On the existence of supergaussian directions on convex bodies

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1 On the existence of supergaussian directions on convex bodies Grigoris Paouris Abstract We study the question whether every centered convex body K of volume 1 in R n has supergaussian directions, which means θ S n 1 such that j Z x K : x, θ t x, θ dx ff e ct2, K for all 1 t n, where c > is an absolute constant. We verify that a random direction is indeed supergaussian for isotropic convex bodies that satisfy the hyperplane conjecture. On the other hand, we show that if, for all isotropic convex bodies, a random direction is supergaussian then the hyperplane conjecture follows. 1 Introduction A well known conjecture in the theory of convex bodies is the hyperplane conjecture: there exists c > such that for any n 1 and any convex body K of volume 1 in R n with centre of mass at the origin, there exists θ S n 1 such that (1.1) K θ c. This question was posed explicitly in [9]. A classical reference is the paper of Milman and Pajor [45] (see also [22]). In the late 8 s, J. Bourgain verified the conjecture in the case of ψ 2 bodies. These are the convex bodies for which every direction is subgaussian: if µ is the uniform measure on K, a direction θ S n 1 is called subgaussian for µ with constant r >, if (1.2) µ ({x : x, θ tm θ }) e t2 r 2, for all 1 t r n, where m θ is the median of, θ with respect to µ. The best possible value of r > for which (1.2) holds true is the subgaussian constant b µ (θ) of µ in the direction of θ. Bourgain provided a lower bound in (1.1), depending on the The author is partially supported by an NSF grant 1

2 parameter sup θ b µ (θ) (see [1], [11], [16]). The main idea that the uniform measure µ on K may resemble the Gaussian measure became a central and fruitful line of research: a central limit theorem for convex bodies (proposed in [2], [14]) was proved (see [29], [3], [21], [18], [2]), various concentration results ([49], [5]) have appeared, and several strong conjectures have been proposed ([27], [6], [39], [19]) and have been verified in special cases ([54], [32], [43]). In this setting, the following question was posed by V. Milman: is it true that every convex body has at least one subgaussian direction? An affirmative answer was given by Bobkov and Nazarov ([7], [8]) for the class of 1-unconditional convex bodies; the same is true for the class of zonoids (see [48]). For a general convex body, B. Klartag ([31]) established the existence of a subgaussian direction up to a logarithmic in the dimension factor (see also [23]). The best known estimate is inf θ b µ (θ) = O( log n) [24]. In this paper we consider the following dual question: is it possible to find at least one supergaussian direction on every convex body? We say that a direction θ S n 1 is supergaussian for µ with constant r > if, for all 1 t n r, (1.3) µ ({x : x, θ tm θ }) e r2 t 2. The best possible value of r > for which (1.3) holds true is called the supergaussian constant of µ in the direction of θ and will be denoted by sg µ (θ). The question about supergaussian directions was considered in a recent paper of P. Pivovarov [52], who gave an affirmative answer (up to a logarithmic in the dimension factor) for the class of 1-unconditional bodies. The first main result of this paper provides an affirmative answer to the question for all convex bodies that satisfy the hyperplane conjecture. In fact, we show that, for isotropic convex bodies which satisfy the hyperplane conjecture, a random direction is supergaussian. Here, the randomness is with respect to the rotation invariant probability measure σ on S n 1. In order to give the precise formulation, we recall that an isotropic convex body K in R n is a convex body of volume 1 which has centre of mass at the origin and satisfies (1.4) x, θ 2 dx = L 2 K K for all θ S n 1 and some constant L K >. The question if the isotropic constant L K of K is uniformly bounded from above is an equivalent formulation of the hyperplane conjecture. This is based on the fact, proved by Hensley [25], that if K has volume 1 and centre of mass at the origin, then (1.5) x, θ 2 dx K θ 2. K for all θ S n 1. We write E(sg µ ) for the expectation of sg µ (θ) with respect to σ. Then, we have the following: 2

