A probabilistic take on isoperimetric-type inequalities

A probabilistic take on isoperimetric-type inequalities Grigoris Paouris Peter Pivovarov April 16, 212 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 due to 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 (conv {x 1,..., x }) dx 1... dx (K K n ). K K Thus Φ(K) gives the expected volume of the convex hull of independent random points sampled in K. In [16], 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., [1], [3], [31], [7], [18], [14], [13, Chapter 9], [9]). 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 A. Sloan Foundation and the US ational Science Foundation, grant DMS- 9615. The second-named author holds a Postdoctoral Fellowship award from the atural Sciences and Engineering Research Council of Canada and was a Postdoctoral Fellow at the Fields Institute during the Fall of 21. 1

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 [4]. 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., [26] and the references therein). In our context, if n and x 1,..., x are independent random points with x i distributed according to µ i P [n], we treat the n random matrix [x 1... x ] as a linear operator from R to R n ; applying [x 1... x ] to a convex body C R produces a random convex set in R n, i.e., { } [x 1... x ]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 and µ 1,..., µ P [n] ; denote the density of µ i by f i. Let C be a convex body in R and set F C (f 1,..., f ) =... vol n ([x 1... x ]C) f i (x i )dx... dx 1. (1) R n R n If f i 1 for i = 1,...,, then F C (f 1,..., f ) F C (1 Dn,..., 1 Dn ), where D n R n is the Euclidean ball of volume one. When the latter theorem is viewed as a result about random linear operators, it is perhaps surprising that the uniform measure on the Cartesian product of Euclidean balls appears as a minimizer (and not the Gaussian measure). The freedom to choose the convex body C and the densities f i reveals connections between a family of isoperimetric-type inequalities for convex bodies. For instance, if K R n is a convex body with vol n (K) = 1, we can take f i = 1 K. If C = conv {±e 1,..., ±e }, then [x 1... x ]C = conv {±x 1,..., ±x }, which corresponds to the symmetric analogue of Groemer s result mentioned above. For another example, take C = [ 1, 1]. In this case, { } [x 1... x ]C = α i x i : α i 1 for i = 1,...,, 2

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 concerns the expected volume of random zonotopes. In this way, we recover a result due to Bourgain, Meyer, Milman and Pajor [3] (which we state precisely in 4.1). For the class K n, a general framework, going back to Rogers and Shephard [32], [34], for proofs of results of this type is discussed by Campi and Gronchi in [9]. In addition to the extension to P [n], a benefit of Theorem 1.1 is that it applies to many functionals at once; one need only select C R. Furthermore, one is not limited to choosing a single C R. By taking a sequence of convex bodies C R for = 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 prove the following theorem. 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 (µ), 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. If f = 1 K where K R n is a convex body with vol n (K) = 1, the latter theorem was proved by Lutwak, Yang and Zhang [24]. Thus Theorem 1.2 can be seen as an extension of the Lutwak-Yang-Zhang result to the class P [n] (here we deal only with the symmetric case; cf. Remark 5.3). Despite the fact that Theorem 1.2 involves non-random sets, our proof shows that it is a consequence of Theorem 1.1 and the 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. The techniques used are analytic and depend heavily on approximation and limiting arguments, which puts the equality cases beyond our present reach. When f = 1 K and K R n is a convex body of vol n (K) = 1 (with the origin in its interior) equality holds in Theorem 1.2 if and only if K is a centered ellipsoid [24]. Our motivation for obtaining measure-theoretic extensions of such inequalities comes from questions related to extremal behavior of geometric properties of high-dimensional measures. For instance, this approach has already been used in [27]; further work in this direction can be found in [3]. 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 the Brascamp-Lieb-Luttinger rearrangement inequality [4]. In Section 3, we isolate a condition (which we call Groemer s 3

