Asymptotic shape of the convex hull of isotropic log-concave random vectors

Transcription

1 Asymptotic shape of the convex hull of isotropic log-concave random vectors Apostolos Giannopoulos, Labrini Hioni and Antonis Tsolomitis Abstract Let x,..., x N be independent random points distributed according to an isotropic log-concave measure µ on R n, and consider the random polytope K N := conv{±x,..., ±x N }. We provide sharp estimates for the quermaßintegrals and other geometric parameters of K N in the range cn N exp(n); these complement previous results from [3] and [4] that were given for the range cn N exp( n). One of the basic new ingredients in our work is a recent result of E. Milman that determines the mean width of the centroid body Z q(µ) of µ for all q n. Introduction The purpose of this work is to add new information on the asymptotic shape of random polytopes whose vertices have a log-concave distribution. Without loss of generality we shall assume that this distribution is also isotropic. Recall that a convex body K in R n is called isotropic if it has volume, it is centered, i.e. its center of mass is at the origin, and its inertia matrix is a multiple of the identity: there exists a constant L K > 0 such that (.) x, θ 2 dx = L 2 K K for every θ in the Euclidean unit sphere S n. More generally, a log-concave probability measure µ on R n is called isotropic if its center of mass is at the origin and its inertia matrix is the identity; in this case, the isotropic constant of µ is defined as (.2) L µ := sup x R n ( fµ (x) ) /n, where f µ is the density of µ with respect to the Lebesgue measure. Note that a centered convex body K of volume in R n is isotropic if and only if the log-concave probability measure µ K with density x L n K K/L K (x) is isotropic. A very well-known open question in the theory of isotropic measures is the hyperplane conjecture, which asks if there exists an absolute constant C > 0 such that (.3) L n := sup{l µ : µ is an isotropic log-concave measure on R n } C for all n. Bourgain proved in [9] that L n c 4 n logn (more precisely, he showed that L K c 4 n logn for every isotropic symmetric convex body K in R n ), while Klartag [8] obtained the bound L n c 4 n. A second proof of Klartag s estimate appears in [20]. The study of the asymptotic shape of random polytopes whose vertices have a log-concave distribution was initiated in [3] and [4]. Given an isotropic log-concave measure µ on R n, for every N n we consider N independent random points x,..., x N distributed according to µ and define the random polytope

2 K N := conv{±x,..., ±x N }. The main idea in these works was to compare K N with the L q -centroid body of µ for a suitable value of q; roughly speaking, K N is close to the body Z log(2n/n) (µ) with high probability. Recall that the L q -centroid bodies Z q (µ), q, are defined through their support function h Zq(µ), which is given by ( ) /q (.4) h Zq(µ)(y) :=, y Lq(µ) = x, y q dµ(x). R n These bodies incorporate information about the distribution of linear functionals with respect to µ. The L q -centroid bodies were introduced, under a different normalization, by Lutwak and Zhang in [23], while in [29] for the first time, and in [30] later on, Paouris used geometric properties of them to acquire detailed information about the distribution of the Euclidean norm with respect to µ. It was proved in [3] that, given any isotropic log-concave measure µ on R n and any cn N e n, the random polytope K N defined by N independent random points x,..., x N which are distributed according to µ satisfies, with high probability, the inclusion (.5) K N c Z log(n/n) (µ) (for the precise statement see Fact 3.2). Then, using the fact that the volume of the L q -centroid bodies satisfies the lower bounds Z q (µ) /n c 2 q/n if q n and Zq (µ) /n c 3 L µ q/n if n q n (see Section 2), we see that for n N e n we have (.6) K N /n c 4 log(2n/n) n, while in the range e n N e n we have (.7) K N /n c 5 L µ log(2n/n) n with probability exponentially close to. On the other hand, one can check that for every α > and q, (.8) E [ σ n ({θ : h KN (θ) αh Zq(µ)(θ)}) ] Nα q, where σ n is the rotationally invariant probability measure on the Euclidean unit sphere S n. This estimate is sufficient for some sharp upper bounds. First, for all n N exp(n) one has (.9) E [ w(k N ) ] c 6 w(z log N (µ)), where the mean width w(c) of a convex body C in R n containing the origin, is defined as twice the average of its support function on S n : w(c) = h C (θ) dσ n (θ). S n Second, one has (.0) K N /n c 7 log(2n/n) n with probability greater than N, where C > 0 is an absolute constant. In [4] these results were extended to the full family of quermaßintegrals W n k (K N ) of K N. These are defined through Steiner s formula n ( ) n (.) K + tb2 n = W n k (K)t n k, k k=0 2

3 where W n k (K) is the mixed volume V (K, k; B n 2, n k). It is more convenient to express the estimates using a normalized variant of W n k (K): for every k n we set ( ) ( /k ) /k Wn k (K) (.2) Q k (K) = = P F (K) dν n,k (F ), ω n ω k where the last equality follows from Kubota s integral formula (see Section 2 for background information on mixed volumes). Then, one has the following results on the expectation of Q k (K N ) for all values of k: Theorem. (Dafnis, Giannopoulos and Tsolomitis, [4]). If n 2 N exp(cn) then for every k n we have (.3) L [ µ log N E Qk (K N ) ] w(z log N (K)). In the range n 2 N exp( n) one has an asymptotic formula: for every k n, (.4) E [ Q k (K N ) ] log N. All these estimates remain valid for n +δ N n 2, where δ (0, ) is fixed, if we allow the constants to depend on δ. Working in the range N n is possible, but requires some additional attention (see e.g. [5] for the case of mean width). A more careful analysis (which can be found in [4, Theorem.2]) shows that if n 2 N exp( n) then, for any s, a random K N satisfies, with probability greater than N s, (.5) Q k (K N ) c (s) log N for all k n and, with probability greater than exp( n), (.6) Q k (K N ) c 8 log N for all k n, where c (s) > 0 depends only on s, and c 8 > 0 is an absolute constant. A natural question that arises is whether these results can be extended to the full range cn N exp(n) of values of N. If one decides to follow the approach of [3] and [4] then there are two main obstacles. The first one is that the lower bound Z q (µ) /n c q/n is currently known only in the range q n. In fact, proving the same for larger values of q would lead to improved estimates on L n (for example, see the computation after Lemma 2.2 in [20]). The second one was that, until recently, a sharp estimate on the mean width of Z q (µ) was known only for q n; G. Paouris proved in [29] that for every isotropic log-concave measure µ on R n and any q n one has (.7) w ( Z q (K) ) c 9 q. Recently, E. Milman [25] obtained the same upper bound (modulo logarithmic terms) for q beyond n. Theorem.2 (E. Milman, [25]). For every isotropic log-concave measure µ on R n and for all q [ n, n] we have (.8) w(z q (µ)) c 0 q log 2 ( + q). An immediate consequence of this result is that it provides a new bound for the mean width of an origin symmetric isotropic convex body K in R n. In this case it is known that Z n (K) ck, and we conclude that (.9) w(k) C n log 2 ( + n)l K improving the earlier known bound w(k) C 2 n 3/4 L K of Hartzoulaki, from her PhD thesis [7]. We note here that not all of the logarithmic terms in (.9) can be removed, as the example of B n / B n /n shows. Using E. Milman s theorem we can show the following. 3