3 Theorem 1.1. Let K be a convex body in R n with centre of mass at the origin and volume 1. Assume that L K = O(1). Then, there exists θ S n 1 such that { } (1.6) x K : x, θ t x, θ dx e c1t2 K for all 1 t n. More generally, for any isotropic convex body K in R n, one has (1.7) E(sg K ) c 2 L K, where c 1, c 2 > are absolute constants. The best known upper bound for L K is due to B. Klartag: L K O( 4 n) (see [28]), although the hyperplane conjecture has been verified for a large variety of classes of convex bodies: 1-unconditional bodies [45], projection bodies and polars of projections bodies[26], [4], intersection bodies [36], unit balls of the Schatten classes [37], polytopes with few vertices [1], various random polytopes [34], [17]. Therefore, Theorem 1.1 shows that, at least in all these cases, there exists a supergaussian direction. We give two different proofs of Theorem 1.1. The first one is based on inequalities for the volume of L p centroid bodies, proved by Lutwak, Zhang and Yang [42]. The second proof is based on the techniques developed in [49], [5]. Actually, the proof can be carried out in a setting much broader than the one of log-concave measures. This is somehow expected: a recent result of B. Klartag [33] shows that any non-degenerate n-dimensional measure has at least one direction which behaves in a supergaussian way on a non-trivial interval. The second main result of this paper shows that if, for all isotropic convex bodies, a random direction is supergaussian, then the hyperplane conjecture follows: Theorem 1.2. There exists an absolute constant c > such that for every n 1, (1.8) sup L K c sup [E(sg K )] 3 log [E(sg K )] 2. K isotropic in R n K isotropic in R n The proof of Theorem 1.2 is based on the techniques developed in [16]. The paper is organized as follows: In 2 we introduce notation, terminology and some background material which is needed for the rest of the paper. In 3 we show some basic properties of the supergaussian constant. In 4 we give a first proof of Theorem 1.1. In 5 we give a second proof of Theorem 1.1 and the proof of Theorem 1.2. Acknowledgments. I would like to thank Apostolos Giannopoulos for many interesting discussions. 2 Preliminaries 2.1 Basic notation. We work in R n, which is equipped with a Euclidean structure,. We denote by 2 the corresponding Euclidean norm, and write B n 2 for the 3

4 Euclidean unit ball, and S n 1 for the unit sphere. Volume is denoted by. We write ω n for the volume of B2 n and σ for the rotationally invariant probability measure on S n 1. The Grassmann manifold G n,k of k-dimensional subspaces of R n is equipped with the Haar probability measure µ n,k. We also write à for the homothetic image of volume 1 of a compact set A R n, i.e. à := A. A 1/n The letters c, c, c 1, c 2 etc. denote absolute positive constants which may change from line to line. Whenever we write a b, we mean that there exist absolute constants c 1, c 2 > such that c 1 a b c 2 a. Also, if K, L R n we will write K L if there exist absolute constants c 1, c 2 > such that c 1 K L c 2 K. 2.2 Probability measures. We denote by P [n] the class of all probability measures on R n which are absolutely continuous with respect to the Lebesgue measure. The density of µ P [n] is denoted by f µ. A probability measure µ P [n] is called symmetric if f µ is an even function on R n. We say that µ P [n] is centered if for all θ S n 1, (2.1) x, θ dµ(x) =. R n A measure µ on R n is called log-concave if for any Borel sets A, B and any λ (, 1), (2.2) µ(λa + (1 λ)b) µ(a) λ µ(b) 1 λ. A function f : R n [, ) is called log-concave if log f is concave on its support {f > }. It is known that if µ is log-concave and if µ(h) < 1 for every hyperplane H, then µ P [n] and its density f µ is log-concave (see [13]). 2.3 Convex bodies. A convex body in R n is a compact convex subset C of R n with non-empty interior. We say that C is symmetric if x C implies that x C. We say that C is centered if it has centre of mass at the origin: x, θ dx = C for every θ S n 1. The support function h C : R n R of C is defined by h C (x) = max{ x, y : y C}. The mean width of C is defined by (2.3) W (C) = h C (θ)σ(dθ). S n 1 For each < p <, p, we define the p-mean width of C by ( ) 1/p (2.4) W p (C) = h p C (θ)σ(dθ). S n 1 The radius of C is the quantity R(C) = max{ x 2 : x C} and, if the origin is an interior point of C, the polar body C of C is (2.5) C := {y R n : x, y 1 for all x C}. Note that if K is a convex body in R n then the Brunn-Minkowski inequality implies that 1 ek is the density of a log-concave measure. We refer to the books [53], [46] and 4