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 ) satisfies (GCC). Section 5 concludes with applications; in particular, the proof of Theorem 1.2. otation: The setting is R n with the usual inner-product,, standard Euclidean norm 2 and standard unit vector basis e 1,..., e n ; we assume that n 2; for a subspace F R n, F denotes the orthogonal complement of F. We write vol n ( ) for n-dimensional Lebesgue measure; B2 n for the Euclidean ball of radius one, the volume of which is ω n = vol n (B2 n ); Bp n for the closed unit ball in l n p. We reserve D n for the Euclidean ball of volume one, i.e., D n = ωn 1/n B2 n ; Lebesgue measure restricted to D n is λ Dn. The unit sphere is denoted S n 1 and is equipped with the Haar measure σ. The set of non-negative real numbers is denoted by R +. 2 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 center at the origin, whose volume is equal to the 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) In other words, we obtain f ( θ) by rearranging f along every line parallel to θ. We refer the reader to the book [22] or the introductory notes [5] for further background material on rearrangements of functions. 4

2.1 Brascamp-Lieb-Luttinger and consequences In this section we give an overview of results related to the Brascamp-Lieb-Luttinger rearrangement inequality [4, Theorem 1.2]. The main consequence which we use here was proved by Christ [1, 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 ([4]). Let f 1,..., f M : R R + be non-negative measurable functions. Let u 1,..., u M R n. Then M M f i ( x, u i )dx fi ( x, u i )dx. (3) R n R n The following corollary for symmetric convex sets K = K R n is immediate. 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 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 [11]; see [31] for the case when f i is the indicator of a compact subset of R; related results appear in [2]. We say that F : R R is quasi-concave if for all s the set {x : F (x) > s} is convex. Similarly, F : R 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 R + be an even quasi-concave function and g i be real nonnegative integrable functions. Then F (t)g 1 (t 1 ) g (t )dt F (t)g1(t 1 ) g(t )dt. R R If F : R R + is even and quasi-convex then F (t)g 1 (t 1 ) g (t )dt F (t)g1(t 1 ) g(t )dt. R R 5

Proof. For s, let K(s) := {x : F (x) > s}. Then K(s) is symmetric and convex. Using the layer-cake representation (cf. [22, Theorem 1.13]), Fubini s Theorem, and Corollary 2.2, we have F (t)g 1 (t 1 ) g (t )dt = g 1 (t 1 ) g (t )dtds R K(s) g1(t 1 ) g(t )dtds K(s) = F (t)g1(t 1 ) g(t )dt. R 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 : R n R, set F F (f 1,..., f ) :=... F (x 1,... x )f 1 (x 1 )... f (x )dx 1,..., dx. R n R n In this section we isolate a condition on F from which one can conclude a minimization result such as Theorem 1.1. Definition 3.1. We will say that F : R n R + satisfies Groemer s Convexity Condition, or simply (GCC) in short, if for every z R n \{} and for every Y = {y 1..., y } z the function F Y : R R + defined by is even and convex. F Y (t) = F (y 1 + t 1 z,..., y + t z) (4) For instance, if F (x 1,..., x n ) = det([x 1... x n ]) then F satisfies (GCC) since the determinant is independent of the choice of basis and is a multi-linear function of its rows. These properties of the determinant are used in proving Busemann s random simplex inequality, see e.g. [13, Theorem 9.2.6], a variant of Groemer s theorem mentioned in the introduction when = n + 1. In [16], a main technical step is to show that F (x 1,..., x ) = vol n (conv {x 1,..., x }) satisfies (GCC). We have chosen the latter terminology since a simple modification of Groemer s argument applies in our setting (we verify that the integrand in Theorem 1.1 satisfies (GCC) in the next section). We now indicate how the rearrangement inequalities mentioned so far are useful in the presence of (GCC) and an even weaker condition. Similar connections have appeared in 6