4 Theorem.3. Let x,..., x N be independent random points distributed according to an isotropic log-concave measure µ on R n, and consider the random polytope K N := conv{±x,..., ±x N }. If exp( n) N exp(cn) then for every k n we have (.20) L [ µ log N E Qk (K N ) ] log N ( log log N ) 2. Next we provide estimates for Q k (K N ) for most K N : Theorem.4. Let x,..., x N be independent random points distributed according to an isotropic log-concave measure µ on R n, and consider the random polytope K N := conv{±x,..., ±x N }. For all exp( n) N exp(n) and s we have (.2) Q k (K N ) c 2 (s) log N (log log N) 2, for all k < n, with probability greater than N s. We also provide estimates on the volume radius of a random projection P F (K N ) of K N onto F (in terms of n, k and N) in the range e n N e n ; these extend the sharp estimate v.rad(p F (K N )) log N that was obtained in [4] for the case N e n. Theorem.5. If exp( n) N e cn and s, then a random K N satisfies with probability greater than max{n s, e c N } the following: for every k n there exists a subset M n,k of with ν n,k (M n,k ) e c2k such that (.22) c 3 L µ log N v.rad(pf (K N )) := ( PF (K N ) ω k ) /k c 3 (s) log N ( log log N ) 2 for all F M n,k. In Section 4 we provide an alternative proof of an estimate of Alonso-Gutiérrez, Dafnis, Hernández-Cifre and Prochno from [3] on the k-th mean outer radius (.23) Rk (K N ) = R(P F (K N )) dν n,k (F ) of a random K N, as a function of N, n and k. Theorem.6. Let x,..., x N be independent random points distributed according to an isotropic log-concave measure µ on R n, and consider the random polytope K N := conv{±x,..., ±x N }. If n N exp( n) then, for all k n and s > 0 one has { } (.24) c 4 (s) max k, log(n/n) R { } k (K N ) c 5 (s) max k, log N with probability greater than N s, where c 4 (s), c 5 (s) are positive constants depending only on s. We provide a formula for R k (K N ) which is valid for all cn N exp(n). This allows us to recover (and explain) the sharp estimate of Theorem.6 for small values of N and to obtain its analogue for large values of N; see Theorem 4.5. In Section 5 we obtain estimates on the regularity of the covering numbers and the dual covering numbers of a random K N. In a certain range of values of N, these allow us to conclude that a random K N is in α-regular M-position with α (see Section 5 for definitions and terminology). 4

5 Theorem.7. Let µ be an isotropic log-concave measure on R n. Then, assuming that n 2 N exp ( (n log n) 2/5), we have that a random K N satisfies with probability greater than N the entropy estimates max {log N(K N, tr N B2 n ), log N(r N B2 n n(log n) 2 log( + t), tk N )} c 4 t for every t, where r N = log N and c 4 > 0 is an absolute constant. As an application we estimate the average diameter of k-dimensional sections of a random K N, defined by (.25) Dk (K N ) = R(K N F ) dν n,k (F ). The discussion shows that the behavior of D k (K N ) is not always the same as that of R k (K N ). In order to give an idea of the results, let us mention here the following simplified version. Theorem.8. Let µ be an isotropic log-concave measure on R n and a, b (0, ). (i) If k bn then a random K N satisfies with probability N D k (K N ) c b log N if n 2 N exp( n) and D k (K N ) c b log N(log log N) 2 if exp( n) N exp(n). (ii) If k an and N exp((n log n) 2/5 ) then a random K N satisfies with probability exp( n) c a log N log 3 n D k (K N ). where c a, c b are positive constants that depend only on a and b respectively. We conclude this paper with a brief discussion of the interesting (open) question whether the isotropic constant of a random K N is bounded by a constant independent from n and N. The first class of random polytopes K N in R n for which uniform bounds were established was the class of Gaussian random polytopes. Klartag and Kozma proved in [9] that if N > n and if G,..., G N are independent standard Gaussian random vectors in R n, then the isotropic constant of the random polytope K N = conv{±g,..., ±G N } is bounded by an absolute constant C > 0 with probability greater than Ce cn. The same idea works in the case where the vertices x j of K N are distributed according to an isotropic ψ 2 -measure µ; the bound then depends only on the ψ 2 -constant of µ. Alonso-Gutiérrez [2] and Dafnis, Giannopoulos and Guédon [2] have applied the same more or less method to obtain a positive answer in the case where the vertices of K N are chosen from the unit sphere or an unconditional isotropic convex body respectively. We show that, in the general isotropic log-concave case, the method of Klartag and Kozma gives the bound O( log(2n/n)) if N exp( n) (a proof along the same lines and an extension to random perturbations of random polytopes appear in [4]). 2 Notation and background material We work in R n, which is equipped with a Euclidean structure,. We denote by 2 the corresponding Euclidean norm, and write B2 n for the Euclidean unit ball, and S n for the unit sphere. Volume is denoted by. We write ω n for the volume of B2 n and σ n for the rotationally invariant probability measure on S n. The Grassmann manifold of k-dimensional subspaces of R n is equipped with the Haar probability measure ν n,k. Let k n and F. We will denote the orthogonal projection from R n onto F by P F. We also define B F = B2 n F and S F = S n F. 5