5 [51] for basic facts from the Brunn-Minkowski theory and the asymptotic theory of finite dimensional normed spaces. 2.4 L q centroid bodies. Let µ P [n]. For every q 1 and θ S n 1 we define ( ) 1/q (2.6) h Zq(µ)(θ) := x, θ q f(x) dx, R n where f is the density of µ. If µ is log-concave then h Zq(µ)(θ) < for every q 1 and every θ S n 1. We define the L q -centroid body Z q (µ) of µ to be the centrally symmetric convex set with support function h Zq(µ). L q centroid bodies were introduced, with a different normalization, in [41] (see also [42] where an L q affine isoperimetric inequality was proved). Here we follow the normalization (and notation) that appeared in [49]. The original definition concerned the class of densities 1 K where K is a convex body of volume 1. In this case, we also write Z q (K) instead of Z q (1 K ). If K is a compact set in R n and K = 1, it is easy to check that Z 1 (K) Z p (K) Z q (K) Z (K) for all 1 p q, where Z (K) = conv({k, K}). Note that if T SL n then Z p (T (K)) = T (Z p (K)). Moreover, if K is convex body, as a consequence of the Brunn Minkowski inequality (see, for example, [49]), one can check that (2.7) Z q (K) c q Z 2 (K) for every q 2 and, more generally, (2.8) Z q (K) c q p Z p(k) for all 1 p < q, where c 1 is an absolute constant. Also, if K has its centre of mass at the origin, then (2.9) Z q (K) c K for all q n, where c > is an absolute constant. For a proof of this fact and additional information on L q centroid bodies, we refer to [48] and [5]. 2.5 Isotropic probability measures. Let µ be a centered measure in P [n]. We say that µ is isotropic if Z 2 (µ) = B2 n. We say that a centered convex body K is isotropic if Z 2 (K) is a multiple of the Euclidean ball. We define the isotropic constant of K by ( ) 1/n Z2 (K) (2.1) L K := B2 n. So, K is isotropic if and only if Z 2 (K) = L K B n 2. Note that K is isotropic if and only if L n K 1 K L K is isotropic. We define the isotropic constant of µ P [n] by L µ := f() 1 n. We refer to [45], [22] and [5] for additional information on isotropic convex bodies. 5

6 2.6 The bodies K p (µ). A natural way to pass from log-concave measures to convex bodies was introduced by K. Ball in [3]. Here, we will give the definition in a somewhat more general setting: Let µ P [n] and assume that supp(µ). For every p > we define a set K p (µ) as follows: { } (2.11) K p (µ) := x R n : p f µ (rx)r p 1 dr f µ (). It is clear that K p (µ) is a star shaped body with gauge function (2.12) x Kp(µ) := ( p f µ () f µ (rx)r p 1 dr) 1/p. 2.7 The parameter k (C). Let C be a symmetric convex body in R n. Define k (C) as the largest positive integer k n for which µ n,k (F G n,k : 1 ) 2 W (C)(Bn 2 F ) P F (C) 2W (C)(B2 n F ) n n + k. Thus, k (C) is the maximal dimension k such that a random k-dimensional projection of C is 4-Euclidean. The parameter k (C) is completely determined by the global parameters W (C) and R(C): There exist absolute constants c 1, c 2 > such that (2.13) c 1 n W (C)2 R(C) 2 k (C) c 2 n W (C)2 R(C) 2 for every symmetric convex body C in R n. The lower bound appears in Milman s proof of Dvoretzky s theorem (see [44]) and the upper bound was proved in [47]. We will need the following result (see [4], [35], [38] for a proof): Proposition 2.1. Let C be a symmetric convex body in R n and p >. Then, (i) W p (C) W (C) for all p k (C). (ii) W p (C) p/n R(C) for all k (C) p n. (iii) W p (C) R(C) for all p n. (iv) W cp (C) W p (C), where p = k (C) and c > is an absolute constant. 2.8 The parameters q (µ), q c (µ). Let µ P [n] and p > n, p. We define the quantities I p (µ) by ( ) 1/p (2.14) I p (µ) := x p 2 dµ(x). R n As before, if K is a compact set of volume 1, we write I p (K) instead of I p (1 K ). The following result is proved in [5]: 6

7 Proposition 2.2. Let µ be a log-concave centered measure on R n and let 1 p n/2. Then, n (2.15) I p (µ) p W p(z p (µ)) and (2.16) I p (µ) n p W p(z p (µ)). It is known (see [5, Proposition 4.7]) that for any convex body K of volume 1 and for all p > n/2, p, (2.17) I p (K) I p ( B n 2 ) n. Then, Proposition 2.2 implies that (2.18) W (Z p (K)) W p (Z p (K)) c p, for 1 p n/2, where c > is an absolute constant. Let µ be a centered log-concave measure in R n and let δ 1. The parameters { (2.19) q c (µ, δ) := sup < r n 2 : I r(µ) 1 } δ I 2(µ) and (2.2) q (µ, δ) := sup { < r n : k (Z r (µ)) r } δ played a crucial role in [49], [16]. In the sequel, we collect some basic facts for these two parameters. Lemma 2.3. Let C be a symmetric convex body in R n and assume that p, δ 1 satisfy W (C) 1 δ W p(c). Then, k (C) c p δ 2. Proof. Then, We may assume that p k (C) (otherwise, we have nothing to prove). (2.21) δw (C) W p (C) p/nr(c). But this implies that (2.22) p cδ 2 n ( ) 2 W (C). R(C) This completes the proof. 7