[1, Theorem 4.2], [2, page 15], [15, Lemma 3.3]. The proof of the next proposition is the same as that of [1, Theorem 4.2]; the formulation given here is that which best serves our purpose. Proposition 3.2. Let f 1,..., f be non-negative integrable functions on R n. Suppose that F : R n R + satisfies the following condition: for each z S n 1 and for each Y = {y 1,..., y } z, the function F Y defined by (4) is even and quasi-convex. Then Proof. Let θ S n 1. We will first show that F F (f 1,..., f ) F F (f 1,..., f ). (5) F F (f 1,..., f ) F F (f 1 ( θ),..., f ( θ)), where f ( θ) is the Steiner symmetrization of f with respect to θ (cf. (2)). For fixed y 1,..., y θ, we set h i (t i ) = f i (y i + t i θ). To condense the notation, we also write dt = dt 1... dt and dy = dy 1... dy. Using Fubini s Theorem and Corollary 2.3, F F (f 1,..., f ) = F (y 1 + t 1 θ,... y + t θ) f i (y i + t i θ)dtdy (θ ) R = F Y (t 1,..., t )h 1 (t 1 ) h (t )dtdy (θ ) R F Y (t 1,..., t )h 1(t 1 ) h (t )dtdy (θ ) R = F F (f1 ( θ),..., f( θ)), since fi ( θ) is the function obtained by rearranging f i along every line parallel to θ (cf. (2)). Suitable successive symmetrizations with respect to n 1 dimensional subspaces yield the symmetric decreasing rearrangement fi for each f i, i. In particular, we will make use of the following fact, proved in [4]: if g : R n R + is a measurable function with compact support, there exists a sequence of functions g k, where g = g and g k+1 = gk ( θ k), for some θ k S n 1, such that lim g k g L1 =. k The proposition follows by approximation as in [4]. Remark 3.3. For a thorough exposition of the Brascamp-Lieb-Luttinger rearrangement inequalities (in particular, approximation by iterated Steiner symmetrizations) see [35, Chapter 14]. For recent developments on iterated Steiner symmetrizations, see [6] and the references therein. 7

For completeness, we also state the reverse inequality due to Christ mentioned above (see [1, Theorem 4.2]). Proposition 3.4. Let f 1,..., f be non-negative integrable functions on R n. Suppose that F : R n R + satisfies the following condition: for each z S n 1 and for each Y = {y 1,..., y } z, the function F Y defined by (4) is even and quasi-concave. Then 3.1 From rotational invariance to the ball F F (f 1,..., f ) F F (f 1,..., f ). (6) Let f be an integrable function with R n f(x)dx = 1. We will say that f 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 of measures with rotationally invariant densities. Proposition 3.2 shows that if F satisfies (GCC), then inf F F (f 1,..., f ) = inf F F (f 1,..., f ), 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 ) RP [n] under the additional assumption that f i 1, for 1 i. The following lemma is standard; the proof is 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 +, φ(t)f(t)t n 1 dt φ(t)g(t)t n 1 dt. Here and elsewhere we use the term increasing in the non-strict sense. Proof. Set B = (na) 1/n and note that f(t)t n 1 dt = 8 B t n 1 dt.

Then φ(t)f(t)t n 1 dt = = = B B B B B φ(t)f(t)t n 1 dt + B φ(t)f(t)t n 1 dt + φ(b) φ(t)f(t)t n 1 dt + φ(b) φ(t)f(t)t n 1 dt + φ(t)t n 1 dt. B φ(t)f(t)t n 1 dt B B f(t)t n 1 dt (1 f(t))t n 1 dt φ(t)(1 f(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 is increasing and hence so is r 1 H(φ,s) (rθ) r 1 H(φ,s) (rθ)dσ(θ). S n 1 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) 9

Lemma 3.7. Let (Ω, Σ, P) be a probability space and let E denote expectation with respect to P. Let X : Ω R n be a symmetric random vector. Let ρ : R n R be a function such that R s ρ(sx) is convex for each x R n. Then is an increasing function. Proof. It is sufficient to show that R + s Eρ(sX) Eρ(aX) Eρ(X) (7) 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 (7) follows on taking expectations. In the sequel we will use only the following consequence of Lemma 3.7: for any such ρ, the function R + s ρ(sθ)dσ(θ) S n 1 is increasing. We have formulated the lemma using random vectors as it applies in other situations as well, e.g., [29]. Lemma 3.8. If F : R n R + satisfies (GCC) then for any x 1,..., x R n and any 1 j, the function R s F (x 1,..., sx j,..., x ) (8) is convex. Proof. The lemma follows from the fact that the restriction of a convex function to a line is itself convex. Formally, we fix j as in the assumption and for each i j, we write x i = x i + s i x j with s i R and x i x j. Since F satisfies (GCC), we can take z = x j, y j = and y i = x i for all i j and Y = {y 1,..., y }. Then the function G Y : R R + defined by G Y (t) := F (y 1 + t 1 s 1 z,..., t j z,..., y + t s z) = F Y (t 1 s 1,..., t j,..., t s ) 1