6 The letters c, c, c, c 2 etc. denote absolute positive constants whose value may change from line to line. Whenever we write a b, we mean that there exist absolute constants c, c 2 > 0 such that c a b c 2 a. Similarly, if K, L R n we will write K L if there exist absolute constants c, c 2 > 0 such that c K L A A /n. c 2 K. We also write A for the homothetic image of volume of a convex body A R n, i.e. A := A convex body is a compact convex subset C of R n with non-empty interior. We say that C is symmetric if x C whenever x C. We say that C is centered if it has center of mass at the origin i.e. x, θ dx = 0 C for every θ S n. The support function h C : R n R of C is defined by h C (x) = max{ x, y : y C}. For each < p <, p 0, we define the p-mean width of C by ( ) /p (2.) w p (C) := h p C (θ)dσ n(θ). S n The mean width of C is the quantity w(c) = w (C). The radius of C is defined as R(C) = max{ x 2 : x C} and, if the origin is an interior point of C, the polar body C of C is (2.2) C := {y R n : x, y for all x C}. Finally, if C is a symmetric convex body in R n and C is the norm induced to R n by C, we set M(C) = x C dσ n (x) S n and write b(c) for the smallest positive constant b with the property x C b x 2 for all x R n. From V. Milman s proof of Dvoretzky s theorem (see [6, Chapter 5]) we know that if k cn(m(c)/b(c)) 2 then for most F we have C F M(C) B F. 2. Quermaßintegrals Let K n denote the class of non-empty compact convex subsets of R n. The relation between volume and the operations of addition and multiplication of compact convex sets by nonnegative reals is described by Minkowski s fundamental theorem: If K,..., K m K n, m N, then the volume of t K + + t m K m is a homogeneous polynomial of degree n in t i 0: (2.3) t K + + t m K m = V (K i,..., K in )t i t in, i,...,i n m where the coefficients V (K i,..., K in ) can be chosen to be invariant under permutations of their arguments. The coefficient V (K i,..., K in ) is called the mixed volume of the n-tuple (K i,..., K in ). Steiner s formula is a special case of Minkowski s theorem; if K is a convex body in R n then the volume of K + tb n 2, t > 0, can be expanded as a polynomial in t: (2.4) K + tb n 2 = n k=0 ( ) n W n k (K)t n k, k where W n k (K) := V (K, k; B n 2, n k) is the (n k)-th quermaßintegral of K. It will be convenient for us to work with a normalized variant of W n k (K): for every k n we set ( ) /k (2.5) Q k (K) = P F (K) dν n,k (F ). ω k Note that Q (K) = w(k). Kubota s integral formula (2.6) W n k (K) = ω n P F (K) dν n,k (F ) ω k 6

7 shows that ( ) /k Wn k (K) (2.7) Q k (K) =. The Aleksandrov-Fenchel inequality states that if K, L, K 3,..., K n K n, then (2.8) V (K, L, K 3,..., K n ) 2 V (K, K, K 3,..., K n )V (L, L, K 3,..., K n ). This implies that the sequence (W 0 (K),..., W n (K)) is log-concave: we have (2.9) W k i j ω n W k j i W j i k if 0 i < j < k n. Taking into account (2.7) we conclude that Q k (K) is a decreasing function of k. For the theory of mixed volumes we refer to [33]. 2.2 L q -centroid bodies of isotropic log-concave measures We denote by P n the class of all Borel 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. We say that µ P n is centered if, for all θ S n, (2.0) x, θ dµ(x) = R n x, θ f µ (x)dx = 0. R n A measure µ on R n is called log-concave if µ(λa + ( λ)b) µ(a) λ µ(b) λ for all compact subsets A and B of R n and all λ (0, ). A function f : R n [0, ) is called log-concave if its support {f > 0} is a convex set and the restriction of log f to it is concave. Borell has proved in [8] that if a probability measure µ is log-concave and µ(h) < for every hyperplane H, then µ P n and its density f µ is log-concave. Note that if K is a convex body of volume in R n then the Brunn-Minkowski inequality implies that K is the density of a log-concave measure. If µ is a log-concave measure on R n with density f µ, we define the isotropic constant of µ by (2.) L µ := ( ) supx R n n f µ (x) [det Cov(µ)] 2n, f R n µ (x)dx where Cov(µ) is the covariance matrix of µ with entries x R (2.2) Cov(µ) ij := n i x j f µ (x) dx x R n i f µ (x) dx f R n µ (x) dx f R n µ (x) dx R n x j f µ (x) dx R n f µ (x) dx. Note that L µ is an affine invariant of µ and does not depend on the choice of the Euclidean structure. We say that a log-concave probability measure µ on R n is isotropic if it is centered and Cov(µ) is the identity matrix. Recall that if µ is a log-concave probability measure on R n and if q then the L q -centroid body Z q (µ) of µ is the symmetric convex body with support function ( ) /q (2.3) h Zq(µ)(y) := x, y q dµ(x). R n Observe that µ is isotropic if and only if it is centered and Z 2 (µ) = B n 2. From Hölder s inequality it follows that Z (µ) Z p (µ) Z q (µ) for all p q <. Conversely, using Borell s lemma (see [28, Appendix III]), one can check that (2.4) Z q (µ) c q p Z p(µ) 7

8 for all p < q. In particular, if µ is isotropic, then R(Z q (µ)) c 2 q. For any α and any θ S n we define the ψ α -norm of x x, θ as follows: { ( ( x, θ ) ) } α (2.5), θ ψα := inf t > 0 : exp dµ(x) 2, R t n provided that the set on the right hand side is non-empty. constant b α = b α (θ) in the direction of θ if we have, θ ψα b α, θ 2. We say that µ is a ψ α -measure with constant B α > 0 if sup θ S n, θ ψα, θ 2 B α. We say that µ satisfies a ψ α -estimate with From Borell s lemma it follows that every log-concave measure is a ψ -measure with constant C, where C is an absolute positive constant. From [29] and [30] one knows that the q-moments ( ) /q (2.6) I q (µ) := x q 2 dx, q ( n, + ) \ {0}, R n of the Euclidean norm with respect to an isotropic log-concave probability measure µ on R n are equivalent to I 2 (µ) = n as long as q n. Two main consequences of this fact are: (i) Paouris deviation inequality (2.7) µ({x R n : x 2 c 3 t n}) exp ( t n ) for every t, where c 3 > 0 is an absolute constant, and (ii) Paouris small ball probability estimate: for any 0 < ε < ε 0, one has (2.8) µ({x R n : x 2 < ε n}) ε c4 n, where ε 0, c 4 > 0 are absolute constants. The next theorem summarizes our knowledge on the mean width of Z q (µ). The first statement was proved by Paouris in [29], while the second one is E. Milman s Theorem.2. Theorem 2.. Let µ be an isotropic log-concave measure on R n. If q n, then (2.9) w(z q (µ)) q. Moreover, for all q [ n, n] we have (2.20) w(z q (µ)) c 5 q log 2 ( + q). The next theorem summarizes our knowledge on the volume radius of Z q (µ). The first statement follows from the results of [29] and [20], while the left hand-side in the second one was obtained in [24] and the right hand-side in [29]. Theorem 2.2. Let µ be an isotropic log-concave measure on R n. If q n then (2.2) Z q (µ) /n q/n, while if n q n then (2.22) c 6 L µ q/n Zq (µ) /n c 7 q/n. The reader may find a detailed exposition of the theory of isotropic log-concave measures in the book []. 8