8 We say that the centered convex body K of volume 1 has small diameter (with constant α 1) if (2.23) R(K) α I 2 (K). Let K be a centered convex body of volume 1 in R n. We define V = K 4I 2 (K)B n 2 and K = Ṽ. It is easy to check that K is a body of small diameter and I 2 (K) I 2 ( K). Moreover, we have the following. Lemma 2.4. Let K be a centered convex body of volume 1 in R n and let 1 q n/2. Then, (2.24) I q (K) 2I q ( K). Proof. Recall that K := V 1 n V, where V := K 4I 2 (K)B2 n. Note that V So, we have that (2.25) I q q ( K) = K 1 x q 2 This implies that I q (K) 2I q ( K). dx = V q n V V 1 1 x q dx 2 V 1 q n I q q (K). Proposition 2.5. Let K be a centered convex body of volume 1 in R n and let δ 1. Then, (i) q ( K, c 1 δ 2 ) q c ( K, δ) q c (K, c 2 δ). (ii) q c (K, cδ) q (K,δ) cδ. (iii) If p q (K, δ) then I 2 (K) cr(z p (K)) R(K). (iv) If p 2 p 1 q ( K, δ) then k (Z p2 ( K)) ck (Z p1 ( K)). (v) If δ 1, δ 2 1 and q ( K, δ 1 δ 2 ) cn, then q ( K, δ 1 δ 2 ) δ 2 q ( K, δ 1 ). (vi) If K is isotropic, then q (K, δ) c n. (vii) q c (K, cl(k)) n/2 and q ( K, cl(k) 2 ) n, where L(K) := I2(K) n. Proof. We first note that since K has small diameter we have (2.26) W p (Z p ( K)) p n I p( K) pl(k) for all 1 p n. (i) Let q := q c ( K, δ). Then, W (Z q ( K)) W q (Z q ( K)) q/ni q ( K) nl( K) ql( K) q/n δ δ W q(z q ( K)). δ 8

9 Then, Lemma 2.3 shows that k (Z q ( K)) c q δ 2, and this implies that q ( K, c 1 δ 2 ) q = q c ( K, δ). Moreover, by Lemma 2.4 we have that q c ( K, δ) q c (K, c 2 δ). (ii) Let r := q (K, δ) (we may assume that r 2δ). Then, k (Z r (K)) r/δ. So, r nl(k) δn r δn I r r r δ δ δ W r δ r(k)) cw c r δ r(k)) cδw c r δ (Z c r δ (K)) cδ r δn I c r δ (K). This implies that q c (K, cδ) q (K,δ) cδ (iii) Since as claimed. (2.27) p p p nl(k) δn δn I p (K) W p (Z p (K)) δ δ δ δn R(Z p(k)), we get (iii). (iv) From (iii) we see that, for p q ( K, δ), we have R(Z p ( K)) nl(k). It follows that, for p 2 p 1 q ( K, δ), (2.28) k (Z p2 ( K)) W (Z p 2 ( K)) 2 L(K) 2 c W (Z p 1 ( K)) 2 L(K) 2 k (Z p1 ( K)). (v) Let r 1 := q ( K, δ 1 ). Then, k (Z r1 ( K)) r1 δ 1. We may assume that, if r 2 := q ( K, δ 1 δ 2 ), then k (Z r2 ( K)) = r2 δ 1δ 2. The result follows from (iv). (vi) From (iii) we see that nl K cq (K, δ)l K. (vii) For 1 p n 2 we have (2.29) I p (K) I p ( B n 2 ) n I 2(K) L(K), and hence, q c (K, cl(k)) n/2. Moreover, using (iii), (2.13) and (2.18), we have that for p q ( K, δ), (2.3) k (Z p ( K)) W (Z p( K)) 2 L(K) 2 cp L(K) 2. This completes the proof. Finally, we will need the main result of [16]: Proposition 2.6. There exists an absolute constant c > such that for every n 1 and δ 1, n n (2.31) sup L K sup cδ K isotropic in R n K isotropic in R n q c ( K, log c δ) q c ( K, δ). 9