is convex by (GCC). On the other hand, G Y (t) = F (x 1 + (t 1 1)s 1 x j,..., x j + (t j 1)x j,..., x + (t 1)s x j ) and hence the restriction of G Y just the function in (8). to the line {t R : t j = s R, t i = 1 for each i j} is Proposition 3.9. Let f i : R n [, 1] be rotationally invariant probability densities. Suppose F : R n R + satisfies (GCC). Then F F (f 1,..., f ) 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,...,. To condense the notation, write dr = dr 1... dr and dθ = dσ(θ 1 )... dσ(θ ). Then F F (f 1,..., f ) = (nω n ) F (r 1 θ 1,..., r θ ) f i (r i θ i )r n 1 i dθdr. (R + ) (S n 1 ) Fix 1 j and suppose r 1,..., r j 1, r j+1,..., r are fixed non-negative scalars. Suppose momentarily that θ 1,..., θ S n 1 are fixed vectors. By Lemma 3.8, the function R + r j F (r 1 θ 1,..., r j θ j,..., r θ ) is convex. Regarding θ j as a random vector uniformly distributed on S n 1 and averaging, Lemma 3.7 implies that R + r j F (r 1 θ 1,..., r j θ j,..., r θ )dσ(θ j ) S n 1 is increasing. By assumption, we have 1 = f j (x)dx = nω n f j (r j θ j )r n 1 j dσ(θ j )dr j. R n S n 1 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 1/n ω n F (r 1 θ 1,..., r j θ j,..., r θ )f j (r j θ j )r n 1 j S n 1 dσ(θ j )dr j S n 1 F (r 1 θ 1,..., r j θ j,..., r θ )r n 1 j dσ(θ j )dr j. 11

Applying Fubini s theorem iteratively, we have F F (f 1,..., f ) (nω n ) [,ωn 1/n ] = F F (1 Dn,..., 1 Dn ). (S n 1 ) F (r 1 θ 1,..., r θ ) r n 1 i dθdr We summarize the results of this section with the following theorem. Theorem 3.1. Let µ 1,..., µ P [n] ; denote the density of µ i by f i. Suppose F : R n R + satisfies (GCC) and set Then Moreover, if f i = f i F F (f 1,..., f ) :=... F (x 1,..., x ) f i (x i )dx 1... dx. (9) R n R n F F (f 1,..., f ) F F (f 1,..., f ). and f i 1 for i = 1,...,, we also have F F (f 1,..., f ) F F (1 Dn,..., 1 Dn ). 4 Verifying GCC Let C be a symmetric convex body in R. For x 1,..., x R n, let T (x 1,... x ) = [x 1 x ] be the n matrix with columns the x i s. Throughout this section, we let F : R n R + be the function F (x 1,..., x ) := vol n (T (x 1,..., x )C). (1) ote that for any S SL n, Indeed, for any n n matrix M, we have F (S(x 1 ),..., S(x )) = F (x 1,..., x ). (11) F (M(x 1 ),..., M(x )) = vol n ([M(x 1 ) M(x )]C) = vol n (M[x 1 x ]C) = det(m) F (x 1,..., x ). Our goal is to show that F satisfies (GCC) so that we can apply Theorem 3.1. 12

Proposition 4.1. Let F be as defined in (1). Let θ S n 1 and y 1,..., y θ. Set Y := {y 1,... y }. Let T Y (t) := [y i + t i θ] and define F Y : R 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 [16, Lemma 3]. ote that { } [y 1 + t 1 θ... y + t θ]c = c i (y i + t i θ) : (c i ) C, while { } [y 1 t 1 θ... y t 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 and consider the functions and f A (y) := sup{λ : y + λθ A} (12) g A (y) := inf{λ : y + λθ A}. (13) 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 (12) and (13). 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. 13

Set We claim that Ĉ := {y + λθ : y D, g(y) λ f(y)}. T Y (s/2 + t/2)c Ĉ. (14) Indeed, let x T Y (s/2 + t/2)c so that for some c = (c 1,..., c ) C, we have x = c i (y i + (s i /2 + t i /2)θ) = y + with y := c iy i D. ote that ( ) y + c i s i θ = and hence Similarly, Thus g s (y) g t (y) c i (s i /2 + t i /2) θ, c i (y i + s i θ) T Y (s)c c i s i f s (y). c i t i f t (y). g(y) = (1/2)g s (y) + (1/2)g t (y) (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 + c i(s i /2 + t i /2)θ Ĉ and establishes (14). ext, 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 14