9 3 Estimates for the Quermaßintegrals We start with the proof of Theorem.3. Recall that the equivalence E [ Q k (K N ) ] log N in the range n 2 N exp( n) was proved in [4] (see Theorem.). What is new is the right hand-side estimate in (.20). However, in [4] it was proved that E[Q k (K N )] w(z log N (K)) for the full range of N. So the result follows immediately by applying Theorem 2.. To prove Theorem.4 we will need Lemma 4.2 from [4] which holds true in the more general setting of isotropic log-concave random vectors. Lemma 3.. Let µ be an isotropic log-concave measure on R n. For every n 2 N exp(cn) and for every q log N and r, we have (3.) S n q hk N (θ) h q Z (θ) dσ n(θ) (c r) q q(µ) with probability greater than r q, where c > 0 is an absolute constant. Proof of Theorem.4. Let exp( n) N exp(n). Applying Hölder s inequality we get w(k N ) = h KN (θ) dσ n (θ) S ( n ( ) ) /p ( ( ) q /q p hkn (θ) hzq(µ)(θ) dσn (θ) dσ n (θ)) S n S h Zq(µ)(θ) n ( = w p Zq (µ) ) ( ( ) q ) /q hkn (θ) dσ n (θ), S h Zq(µ)(θ) n where p is the conjugate exponent of q. If we now choose q = log N n and use Lemma 3. we arrive at w(k N ) c rw p ( Zq (µ) ) with probability greater than r q. Since q = log N it follows that p < 2 and thus w p (Z q (µ)) is equivalent to w(z q (µ)) (see [6, Chapter 5]). Using this and applying Theorem.2 we conclude that w(k N ) c 2 r log N ( log log N ) 2 with probability greater than r log N. Choosing r = e we complete the proof of (.2). We can also give estimates on the volume radius of a random projection P F (K N ) of K N onto F in terms of n, k and N. In [4] it was shown that if n 2 N exp( n) then, a random K N satisfies with probability greater than N s the following: for every k n, (3.2) c 3 log N v.rad(pf (K N )) c 4 (s) log N with probability greater than e c5k with respect to the Haar measure ν n,k on. We extend this result to the case exp( n) N exp(n). For the proof we will use Theorem. from [3], which was already mentioned in the introduction. We formulate it in the more general setting of isotropic log-concave random vectors (the probability estimate in the statement makes use of [, Theorem 3.3]: if γ > and Γ : l n 2 l N 2 is the random operator Γ(y) = ( x, y,... x N, y ) defined by the vertices x,..., x N of K N then P( Γ : l n 2 l N 2 γ N) exp( c 0 γ N) for all N cγn see [3] for the details). 9

10 Fact 3.2. Let µ be an isotropic log-concave measure on R n and let x,..., x N be independent random vectors distributed according to µ, with N c n where c > is an absolute constant. Then, for all q c 2 log(n/n) we have that (3.3) K N c 3 Z q (µ) with probability greater than exp( c 4 N). Proof of Theorem.5. For the upper bound we use (.2) and Kubota s formula to get ( ω k P F (K N ) dν n,k (F ) ) /k c 6 (s) log N ( log log N ) 2 LK. Applying now Markov s inequality we get that with probability greater than t k with respect to the Haar measure ν n,k on we have ( ) /k PF (K N ) c 6 (s)t log N ( log log N ) 2. ω k Choosing t = e proves the result. For the lower bound integrating in polar coordinates and using Hölder s inequality we have PF (K N) (3.4) dν n,k (F ) = ω k S F h k P F (K N ) (θ)dσ F (θ) dν n,k (F ) = S F h k K N (θ) dσ F (θ) dν n,k (F ) ( ) k/n S F h n K N (θ) dσ F (θ) dν n,k (F ) ( = S h n n K N (θ) dσ n(θ) ( ) K k/n = N. ω n Apply now the Blaschke-Santaló inequality and the fact that K N c 7 Z log N (µ) (with probability greater than exp( c N) (notice that log N log N/n for the range of N we use)) to get (3.5) ( ) K k/n N ω n ( ) k/n ( ωn K N ) k/n ω n c 7 Z log N (µ) ) k/n. Since log N is greater than n we can apply the inequality Z log N (K) /n cl µ (log N)/n to arrive at (3.6) PF (K N ) ω k dν n,k (F ) ( c8 L µ log N ) k. Finally, we apply Markov s inequality and the reverse Santaló inequality of Bourgain and V. Milman [0] to complete the proof. 0

11 4 Mean outer radii For any convex body C in R n and any k n, the k-th mean outer radius of C is defined by (4.) Rk (C) = R(P F (C)) dν n,k (F ). Alonso-Gutiérrez, Dafnis, Hernández-Cifre and Prochno studied in [3] the order of growth of R k (K N ) as a function of N, n and k. Their main result is Theorem.6: If n N exp( n) then, for all k n and s > 0 one has { } (4.2) c (s) max k, log(n/n) R { } k (K N ) c 2 (s) max k, log N with probability greater than N s, where c (s), c 2 (s) are positive constants depending only on s. In this section we give an alternative (and simpler) proof of this result. We also extend the estimates to the range exp( n) N exp(n). Our approach is based on the next general fact, which is a standard application of concentration of measure on the Euclidean sphere (see [6, Section 5.7] for the details). If C is a symmetric convex body in R n then, for any k < n and any s > there exists a subset Γ n,k with measure greater than e cs2k such that the orthogonal projection of C onto any subspace F Γ n,k satisfies (4.3) R(P F (C)) w(c) + c 2 s k/nr(c), where c > 0, c 2 > are absolute constants. In fact, one has that the reverse inequality R(P F (C)) c max{w(c), k/nr(c)} holds for most F. To see this, first note that if x C and x 2 = R(C) then, for most F we have P F (x) 2 c k/n x 2, and hence R(P F (C)) c k/nr(c); integrating with respect to ν n,k we get R k (C) c k/nr(c). On the other hand, if k/nr(c) c w(c) for a small enough absolute constant 0 < c < then V. Milman s proof of Dvoretzky s theorem shows that most k-dimensional projections of C are isomorphic Euclidean balls of radius w(c), which implies that R k (C) cw(c). These observations lead to the next asymptotic formula. Proposition 4.. Let C be a symmetric convex body in R n. For any k n one has (4.4) Rk (C) w(c) + k/nr(c). We will exploit this formula for a random K N. Because of (4.4) we only need to estimate w(k N ) and R(K N ) for a random K N. This is done in Proposition 4.2 and Proposition 4.4 below. Essential ingredients are the deviation and small ball probability estimates (2.7) and (2.8) of Paouris, as well as Fact 3.2. We start with the case N exp( n). Proposition 4.2. If n 2 N exp( n) then, for any s, a random K N satisfies and with probability greater than max{n s, e cn }. c log N w(kn ) c 2 s log N c 3 n R(KN ) c 4 s n Proof. In the proof of Theorem.4 we saw that, for any n N exp(n), (4.5) w(k N ) c sw ( Z log N (µ) ) with probability greater than N s. Theorem 2. and (4.5) show that Assuming that N exp( n) we have that log N n; then (4.6) w(k N ) c 2 s log N