10 3 Supergaussian directions Let µ be a probability measure on R n, and let θ S n 1 and 1 p n. We define the supergaussian constant sg µ,p (θ) in the direction of θ at level p, as the least value of r > for which (3.1) µ ({x R n : x, θ tm θ }) e r2 t 2 for all 1 t p r. We also write m θ for the median of, θ with respect to µ. In the case p = n we simply write sg µ (θ); we call this value supergaussian constant of θ. If C is a compact set of volume 1 we write sg C (θ) instead of sg 1C (θ). Let µ be a log-concave centered probability measure in R n and let 1 p n. We define a quantity ḡ µ,p (θ) as follows: { } q (3.2) ḡ µ,p (θ) := inf δ > : h Zq(µ)(θ) δ h Z 1(µ)(θ) for all 1 q p. Again, we define ḡ µ (θ) = ḡ µ,n (θ) and if K is a centered convex body of volume 1, we write ḡ K (θ) instead of ḡ 1K (θ). We have the following: Lemma 3.1. Let µ be a log-concave centered probability measure on R n. For all 1 p n, (3.3) sg µ,p (θ) ḡ µ,p (θ). Proof. Let 1 p n and set r 1 := ḡ µ,p (θ) and r 2 := sg µ,p (θ). For every q p we have q (3.4) h Zq(µ)(θ) r h Z 1(K)(θ) and (3.5) h Z2q(µ)(θ) h Zq(µ)(θ). Using the Paley Zygmund inequality we see that, for every q 1, ({ (3.6) µ x R n : x, θ 1 }) 2 h Z q(µ)(θ) e q. Under our assumptions, we have ({ }) q (3.7) µ x R n : x, θ h Z1(K)(θ) e q 2r 1 for every 1 q p. We set t := q 1 2r 1. Then, if 2r1 t p 2r 1, we have that (3.8) µ({x R n : x, θ th Z2(µ)(θ)} e 4r2 1 t2. 1

11 This implies that r 2 cr 1. On the other hand, for all q 2, h q Z (θ) = q(µ) qhq Z (θ) 2(µ) s q 1 µ({x R n : x, θ sh Z2(µ)(θ)})ds = p qh q Z (θ) r 2 2(µ) q hq Z 2(µ) (θ) r q 2 p s q 1 e s2 r 2 2 ds s q 1 e s2 ds ( ) q ( hz2(µ)(θ) q r 2 ( ) q qhz2(µ)(θ) c, taking into account the fact that, for every β q 2 (3.9) 2 r 2 β s q 1 e s2 ds s q 1 e s2 ds p 1, s q 1 e s2 ds e β2 +1 β q. Since Z 2 (µ) Z 1 (µ), we conclude that r 1 cr 2. Working with the quantities ḡ µ,p (θ) instead of sg µ,p (θ) is much more convenient for us. Lemma 3.1 shows that, in the case of log-concave measures, these two quantities are equivalent up to an absolute constant. We also define the following quantities: Eḡ µ,p := ḡ µ,p (θ)dσ(θ) S n 1 a µ,p := sup ḡ µ,p (θ) θ S n 1 ḡ µ,p := inf ḡµ,p(θ). θ Sn 1 Note that the quantities ḡ µ,p and a µ,p are SL n -invariant: for every T SL n, we have (3.1) ḡ µ T,p = ḡ µ,p, a µ T,p = a µ,p. Indeed, one can easily check that if T SL n then ḡ µ T (θ) = ḡ µ ( T θ), where x := x x 2. Intuitively, the directions in which a convex body has large subgaussian constant are the ones in which K resembles a cone. In particular, it follows by a result of Berwald (see [5]) that for all symmetric convex bodies of volume 1 and for all < p q, one has that (3.11) h Zq(K)(θ) h h Z q(c θ )(θ) Zp(K)(θ) h Zp(C θ )(θ), 11 )