As we mentioned in the previous section, establishing (GCC) is a main technical step in proving Groemer s theorem (stated in the introduction). Groemer s theorem and related results can also be proved using linear parameter systems (or shadow systems) [32], [34]; see, e.g., [7], [8], [9]. We have chosen to adapt Groemer s proof since it fits with the rearrangement inequalities in the previous section. Proof of Theorem 1.1. The desired inequality follows from Theorem 3.1 and Proposition 4.1. 4.1 Further Extensions of Theorem 1.1 Let g 1 : (, ) (, ) be a strictly increasing continuous function and suppose that F : R n R + satisfies (GCC). Then g 1 F satisfies the assumption in Proposition 3.2 since for any z S n 1 and any Y = {y 1,..., y } z, {t R : g 1 F Y (t) > α} = {t R : F Y (t) > g 1 1 (α)} for each α >. Similarly, if g 2 : (, ) (, ) is strictly decreasing and continuous, then g 2 F satisfies the assumption in Proposition 3.4. Thus if f 1,..., f are non-negative integrable functions on R n, then and F g1 F (f 1,..., f ) F g1 F (f 1,..., f ) (15) F g2 F (f 1,..., f ) F g2 F (f 1,..., f ). (16) As a special case of the preceeding considerations, if we take C = [ 1, 1], g 1 (t) = t p for p and F (x 1,..., x ) = vol n ([x 1... x ]C), then we obtain the following theorem. Theorem 4.2. Let n and suppose µ 1,..., µ P [n] ; denote the density of µ i by f i. For p, set ( I p (f 1,..., f ) = vol n [x1... x ][ 1, 1] ) p f i (x i )dx 1... dx. R n R n If p > and f i 1 for each i = 1,...,, then Moreover, if p < then the inequality is reversed. I p (f 1,..., f ) I p (1 Dn,..., 1 Dn ). (17) 15

The particular case when p >, f i = 1 Ki and K i R n is a compact set of volume one, for i = 1,...,, was proved by Bourgain, Meyer, Milman and Pajor in [3], which we mentioned in the introduction. Remark 4.3. If g 2 (t) = t p for p >, then (16) gives upper bounds for F F p(f 1,..., f ) provided that one can compute the corresponding quantity in the rotationally invariant case. This is possible in several cases but beyond our present scope; see [3]. Remark 4.4. In Theorem 1.1, one can replace vol n ( ) by intrinsic volumes (refer to e.g., [33] for background on intrinsic volumes) by using the argument in [18, Lemma 2.3]. We omit the details. 5 Applications In this section we prove a corollary of Theorem 1.1 and use it to derive various isoperimetric inequalities. Rather than applying Theorem 1.1 for a fixed convex body C R, we consider a sequence of convex bodies (C ) =n with C R. By a simple application of the classical strong law of large numbers, we will show that for suitable choice of (C ), we obtain isoperimetric inequalities for non-random sets. We start by describing the probabilistic setting. Assume that µ 1, µ 2,... are probability measures in P [n] and f i denotes the density of µ i for i = 1, 2,.... Suppose that we have the following sequences of independent random vectors: (1) X 1, X 2,... with X i distributed according to f i ; (2) X 1, X 2,... with X i distributed according to f i ; (3) Y 1, Y 2,... with Y i distributed according to 1 Dn. We adopt the common convention that all random vectors are defined on a common underlying probability space (Ω, Σ, P) and E denotes expectation with respect to P. For k = 1, 2, 3 and for each n, we denote the corresponding random linear operators T (k) : R R n represented by n matrices as follows: T (1) = [X 1 X ] T (2) = [X 1 X ] T (3) = [Y 1 Y ]. (18) Thus for each k, (T (k) C ) =n is a sequence of random convex bodies in Rn. In the notation of the present section, Theorem 1.1 and its proof imply that for each n, E vol n ( T (1) C ) E vol n ( and if f i 1 for each i = 1, 2,..., then ( ) ( E vol n T (2) E vol n C T (2) T (3) C C ) ). 16