12 with probability greater than N s. For the lower bound we use Fact 3.2: we know that for all N c 3 n we have (4.7) K N c 4 Z log(n/n) (µ) with probability greater than exp( c 5 N). It follows that if N exp( n) then w(k N ) c 4 w(z log(n/n) (µ)) c 6 log(n/n) with probability greater than exp( c 7 N). For the radius of K N, applying (2.7) we see that, for any t 2, (4.8) R(K N ) = max j N x j 2 c 8 t n with probability greater than N exp ( t n) exp( (t ) n) N (t ). For the lower bound, if n 2 N exp( n) we use (2.8) to write ( ) Prob(R(K N ) ε 0 n) = Prob max x j 2 ε 0 n j N = [ µ({x R n : x 2 < ε 0 n}) ] N e c 9 nn, which shows that R(K N ) ε 0 n with probability greater than e c 9 nn. Remark 4.3. In fact, for the proof of the lower bound R(K N ) c n we do not really need the small ball probability estimate of Paouris. Lata la has proved in [22] that if µ is a log-concave probability measure on R n then, for any norm on R n and any 0 t one has (4.9) µ({x : x te µ ( x )}) Ct, where C > 0 is an absolute constant. If we assume that µ is isotropic then we easily see that E µ ( x 2 ) n, and hence, choosing a small enough absolute constant ε 0 we have by (4.9) that This information is enough for our purposes. µ({x R n : x 2 < ε 0 n}) e. Proof of Theorem.6. Let N exp( n). From (4.4) and Proposition 4.2 we get that K N probability greater than max{n s, e cn } the following: for any k n R k (K N ) = R(P F (K N )) dν n,k (F ) w(k N ) + k/nr(k N ) ( ) { } c log(n/n) + k/n n max log(n/n), k satisfies with and similarly, R k (K N ) = R(P F (K N )) dν n,k (F ) w(k N ) + k/nr(k N ) ( ) { } c 2 (s) log N + k/n n 2c 2 (s) max log N, k, as in [3]. The next proposition will allow us to handle the case exp( n) N exp(n). 2

13 Proposition 4.4. If exp( n) N exp(n) then, for any s, a random K N satisfies c L µ log N w(kn ) c 2 s log N(log log N) 2 and with probability greater than max{n s, e cn }. c 3 max{ n, R(Z log N (µ))} R(K N ) c 3 s log N Proof. Applying again (4.5) in the range exp( n) N exp(n) we have that (4.0) w(k N ) c 2 s log N(log log N) 2 from Theorem.2. For the lower bound we use again Fact 3.2, Urysohn s inequality and (2.22) from Theorem 2.2 to write w(k N ) c 4 w(z log N (µ)) c 4 ( Z log N (µ) / B2 n ) /n c 6 L µ log N with probability greater than exp( c 5 N). For the radius of K N we first use the estimate R(K N ) ct n from (4.8) with t s log N/ n to obtain the bound c log N with probability greater than N s. For the lower bound, we show that R(K N ) c n exactly as in the proof of Proposition 4.2, and we also use the bound R(K N ) R(Z log N (µ)). Using Proposition 4.4 and Proposition 4. as in the proof of Theorem.6, we arrive at the following estimate: Theorem 4.5. Let x,..., x N be independent random points distributed according to an isotropic log-concave measure µ on R n, and consider the random polytope K N := conv{±x,..., ±x N }. If exp( n) N exp(n) then, for any s and for all k n one has { c max L µ log N, k, k/nr(zlog N (µ))} R k (K N ) Cs max{ log N(log log N) 2, k/n log N} with probability greater than N s, where c, C > 0 are absolute constants. In full generality one cannot expect something significantly better: for example, if µ = µ n is the uniform measure on B n / B n then R(Z log N (µ n )) log N, and for large values of N (i.e. exponential in N) we get R k (K N ) k/n log N. On the other hand, if µ satisfies a ψ 2 estimate with constant b then we know that L µ C b (see [20]) and we also know that I n (µ) cb n (see [29]), which implies that w(k N ) R(K N ) C 2 b n. Moreover, Z log N (µ) b log NB2 n. Thus, in this case (which e.g. includes the case of the standard Gaussian measure) we get: Theorem 4.6. Let x,..., x N be independent random points distributed according to an isotropic log-concave measure µ on R n which satisfies a ψ 2 -estimate with constant b, and consider the random polytope K N := conv{±x,..., ±x N }. If n N exp(n) and s then K N satisfies with probability greater than N s { } (4.) c b max k, log(n/n) R { } k (K N ) c 2 (s)b max k, log N for all k n, where c 2 (s) is a positive constant depending only on s. 3