12 where C θ is a double cone in the direction of θ, i.e C θ := conv{a, θ, θ} for some A θ, normalized to have volume 1. On the other hand, the directions in which a convex body has large supergaussian constant are the ones in which K resembles a cylinder Q θ in the direction of θ, i.e. Q θ := aa rθ for some A θ, where A = 1 and a, r > satisfy ar = 1. More precisely, we have the following: Proposition 3.2. Let K be a symmetric convex body of volume 1 in R n and let θ S n 1. Then, (i) For all < p q, (3.12) h Zq(K)(θ) h h Z q(q θ )(θ) Zp(K)(θ) h Zp(Q θ )(θ) with equality if and only if K = Q θ. (ii) For all 1 p n, 1 (p + 1) p =, (q + 1) 1 q (3.13) ḡ K,p (θ) ḡ Qθ,p(θ) c p, where c > is an absolute constant. For the proof of Proposition 3.2 we will use the next lemma; a proof can be found in [12]. Lemma 3.3. Let g be a non-negative, finite, not identically zero, integrable, strictly decreasing on its support, convex function on [, ). Then, the function (3.14) q ( q(q + 1) ) 1/q q 1 g(t) t g() dt is strictly increasing in (, ), unless if g(t) := a1 [, ) (1 rt) for some a, r >. Proof of Proposition 3.2. (i) Let t [, h K (θ)]. We define a function g by (3.15) g(t) := g K (t) := {x K : x, θ t} = 2 t f(s)ds, where f(s) := {x K : x, θ = s}. By the Brunn Minkowski inequality, f is log-concave and, since K is symmetric, f is decreasing and attains its maximum at. So, g is a strictly decreasing convex function on its support. Note that g() = K = 1. Moreover, (3.16) h q Z (θ) = q q(k) t q 1 g(t)dt. Note that g Qθ (t) := a1 [, 1 r ] (1 rt) and h Zq(Q θ )(θ) = (q + 1) 1 q. So, (3.17) h Zq(K)(θ) h Zq(Q θ )(θ) = ( q(q + 1) 12 ) 1/q t q 1 g(t)dt

13 and the result follows from Lemma 3.3. Assertion (ii) follows from the definition of ḡ K,p (θ). As an immediate consequence of Proposition 3.2 we get that, for all θ S n 1, (3.18) ḡ K (θ) c n. Intuition suggests that it is not possible that all directions are cylindrical. So, we expect that there exist directions with supergaussian constant much smaller than n. We conclude this section by pointing out the following immediate consequence of the definitions: For all 1 q 1 q 2 n and θ S n 1, q2 (3.19) ḡ µ,q1 (θ) ḡ µ,q2 (θ) ḡ µ,q1 (θ). q 1 4 Proof of Theorem 1.1 We will use the L p versions of the Blaschke Santaló inequality and of the Busemann Petty inequality obtained by Lutwak Zhang [41] and Lutwak, Yang and Zhang [42] respectively (see also [15] for a proof in the convex case). Theorem 4.1. Let K be a star-shaped, with respect to the origin, body of volume 1 in R n. Then, (4.1) Z p(k) 1/n Z p( B n 2 ) 1/n and (4.2) Z p (K) 1/n Z p ( B n 2 ) 1/n, with equality if and only if K is a centered ellipsoid of volume 1. Using K. Ball s bodies K p (µ) we can extend Theorem 4.1 to the case of probability measures µ P [n]. We will use the following lemma (see [45, p.76] for a proof). Lemma 4.2. Let f be a bounded measurable, non-negative function on [, ]. Then, for every < p q we have ( p 1/p ( q 1/q (4.3) t f(t)dt) p 1 t f(t)dt) q 1. f f There is equality if and only if f := f 1 [,a] for some a >. Proposition 4.3. Let µ P [n] and let f µ be its density. If supp(µ) and f µ 1, then (4.4) Z p (µ) Z p ( B n 2 ) and Z p(µ) Z p( B n 2 ), with equality if and only if f µ := 1 E for some centered ellipsoid E of volume 1. 13

14 Proof. Recall that for any µ P [n], the set K p (µ) is star-shaped with respect to the origin and (4.5) x Kp(µ) := ( p f µ () Integration in polar coordinates gives (4.6) K n (µ) = 1 f µ () and f µ (rx)r p 1 dr) 1/p. R n f µ (x)dx = 1 f µ () (4.7) Z p ( K n+p (µ)) K n+p (µ) 1 p + 1 n fµ () 1/p = Z p (µ). Lemma 4.2 implies that if q p > 1, then (4.8) x Kq(µ) ( ) 1 fµ p 1 q x Kp(µ). f µ () We have equality only if f µ,x (r) := f µ (rx) is constant for every x S n 1, or f equivalently, if µ f µ is the indicator function of some star-shaped set in R n. It follows that ( ) n( 1 fµ n () 1 n+p ) ( ) p fµ n+p () 1 (4.9) K n+p (µ) K n (µ) = f µ f µ f µ (), which implies ( ) 1 (4.1) K n+p (µ) 1 p n f p fµ n () n+p µ () f np µ ()f 1 p µ () = f µ 1 n. f µ This shows that (4.11) Z p (µ) f µ 1 n Z p ( K n+p (µ)) and Z p(µ) f µ 1 n Z p( K n+p (µ)). Taking volumes and using Theorem 4.1, we conclude that (4.12) Z p (µ) f µ 1 Z p ( B n 2 ) and Z p(µ) f µ Z p( B n 2 ). This proves the Proposition. Using Theorem 4.1 we can show that if K is an isotropic convex body, then ḡ K (θ) CL K for a random direction θ. Proposition 4.4. Let K be an isotropic convex body in R n. Then, for every t 1, (4.13) σ{θ S n 1 : ḡ K (θ) c 1 tl K } 1 log 2 n t n, where C 1 > is an absolute constant. 14

