) vol n ( ) is continuous with respect to convergence of convex bodies in the Hausdorff metric, (2) the Lebesgue Dominated Convergence Theorem, and (3) Theorem 1.1. E vol n (C) = E lim n (T N C N ) N = lim n (T N C N ) N lim n (T sym N C N) N = E lim n (T sym N C N) N = E vol n (C ) To use the corollary, it is convenient to have several basic facts from convexity at hand. We record them here for the reader s convenience. We refer to the introductory chapters of [23] or [9] for additional background material on convexity. Verifying convergence in the Hausdorff metric is often done by using support functions. Recall that if K K n, its support function is defined by h(k, y) = sup{ x, y : x K}. We will use the following standard lemma (see, e.g., [23, page 53]). 15

16 Lemma 5.2. Let K, L K N. Then δ H (K, L) = sup h(k, y) h(l, y). y S n 1 If T : R N R n is any linear operator, denote its adjoint by T t : R n R N. In particular, if T N = [x 1... x N ], then T t N : Rn R N is given by T t Ny = ( x 1, y,..., x N, y ) (y R n ). Using this fact, we can write an explicit formula for the support function of T N C N. Lemma 5.3. Let T : R N R n be a linear operator. Suppose C R N is a convex body. Then for any y R n, h(t C, y) = h(c, T t y). Proof. h(t C, y) = sup{ T x, y : x C} = sup{ x, T t y : x C} = h(c, T t y). Before proving Theorem 1.2, we mention one special case. 5.1 L p -centroid bodies Let K R n be a bounded Borel measurable set with vol n (K) = 1. Let Z p (K) denote the L p -centroid body of K, i.e., the body with support function ( 1/p h(z p (K), y) = x, y dx) p. L p -centroid bodies were introduced by Lutwak, Yang and Zhang [17] (under a different normalization). L p -centroid bodies play an important role in concentration of measure for convex bodies, e.g., [21], [15], [12], [14]. In this section we show how Corollary 5.1 gives a short proof of the following result. Corollary 5.4. Let K R n be a bounded Borel measurable set with vol n (K) = 1. Then vol n (Z p (K)) vol n (Z p (D n )), where D n is the Euclidean ball of volume one. For star-shaped bodies K R n the latter inequality, together with the equality conditions, is proved in [17]. In [2], the latter result is extended to measures µ P [n], although it makes use of the result for star-shaped bodies. In the next section, we prove the more general Orlicz version (also for measures µ P [n] ); the proof of this special case is given here to illustrate the direct connection to the Law of Large Numbers. K 16

17 Proof. Taking C N = N 1/p B N q in Lemma 5.3, we have h(n 1/p T N B N q, y) p = h(n 1/p B N q, T t Ny) p = 1 N N X i, y p for each y S n 1. By the Strong Law of Large Numbers, lim h(n 1/p T N Bq N, y) p = x, y p dx (a.s.). N Thus for any y S n 1, lim h(n 1/p T N Bq N, y) = N ( K K x, y p dx) 1/p (a.s.). Pointwise convergence of support functions in fact implies uniform convergence (see, e.g., [23, page 54]). Therefore, in the Hausdorff metric, Z p (K) = lim N N 1/p T N B N q Finally, let R(K) denote the circumradius of K, i.e., (a.s.). R(K) = inf{r > : K RB n 2 }. Since X i, y R(K), we have N 1/p T N B N q R(K)B n 2 and hence Corollary 5.1 gives the desired result. 5.2 Orlicz centroid bodies Here we use Corollary 5.1 to prove Theorem 1.2 stated in the introduction. As in the statement of said theorem, let ψ : [, ) [, ) be a Young function, i.e., convex, strictly increasing with ψ() =. Let µ P [n]. Define the Orlicz-centroid body Z ψ (µ) of µ corresponding to ψ by its support function h(z ψ (µ), y) = inf { λ > : R n ψ ( x, y λ ) } dµ(x) 1. Remark 5.5. By our definition, Z ψ (µ) is centrally-symmetric. In [18], Orlicz-centroid bodies are defined and studied for more general functions ψ. The idea of the proof is the same as that of Corollary 5.4. Set { } B ψ/n := t = (t 1,..., t N ) R N 1 N : ψ( t i ) 1. N 17