14 5 Entropy estimates and diameter of sections For every pair of convex bodies A and B in R n, the covering number N(A, B) of A by B is defined to be the smallest number of translates of B whose union covers A. A fundamental theorem of V. Milman states that there exists an absolute constant β > 0 such that every symmetric convex body K in R n has a linear image K which satisfies K = B n 2 and (5.) max { N( K, B n 2 ), N(B n 2, K), N( K, B n 2 ), N(B n 2, K ) } exp(βn). A convex body which satisfies the above is said to be in M-position with constant β. Pisier has offered in [3] a refined version of this result: for every 0 < α < 2 and every symmetric convex body K in R n there exists a linear image K α of K such that (5.2) max { N( K α, tb2 n ), N(B2 n, t K α ), N( K α, tb2 n ), N(B2 n, t K α) } ( ) c(α)n exp for every t, where c(α) depends only on α, and c(α) = O ( (2 α) α/2) as α 2. One says that K α is an α-regular M-position of K (we refer to [6, Chapter 8] and [32] for a detailed exposition of these results). In this section we will first show that if µ is an isotropic log-concave measure on R n then, for a considerably large range of values of N, a random K N is in α-regular M-position with α. To this end, it is convenient to set r N = log N: recall that if n 2 N exp( n) then v.rad(k N ) r N for a random K N (in the case N exp( n) one has the weaker estimate c L µ r N v.rad(k N ) c 2 r N ). We provide estimates for the covering numbers N(K N, tr N B2 n ) and N(r N B2 n, tk N ) for a random K N and for all t ; by the duality of entropy theorem of Artstein-Avidan, V. Milman and Szarek [7], these also determine the covering numbers N(r N KN, tbn 2 ) and N(B2 n, tr N KN ), thus completing the proof of the four required entropy estimates in (5.2). Proposition 5.. Let µ be an isotropic log-concave measure on R n. Then a random K N satisfies with probability greater than N the entropy estimate { cn log N(K N, tr N B2 n t if n 2 N exp( n) ) if exp( n) N exp(cn). cn log 4 n t 2 for every t, where c > 0 is an absolute constant. Proof. We simply recall that a random K N satisfies w(k N ) c log N rn for small N, and w(k N ) c 2 log N(log log N) 2 r N (log log N) 2 for large N, by Proposition 4.2 and Proposition 4.4 respectively. The bound for N(K N, tr N B n 2 ) is then a direct consequence of Sudakov s inequality log N(C, tb n 2 ) cn(w(c)/t) 2 which is true for every convex body C in R n and every t > 0 (see e.g. [6, Chapter 4]). We turn to estimates for the dual covering numbers N(r N B2 n, tk N ). We will make use of the following fact (see [6] and [, Proposition 9.2.8] or [5] for the stronger statement below): If µ is an isotropic log-concave measure on R n, then for any 2 q { q, } n log q n and for any t min c q we have 2 (5.3) log N ( qb n 2, tz q (µ) ) n(log q) 2 log t c 2, t where c, c 2 > 0 are absolute constants. Moreover, if q (n log n) 2/5 then (5.3) holds true for all t. Analogous estimates are available for larger values of q, but they are weaker and do not seem to be final; so, we prefer to restrict ourselves to the next case. t α 4

15 Proposition 5.2. Let µ be an isotropic log-concave measure on R n. Then, assuming that n 2 N exp ( (n log n) 2/5), we have that a random K N satisfies with probability greater than exp( c N) the entropy estimate log N(r N B2 n n(log n) 2 log( + t), tk N ) c 2 t for every t, where c, c 2 > 0 are absolute constants. Proof. It is an immediate consequence of the fact that K N exp( c N). Then, we clearly have c 3 Z log N (µ) with probability greater than log N(r N B n 2, tk N ) log N(r N B n 2, c 3 tz log N (µ)), and the result follows from (5.3). Proof of Theorem.7. By Proposition 5.2 By Proposition 5., since for a suitable absolute constant θ > 0, we have log N(r N B2 n, tk N ) c n log2 n log( + t). t N exp ( (n log n) 2/5) exp(θ n), log N(K N, tr N B n 2 ) cn t (here we can compensate for the extra factor θ in the exponent since for the proof of Proposition 5. we can use the fact that Z θ n (µ) θz n(µ)). Combining the above bounds we get the result. Remark 5.3. Following the reasoning of [4] one can also check that there exist absolute positive constants c, c 2, c 3 and c 4 so that for every 0 < t < a random K N satisfies with probability greater than N the next entropy estimates: (i) If n 2 N exp( n) then (5.4) (ii) If exp( n) N exp(n) then c n log c 2 t log N(K N, tr N B n 2 ) c 3 n log c 4 t, (5.5) c n log c 2 t log N(K N, t r N B2 n ) c 3 n log c ( ) 2 4 log log N, t where r N := v.rad(k N ) satisfies c 5 L µ r N r N c 6 r N. As an application we provide estimates for the average diameter of k-dimensional sections of a random K N. This parameter can be defined for any convex body C in R n and any k n as follows: (5.6) Dk (C) = R(C F ) dν n,k (F ). We shall use the next lemma that (in the case α = 2) can be essentially found in the article [26] of V. Milman (see also [, Lemma 9.2.5]): 5

16 Lemma 5.4. Let C be a symmetric convex body in R n and assume that (5.7) log N(C, tb n 2 ) γn t α for all t and some constants α > 0 and γ. Then, for every integer d < n, a subspace H G n,d satisfies ( (5.8) C H c α γn ) /α ( n ) log B d d H with probability greater than exp( c 2 d), where c, c 2 > 0 are absolute constants. From Proposition 5. we know that a random r N K N satisfies the assumption of Lemma 5.4 with γ if N exp( n) and γ log 4 n if N exp( n). Therefore, for any k < n we have that if N exp( n) then a k-dimensional section of K N has radius (5.9) R(K N F ) c log N n n k log while if exp( n) N exp(n) then the bound becomes (5.0) R(K N F ) c log N(log n) 2 n n k log ( ) n, n k ( ) n, n k both with probability greater than exp( c 2 (n k)), where c, c 2 > 0 are absolute constants. Proposition 4.2 and Proposition 4.4 we also know that a random K N has radius From R(K N ) c max{ n, log N}, and the same bound is clearly true for all its sections K N F. Therefore, if n exp( c 2 (n k)) (which is true provided that k < n c 3 log n) integration on shows that the bounds (5.9) and (5.0) hold for D k (K N ) as well. Taking into account the fact that D k (K N ) R k (K N ) we conclude the following. Proposition 5.5. Let µ be an isotropic log-concave measure on R n. Then a random K N satisfies with probability greater than N the following: (i) If n 2 N exp( n) then:. If k log N then D k (K N ) c log N. 2. If k log N then D { k, ( k (K N ) c min log N n n k log (ii) If exp( n) N exp(n) then: n n k. If k n(log log N) 4 / log N then D k (K N ) c 2 log N(log log N) If k n(log log N) 4 / log N then { k/n D k (K N ) c 2 min log N, log N(log n) 2 where c, c 2 > 0 are absolute constants. )}, n n k log ( )} n, n k Remark 5.6. An alternative way to estimate the average radius of K N F on for some values of k is given by the next theorem of Klartag and Vershynin from [2]: If k c n(m(c)/b(c)) 2, then (5.) c 2 M(C) ( ) /k R(C F ) k dν n,k (F ) c 3 M(C), 6