15 Proof. Since K is convex, we have that (4.14) h Zp(K)(θ) h Z2p(K)(θ), for every p 1 and θ S n 1. So, we can write 2 k 2 (4.15) ḡ K (θ) L K sup k log 2 n h Z2 k (K)(θ), where we have also used the fact that K is isotropic. A direct computation shows that for p n, (4.16) Zp( B 2 n) 1/n ω1/n n. p Using polar coordinates, we get from Theorem 4.1 that dσ(θ) ω n (4.17) ω n (θ) c n ( p) n. S h n n 1 Z p(k) By Markov s inequality we get that, for every t 1, { (4.18) σ θ S n 1 : h Zp(K)(θ) C } p 1 t t n. This implies that { (4.19) σ θ S n 1 : h Z2 k (K)(θ) C } 2 k 2 LK, k = 1,..., log tl 2 n 1 log 2 n K t n, and hence, (4.2) σ{θ S n 1 : ḡ K (θ) C 1 tl K } 1 log 2 n t n, as claimed. Proof of Theorem 1.1. Let K be an isotropic convex body in R n. From Proposition 3.2 we have that ḡ K (θ) c n. Let (4.21) A := {θ S n 1 : g K (θ) 2c 1 L K }. From Proposition 4.4 we have that σ(a) 1 log 2 n 2. Therefore, n Eḡ K = ḡ K (θ)dσ(θ) S n 1 = ḡ K (θ)dσ(θ) + ḡ K (θ)dσ(θ) A S n 1 \A 2c 1 L K + c nσ(s n 1 \ A) n log2 n 2c 1 L K + 3c 1 L K, 2 n 15

16 since L K c > for all convex bodies (see [45]). Moreover, if K is a centered convex body of volume 1 and K iso is the isotropic image of K, by (3.1) we have that (4.22) g K = g Kiso = inf θ S n 1 g Kiso (θ) E (ḡ Kiso ) 3c 1 L K. Now, Lemma 3.1 completes the proof. Note. (i) Using Proposition 4.3 instead of Theorem 4.1 one can prove the same result for any log-concave measure. Recall that L µ = f() 1 n. We omit the details. (ii) The only place where convexity is needed in the proof of Proposition 4.4 is in assuming (4.14). So, the same result holds true for any µ P [n] which has some regularity as the one expressed in (4.14). 5 Proof of Theorem 1.2 As we have already mentioned in the Introduction, J. Bourgain ([11]) has proved that convex bodies for which every direction is subgaussian have bounded isotropic constant (by a function which depends only on the subgaussian constant). We will show that the same holds true in the supergaussian case; the proof is much easier. Proposition 5.1. Let K be a convex body in R n with volume 1 and centre of mass at the origin. Let a K := sup θ S n 1 ḡ K (θ). Then, (5.1) L K c 1 a K, where c 1 > is an universal constant. Proof. Recall that a K is SL n -invariant. So, we can assume that K is isotropic. Then, for every θ S n 1, nlk (5.2) h K (θ) c 1 h Zn(K)(θ) c 2 ḡ K (θ) c nlk 2. a K Since K = 1, there exists θ S n 1 such that (5.3) h K (θ ) h fb n 2 (θ ) n. It follows that L K c 3 a K. Let K be an isotropic convex body in R n. We write K for the isotropic convex body of small diameter created by K, i.e. the isotropic image of K 4 nl K B2 n. Note that R( K) c nl K. Moreover, if A K with A 1 2, then for every p 1 and θ S n 1, ( (5.4) h Zp(K)(θ) x, θ p dx A ) 1 p = A 1 16 p + 1 n hzp( e A) (θ) ch Z p( e A) (θ).