18 One can check that B ψ/n is convex, symmetric, bounded and the origin is an interior point, hence t Bψ/N := inf{λ > : t λb ψ/n } defines a norm on R N, commonly called the Orlicz norm associated with ψ. In particular, Bψ/N is the support function for Bψ/N, the polar of B ψ/n. If T : R N R n is a linear operator, the support function of T Bψ/N is cf. Lemma 5.3. h(t B ψ/n, y) = h(b ψ/n, T t y) = T t y Bψ/N (y S n 1 ); Lemma 5.6. Let µ P [n]. Let x 1, x 2,... be a sequence of vectors in R n and suppose that span{x 1,..., x n } = R n. (15) Let ψ be a Young function. Assume that for each y S n 1 and each λ >, we have 1 N ( ) ( ) xi, y x, y lim ψ dµ(x) =. (16) N N λ λ R n ψ Let T N = T N (x 1,..., x N ) be the n N matrix with columns x 1,..., x N. Then Z ψ (µ) = lim N T NB ψ/n. (17) Proof. It will be shown that for each y S n 1, we have pointwise convergence of support functions lim h(t NBψ/N, y) = h(z ψ (µ), y). (18) N This is sufficient as pointwise convergence implies uniform convergence (as noted in the proof of Corollary 5.4) Fix y S n 1. For simplicity of notation, for each N n, let g N : (, ) (, ) be defined by g N (λ) := 1 N ( ) xi, y ψ. N λ By (15), there exists i {1,..., n} such that x i, y, hence g N Consider also g : (, ) (, ) defined by ( ) x, y g(λ) := dµ(x). λ R n ψ is strictly positive. 18

19 Since ψ is convex and strictly increasing, g and g N are continuous and strictly decreasing. Let us also set λ(n) := h(t N B ψ/n, y) = inf{λ > : g N (λ) 1} and λ := h(z ψ (µ), y) = inf{λ > : g(λ) 1}. Suppose towards a contradiction that (18) is false. Then there exists ε > and a subsequence (N(j)) j=1 N such that either (i) λ(n j ) λ + ε for each j = 1, 2,..., or (ii) λ(n j ) λ ε for each j = 1, 2,.... Suppose first that (i) holds. Set so that λ := inf j λ(n(j)) λ λ + ε. (19) Let η >. For each j = 1, 2,..., by definition of λ(n(j)) and the fact that g N(j) is decreasing, we have Thus by (16), 1 < g Nj (λ(n j ) η) g Nj (λ η). 1 lim j g Nj (λ η) = g(λ η) As η > was arbitrary, and g is continuous, we have 1 g(λ ). If 1 < g(λ ), then λ < λ, contradicting (19). On the other hand, if 1 = g(λ ), then as g is a strictly decreasing continuous function, we have λ = λ, contradicting (19). Suppose now that (ii) holds. Set so that λ := sup λ(n(j)) j λ λ ε. (2) Let η >. For each j = 1, 2,..., by the definition of λ(n(j)) and the fact that g Nj is decreasing, we have g Nj (λ + η) g Nj (λ(n j ) + η) 1. 19

20 Thus by (16), g(λ + η) = lim j g Nj (λ + η) 1. Thus λ λ + η. As η > was arbitrary, we in fact have λ λ, contradicting (2). Proof of Theorem 1.2. By standard approximation arguments, we can assume that µ is compactly supported, say supp(µ) RB n 2. Let X 1, X 2,... be independent random vectors distributed according to µ. Let T N = T N (X 1,..., X N ) be the matrix with columns X 1,..., X N. Set λ := R ψ 1 (1) and observe that for any N and for any y S n 1, 1 N ( ) Xi, y ψ 1 N ψ(ψ 1 (1)) 1, N λ N hence Thus for any N, we have h(t N B ψ/n, y) = T t N y Bψ/N λ. T N B ψ/n λb n 2. (21) This shows that (5.1) in Corollary 5.1 is satisfied. On the other hand, by the Strong Law of Large Numbers, the X i s satisfy the assumption (16) in Lemma 5.6 almost surely. Hence, in the Hausdorff metric, Z ψ (µ) = lim N T NB ψ/n (a.s.), and Corollary 5.1 applies. Acknowledgements P. Pivovarov would like to thank N. Tomczak-Jaegermann for her continuing support. We are indebted to A. Burchard, M. Fradelizi, A. Giannopoulos, R. Lata la and V. Yaskin for helpful discussions. Our research started during the Workshop in Analysis and Probability in August 29 at Texas A & M University. It was completed at the Fields Institute during the Fall 21 thematic program on Asymptotic Geometric Analysis. We are grateful to the organizers of both programs and the institutions for their hospitality and financial support. 2

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23 Grigoris Paouris Department of Mathematics Texas A & M University College Station TX grigoris paouris@yahoo.co.uk Peter Pivovarov Department of Mathematics Texas A & M University College Station TX ppivovarov@math.tamu.edu 23