17 where c, c 2, c 3 > 0 are absolute constants. Note that a random K N satisfies K N Z 2 (µ) = B2 n and integration in polar coordinates combined with Hölder s inequality shows that M(K N ) v.rad(k N ). log N Therefore, we may apply (5.) to K N : for all k cn/ log N we have ( ) /k (5.2) Dk (K N ) R(C F ) k dν n,k (F ) c 3 M(C) c 4 log N. We pass now to lower bounds for D k (K N ). In fact, we will give a lower bound which is valid for the radius of every section K N F, F. We need the next lemma. Lemma 5.7. Let C be a symmetric convex body in R n and assume that (5.3) log N(B n 2, tc) γn t α for all t and some constants α > 0 and γ. Then, for every k < n and any subspace F we have (5.4) R(C F ) cαγ /α (k/n) /α. where c > 0 is an absolute constant. Proof. Let k < n and consider any F. By the duality of entropy theorem of S. Artstein-Avidan, V. Milman, and S. Szarek (see [7]) the projection P F (C ) of C onto F satisfies (5.5) N(P F (C ), tb F ) N(C, tb n 2 ) exp( γn k for every t. We apply Lemma 5.4 for the body P F (C ) (with γ = γn/k): there exists H G k, k/2 (F ) such that (5.6) P F (C ) H c α(γn/k) /α B H. Taking polars in H we see that P H (C F ) c α(k/γn) /α B H. Using the fact that for every symmetric convex body A in R k and every H G k,s we have M(A H) k/sm(a) (see [6, Chapter 5]) we get k t α ), w(c F ) = M((C F ) ) 2 M((C F ) H) = 2 w(p H (C F )) c 2 α(k/γn) /α. The same lower bound holds for R(C F ). From Proposition 5.2 we know that if e.g. n 2 N exp ( (n log n) 2/5) then a random K N satisfies with probability greater than exp( c N) the entropy estimate log N(B n 2, tr N K N) c 2 n(log n) 2 log( + t) t for every t, where c, c 2 > 0 are absolute constants. Notice that the interesting range for t is up to n (otherwise tr N K N contains B2 n ) so, we may apply Lemma 5.7 with C = r N K N, γ = log 3 n and α = to get: 7

18 Proposition 5.8. Let µ be an isotropic log-concave measure on R n. If n 2 N exp ( (n log n) 2/5) then a random K N satisfies with probability greater than exp( c N) the following: for every k < n and any subspace F, (5.7) R(K N F ) c log N k n log 3 n, where c > 0 is an absolute constant. The same bound holds for D k (K N ). Remark 5.9. The question to give an upper bound for M(K N ) seems open and interesting. Let us note that the analogous question for Z q (µ) is still open. The best known result appears in [5] (see also [6]): For any isotropic log-concave probability measure µ on R n and any 2 q q 0 := (n log n) 2/5 one has log q (5.8) M(Z q (µ)) C. 4 q This estimate does not seem to be optimal; note that since K N cz log N (µ) we also have log log N (5.9) M(K N ) C 4 log N for a random K N, at least in the range log N (n log n) 2/5. 6 Remarks on the isotropic constant In this last section we apply directly the method of Klartag and Kozma in order to estimate the isotropic constant L KN of a random K N. The starting point is the inequality (6.) K N 2/n nl 2 K N x 2 2 dx K N K N (it is well-known that this holds for any symmetric convex body in R n ; see e.g. [27] or [, Chapter 3]). Assuming that N exp( n) we know by (.6) that (6.2) K N /n c log(2n/n) n with probability greater than exp( c 2 n). We write F(K N ) for the family of facets of K N and we denote by [y,..., y n ] the convex hull of y,..., y n. Observe that, with probability equal to, all the facets of K N are simplices and that, for all j n, x j and x j cannot belong to the same facet of K N. Following [9, Lemma 2.5] one can show the next lemma. Lemma 6.. Let F,..., F M be the facets of K N. Then, (6.3) x 2 K N 2dx n K N n + 2 max s M u 2 F s 2du. F s Let y,..., y n R n and define F = [y,..., y n ]. Then, F = T ( n ) where n = [e,..., e n ] and T ij = y j, e i =: y ji. Assume that det T 0. Then, u 2 F 2du = F n T u 2 2du n 2 n n = n y ji u j du. n i= j= 8

19 Using the fact that (6.4) we see that (6.5) n (u j u j2 ) du = + δ j,j 2 n(n + ), n u 2 n n n F 2du = yji 2 + F n(n + ) i= j= j= y ji 2, from where one can conclude that (6.6) u 2 2 F 2du n(n + ) max ε y + + ε n y n 2 ε 2. j=± F Next we use a Bernstein type inequality (for a proof, see e.g. [6, Theorem 3.5.6]): Lemma 6.2. Let g,..., g n be independent random variables with E (g j ) = 0 on some probability space (Ω, µ). Assume that g j ψ A for all j n and some constant A > 0. Then, n ( { }) (6.7) P a j g j t 2 t t 2 exp c min A 2 a 2, 2 A a for every t > 0. j= We first fix θ S n and a choice of signs ε j = ±, and apply Lemma 6.2 to the random variables g j (y,..., y n ) = ε j y j, θ on Ω = (R n, µ) n. Since µ is isotropic, we know that g j ψ C. Choosing α = C 0 log(2n/n) we get (6.8) P { ε y + + ε n y n, θ > αn} 2 exp( cαn). Consider a /2-net N for S n with cardinality N 5 n. Then, with probability greater than exp( c 2 αn) we have (6.9) ε y + + ε n y n, θ αn for every θ N and every choice of signs ε j = ±. Using a standard successive approximation argument, and taking into account all 2 n possible choices of signs ε j = ±, we get that, with probability greater than exp( c 3 αn), (6.0) max ε j=± ε y + + ε n y n 2 C αn. Now, we use the fact that (6.) F(K N ) ( ) 2N exp(c 3 αn/2) n provided that C 0 is large enough. Therefore, taking also Lemma 6. and (6.6) into account, we see that, with probability greater than F(K N ) exp( c 3 αn) exp( c 4 αn), we have (6.2) x 2 K N 2dx C 2 α 2 = C 3 log 2 (2N/n), K N 9