17 In particular, we have that Z p ( K) cz p (K), where c > is a universal constant. Note that L K L K. By the definition of ḡ K,p, we have that for every p 1, (5.5) ḡ K,p cḡ K,p. Lemma 5.2. Let K be an isotropic convex body in R n and let K be defined as above. Then, (i) q ( K, c1 E 2 (ḡ K) ) n, (ii) For any δ 2, E(ḡ K,q ( K,δ)) c 2 δ, where c 1, c 2 > are absolute constants. Proof. (i) Let E := E(ḡ K). By definition we have that, for all 1 p n, p (5.6) h K)(θ) Zp( ḡ K(θ) L K. Integrating over S n 1 we see that (5.7) W 1 1 (Z p( K)) = h 1 1 Z p( (θ)dσ(θ) K) ḡ K(θ)dσ(θ) = E. S pl n 1 K S pl n 1 K By Hölder s inequality we get that, for all 1 p n, (5.8) W (Z p ( K)) W 1 (Z p ( K)) pl K E. Recall that (5.9) R(Z p ( K)) R(conv({ K, K})) c nl K. So, by Milman s formula (2.13), for all 1 p n, (5.1) k (Z p ( K)) n This proves assertion (i). ( W (Zp ( K)) ) 2 R(Z p ( K)) c p E 2. (ii) Let q 1 := q ( K, δ) and q := q ( K, 2). Note that n q q 1. For all 1 p q, (5.11) W (Z p ( K)) W p (Z p ( K)) p n I p( K) c pl K, where we have used Propositions 2.1 and 2.2. Moreover, (5.12) R(Z q ( K)) n W q (Z q ( q K)) I q ( K) c nl K. 17

18 By definition we have that, for all q p q 1, k (Z p ( K)) p δ, or, using (2.13), (5.13) W (Z p ( K)) p R(Z p ( c K)) p R(Z q ( c K)) pl c K. n δ n δ δ Thereore, for all 1 p q 1, (5.14) W (Z p ( K)) c pl K δ. Moreover, recall that k (Z p ( K)) n for all 1 p n. A standard argument, based on the concentration of measure on S n 1, shows that for 1 p q 1, ( ) pl (5.15) σ {θ S n 1 : h K)(θ) Zp( c K } 1 e cn. δ Since Z p ( K) Z 2p ( K), we get ({ }) pl σ θ S n 1 : h K)(θ) Zp( c K for all p [1, q 1 ] δ 1 log 2 (q 1 )e c n In other words, (5.16) σ Let ( {θ S n 1 : ḡ K,q1 (θ) c ) δ} 1 e c n. (5.17) A := {θ S n 1 : ḡ K,q1 (θ) c δ}. 1 e c n. Then, E = = ḡ K,q1 (θ)dσ(θ) S n 1 ḡ K,q1 (θ)dσ(θ) + ḡ K,q1 (θ)dσ(θ) A S n 1 \A c δ + σ(s n 1 \ A) n c δ. This proves assertion (ii). We can give now a second proof of Theorem 1.1. Proposition 5.3. Let K be an isotropic convex body in R n. Then, (i) E(ḡ K, n ) c 1, (ii) E(ḡ K ) c 1 4 n, 18

19 (iii) E(ḡ K ) c 2 L K, where c 1, c 2 > are absolute constants. Proof. Let K be defined as above. Using (5.5) we have that (5.18) E(ḡ K,p ) ce(ḡ K,p ) for all 1 p n. By Proposition 2.5 (vi) we have that q (K, c) n. Then, Lemma 5.2 shows that (5.19) E(ḡ K, n ) ce(ḡ K, n ) c 1. This proves (i), while (ii) follows from (i) and (3.19). Finally, using Proposition 2.5 (i) and (vii), we have that q (K, cl 2 K ) n, and by Lemma 5.2, (5.2) E(ḡ K ) ce(ḡ K) c 2 L K. This completes (iii). Proposition 5.3 (iii) provides an alternative proof of Theorem 1.1, although the reader may notice that the measure estimate for the directions in S n 1 with supergaussian constant bounded by L K is much better in Proposition 4.4. Moreover, the convexity plays more important role in the proof of the Proposition 5.3. The proof of Proposition 5.3 can be certainly carried out in the case of log-concave measures. Proof of Theorem 1.2. By Proposition 2.5 (ii) we have that, for any δ 1, q c ( K, q ( K,cδ) δ) cδ. Then, by Lemma 5.2, ( (5.21) q c K, c1 E 2 (ḡ K) ) q ( K, c1 E 2 (ḡ K) ) cn c 1 E 2 (ḡ K) c 1 E 2 (ḡ K). Now, we apply Proposition 2.6: we have ( sup K isotropic L K c sup K isotropic E 2 (ḡ K) n q c ( K, c1 E 2 (ḡ K) ) c sup E 3 (ḡ K) log E 2 (ḡ K) K isotropic c sup E 3 (ḡ K ) log E 2 (ḡ K ). K isotropic log ) n ( q c K, c1 E 2 (ḡ K) ) This completes the proof. References [1] D. Alonso, J. Bastero, J. Bernues, P. Wolff, On the isotropy constant of projections of polytopes, Journal of Functional Analysis 258 (21),

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