As in the introduction, let K n denote the collection of all convex bodies in R n. Recall that the Hausdorff metric δ H is defined as follows: 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 }. If T (k) C happens to converge in δ H as, almost surely (a.s.), to some random convex body C (k), then ( ) vol n T (k) ( vol ) n C (k) as, (19) C almost surely, since vol n ( ) is continuous with respect to δ H ( in K n. ) In fact, under an additional assumption such as dominated convergence of vol n T (k) (in the sense of Lebesgue s Dominated Convergence Theorem, e.g., [12, Theorem 4.3.5]), (19) implies that ( ) lim E vol n T (k) C ( = E vol ) n C (k). Thus we obtain the following corollary of Theorem 1.1. Corollary 5.1. Suppose that (C ) =n is a sequence of convex bodies with C R. For each k = 1, 2, 3 and n, let T (k) : R R n be the random linear operators defined in (18). Suppose C (k) are (random) convex bodies in R n defined by the following C C (k) := lim T (k) C (a.s.) (2) for k = 1, 2, 3, where the convergence is in the Hausdorff metric. Let M L 1 (Ω, Σ, P) and suppose further that for each k = 1, 2, 3, ( ) vol n T (k) M (a.s.). (21) Then and, if f i 1 for each i = 1, 2,..., then C E vol n ( C (1) ) E vol n ( C (2) ) E vol n ( C (2) ) E vol n ( C (3) ). The assumption (2) in the corollary turns out to be easy to verify in several cases using only basic facts from convexity and the classical strong law of large numbers. We refer the reader to the introductory chapters of [33] or [13] for additional background material on convexity and to, e.g., [12, Chapter 8] for basics on laws of large numbers. See [1] for a law of large numbers for random compact sets. 17

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 fact: if K, L K n, then δ H (K, L) = sup h(k, y) h(l, y) ; y S n 1 see, e.g., [33, page 53]. If T : R R n is a linear operator, denote its adjoint by T t : R n R. If C R is an arbitrary convex body, the support function of T C is given by h(t C, y) = sup{ T x, y : x C} = sup{ x, T t y : x C} = h(c, T t y) (22) for any y S n 1. ote also that if T = [x 1... x ], then T t : Rn R is given by T t y = ( x 1, y,..., x, y ) (y R n ). 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. For p 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 in [25] (under a different normalization). L p -centroid bodies play an important role in concentration of measure for convex bodies, e.g., [28], [21], [17], [19]. In this section we show how Corollary 5.1 gives a short proof of the following result. Corollary 5.2. 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 [23]; see [8] for an alternate proof. In [27], 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 18

Proof. In the notation described at the beginning of the present section, take f i = 1 K for i = 1, 2,.... ote that fi = 1 Dn and hence the random operators T (2) (3) and T have the same distribution. Let Bq denote the closed unit ball in l q, where 1/p + 1/q = 1. Setting C = 1/p Bq, the support function of T (1) C is h( 1/p T (1) B q, y) p = h( 1/p Bq, (T (1) )t y) p = 1 X i, y p for each y S n 1 (cf. (22) and the subsequent comment). By the strong law of large numbers (see, e.g., [12, Theorem 8.3.5]), the empirical mean converges to the actual mean almost surely, i.e., Thus for any y S n 1, lim 1 X i, y p = lim h( 1/p T (1) B q, y) = K ( K x, y p dx (a.s.). x, y p dx) 1/p (a.s.). Pointwise convergence of support functions in fact implies uniform convergence (see, e.g., [33, page 54]). Therefore, in the Hausdorff metric, Z p (K) = lim 1/p T (1) B q (a.s.), which shows that (2) holds. Finally, let R(K) denote the circumradius of K, i.e., R(K) = inf{r > : K RB n 2 }. Since X i, y R(K), we have 1/p T (1) B q R(K)B2 n, hence (21) is satisfied as well. Of course, the same reasoning applies when T (1) (2) is replaced by T and K by D n. Thus 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 19 λ ) } dµ(x) 1 (y R n ).