20 where C 3 > 0 is an absolute constant. From (6.) and (6.2) we get (with probability greater than exp( c n)) (6.3) L 2 c 4 K N x 2 2 dx C 5 log(2n/n) log(2n/n) K N K N and hence L KN C 6 log(2n/n). Acknowledgement. The authors would like to acknowledge support from the programme APIΣTEIA II of the General Secretariat for Research and Technology of Greece. References [] R. Adamczak, A. E. Litvak, A. Pajor and N. Tomczak-Jaegermann, Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles, J. Amer. Math. Soc. 23 (200), No. 2, [2] D. Alonso-Gutiérrez, On the isotropy constant of random convex sets, Proc. Amer. Math. Soc. 36 (2008), [3] D. Alonso-Gutiérrez, N. Dafnis, M. A. Hernandez Cifre and J. Prochno, On mean outer radii of random polytopes, Indiana Univ. Math. J. 63 (204), no. 2, [4] D. Alonso-Gutiérrez, A. E. Litvak and N. Tomczak-Jaegermann, On the isotropic constant of random polytopes, J. Geom. Anal. (to appear). [5] D. Alonso-Gutiérrez and J. Prochno, On the Gaussian behavior of marginals and the mean width of random polytopes, Proc. Amer. Math. Soc. 43 (205), no. 2, [6] S. Artstein-Avidan, A. Giannopoulos and V. D. Milman, Asymptotic Geometric Analysis, Part I, Amer. Math. Soc., Mathematical Surveys and Monographs 202 (205). [7] S. Artstein, V. Milman and S. Szarek, Duality of metric entropy, Annals of Math. 59 (2004), [8] C. Borell, Convex measures on locally convex spaces, Ark. Mat. 2 (974), [9] J. Bourgain, On the distribution of polynomials on high dimensional convex sets, Geom. Aspects of Funct. Analysis (Lindenstrauss-Milman eds.), Lecture Notes in Math. 469 (99), [0] J. Bourgain and V. D. Milman, New volume ratio properties for convex symmetric bodies in R n, Invent. Math. 88 (987), [] S. Brazitikos, A. Giannopoulos, P. Valettas and B-H. Vritsiou, Geometry of isotropic convex bodies, Amer. Math. Soc., Mathematical Surveys and Monographs 96 (204). [2] N. Dafnis, A. Giannopoulos and O. Guédon, On the isotropic constant of random polytopes, Advances in Geometry 0 (200), [3] N. Dafnis, A. Giannopoulos and A. Tsolomitis, Asymptotic shape of a random polytope in a convex body, J. Funct. Anal. 257 (2009), [4] N. Dafnis, A. Giannopoulos and A. Tsolomitis, Quermaßintegrals and asymptotic shape of random polytopes in an isotropic convex body, Michigan Mathematical Journal 62 (203), [5] A. Giannopoulos and E. Milman, M-estimates for isotropic convex bodies and their L q centroid bodies, in Geometric Aspects of Functional Analysis, Lecture Notes in Mathematics 26 (204), [6] A. Giannopoulos, P. Stavrakakis, A. Tsolomitis and B-H. Vritsiou, Geometry of the L q-centroid bodies of an isotropic log-concave measure, Trans. Amer. Math. Soc. 367 (205), no. 7, [7] M. Hartzoulaki, Probabilistic methods in the theory of convex bodies. PhD thesis, University of Crete, March [8] B. Klartag, On convex perturbations with a bounded isotropic constant, Geom. Funct. Anal. 6 (2006), [9] B. Klartag and G. Kozma, On the hyperplane conjecture for random convex sets, Israel J. Math. 70 (2009),

21 [20] B. Klartag and E. Milman, Centroid Bodies and the Logarithmic Laplace Transform A Unified Approach, J. Funct. Anal. 262 (202), [2] B. Klartag and R. Vershynin, Small ball probability and Dvoretzky theorem, Israel J. Math. 57 (2007), [22] R. Lata la, On the equivalence between geometric and arithmetic means for log-concave measures, Convex geometric analysis (Berkeley, CA, 996), Math. Sci. Res. Inst. Publ., 34, Cambridge Univ. Press, Cambridge (999), [23] E. Lutwak and G. Zhang, Blaschke-Santaló inequalities, J. Differential Geom. 47 (997), 6. [24] E. Lutwak, D. Yang and G. Zhang, L p affine isoperimetric inequalities, J. Differential Geom. 56 (2000), 32. [25] E. Milman, On the mean width of isotropic convex bodies and their associated L p-centroid bodies, Int. Math. Res. Not. IMRN (205), no., [26] V. D. Milman, A note on a low M -estimate, in Geometry of Banach spaces, Proceedings of a conference held in Strobl, Austria, 989 (P.F. Muller and W. Schachermayer, Eds.), LMS Lecture Note Series, Vol. 58, Cambridge University Press (990), [27] V. D. Milman and A. Pajor, Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space, Lecture Notes in Mathematics 376, Springer, Berlin (989), [28] V. D. Milman and G. Schechtman, Asymptotic Theory of Finite Dimensional Normed Spaces, Lecture Notes in Mathematics 200 (986), Springer, Berlin. [29] G. Paouris, Concentration of mass in convex bodies, Geometric and Functional Analysis 6 (2006), [30] G. Paouris, Small ball probability estimates for log-concave measures, Trans. Amer. Math. Soc. 364 (202), [3] G. Pisier, A new approach to several results of V. Milman, J. Reine Angew. Math. 393 (989), 5 3. [32] G. Pisier, The volume of convex bodies and Banach space geometry, Cambridge Tracts in Mathematics 94 (989). [33] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Second expanded edition. Encyclopedia of Mathematics and Its Applications 5, Cambridge University Press, Cambridge, 204. MSC: Primary 52A2; Secondary 46B07, 52A40, 60D05. Apostolos Giannopoulos: Department of Mathematics, University of Athens, Panepistimioupolis 57-84, Athens, Greece. apgiannop@math.uoa.gr Labrini Hioni: Department of Mathematics, University of Athens, Panepistimioupolis 57-84, Athens, Greece. lamchioni@math.uoa.gr Antonis Tsolomitis: Department of Mathematics, University of the Aegean, Karlovassi , Samos, Greece. atsol@aegean.gr 2