We assume that h(z ψ (µ), y) is finite for each y S n 1. In this case, it is well-known that h(z ψ (µ), ) defines a norm and hence is the support function of a convex body; see, e.g., [2], [35, Chapter 2] for further information on Orlicz norms. (If h(z ψ (µ), y) = for some y S n 1, then vol n (Z ψ (µ)) = and Theorem 1.2 is trivially true). Remark 5.3. By our definition, Z ψ (µ) is symmetric. In [24], Orlicz-centroid bodies are defined and studied for more general functions ψ. The idea of the proof is the same as that of Corollary 5.2. Set { } B ψ/ := t = (t 1,..., t ) R 1 : ψ( t i ) 1. One can check that B ψ/ is convex, symmetric, bounded and the origin is an interior point, hence t Bψ/ := inf{λ > : t λb ψ/ } (23) defines a norm on R, commonly called the Orlicz norm associated with ψ. In particular, Bψ/ is the support function for B ψ/ = {x Rn : x, y 1 y B ψ/ }. If T : R R n is a linear operator, we can use (22) to express the support function of T B ψ/ as h(t B ψ/, y) = h(b ψ/, T t y) = T t y Bψ/ (y S n 1 ). (24) Lemma 5.4. 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. (25) Let ψ be a Young function. Assume that for each y S n 1 and each λ >, we have 1 ( ) ( ) xi, y x, y lim ψ dµ(x) =. (26) λ λ R n ψ Let T = T (x 1,..., x ) be the n matrix with columns x 1,..., x. Then Z ψ (µ) = lim T B ψ/. (27) Proof. It will be shown that for each y S n 1, we have pointwise convergence of support functions lim h(t Bψ/, y) = h(z ψ (µ), y). (28) This is sufficient as pointwise convergence implies uniform convergence (as noted in the proof of Corollary 5.2) 2

Fix y S n 1. For simplicity of notation, for each n, let g : (, ) (, ) be defined by g (λ) := 1 ( ) xi, y ψ. λ By (25), there exists i {1,..., n} such that x i, y, hence g Consider also g : (, ) (, ) defined by ( ) x, y g(λ) := dµ(x). λ R n ψ is strictly positive. Since ψ is convex and strictly increasing, g and g are continuous and strictly decreasing. Let us also set λ() := h(t B ψ/, y) = inf{λ > : g (λ) 1} and λ := h(z ψ (µ), y) = inf{λ > : g(λ) 1}. Suppose towards a contradiction that (28) is false. Then there exists ε > and a subsequence ( j ) j=1 such that either (i) λ( j ) λ + ε for each j = 1, 2,..., or (ii) λ( j ) λ ε for each j = 1, 2,.... Suppose first that (i) holds. Set so that λ := inf j λ( j ) λ λ + ε. (29) Let η >. For each j = 1, 2,..., by definition of λ( j ) and the fact that g j is decreasing, we have Thus by (26), 1 < g j (λ( j ) η) g j (λ η). 1 lim j g j (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 (29). On the other hand, if 1 = g(λ ), then as g is a strictly decreasing continuous function, we have λ = λ, contradicting (29). 21

Suppose now that (ii) holds. Set so that λ := sup λ( j ) j λ λ ε. (3) Let η >. For each j = 1, 2,..., by the definition of λ( j ) and the fact that g j is decreasing, we have Thus by (26), g j (λ + η) g j (λ( j ) + η) 1. g(λ + η) = lim j g j (λ + η) 1. Thus λ λ + η. As η > was arbitrary, we in fact have λ λ, contradicting (3). Proof of Theorem 1.2. By standard approximation arguments, we can assume that µ is compactly supported, say supp(µ) RB n 2. In the notation defined at the beginning of the present section, take µ i = µ for each i = 1, 2,.... Thus X 1, X 2,... are independent and identically distributed according to f (the density of µ). Set λ := R ψ 1 (1) and observe that for any and for any y S n 1, 1 ( ) Xi, y ψ 1 ψ(ψ 1 (1)) = 1. λ By (23) and (24), h(t (1) B ψ/, y) = (T (1) )t y λ. Bψ/ Thus for any, we have T (1) B ψ/ λb2 n. (31) This shows that (21) in Corollary 5.1 is satisfied. On the other hand, by the strong law of large numbers (e.g. [12, Theorem 8.3.5]) the X i s satisfy the assumption (26) in Lemma 5.4 almost surely. Hence, in the Hausdorff metric, Z ψ (µ) = lim T (1) 22 B ψ/ (a.s.),

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Grigoris Paouris: grigoris@math.tamu.edu Peter Pivovarov: ppivovarov@math.tamu.edu Department of Mathematics, Mailstop 3368 Texas A&M University College Station TX 77843-3368 26