A remark on the slicing problem

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1 A remark on the slicing problem A. Giannopoulos, G. Paouris and B-H. Vritsiou Abstract The purpose of this article is to describe a reduction of the slicing problem to the study of the parameter I 1(, Z q ()) =, x L q()dx. We show that an upper bound of the form I 1(, Z q ()) C 1q s nl, with 1/ s 1, leads to the estimate L n 4 C n logn, q 1 s where L n := max{l : is an isotropic convex body in R n }. 1 Introduction A convex body in R n is called isotropic if it has volume = 1, it is centered, i.e. its center of mass is at the origin, and if its inertia matrix is a multiple of the identity. The last property is equivalent to the existence of a constant L > 0 such that (1.1) x, θ dx = L for every θ in the Euclidean unit sphere S n 1. It is not hard to see that for every convex body in R n there exists an affine transformation T of R n such that T () is isotropic. Moreover, this isotropic image is unique up to orthogonal transformations; consequently, one may define the isotropic constant L as an invariant of the affine class of. The isotropic constant is closely related to the hyperplane conjecture (also known as the slicing problem) which asks if there exists an absolute constant c > 0 such that max θ c for every convex body of volume 1 in R n with θ S n 1 center of mass at the origin. This is because, by Brunn s principle, for any convex body in R n and any θ S n 1, the function t (θ 1 + tθ) n 1 is concave on its support, and this is enough to show that (1.) x, θ dx θ. Using this relation one can check that an affirmative answer to the slicing problem is equivalent to the following statement: There exists an absolute constant C > 0 1

2 such that L C for every convex body. We refer to the article [13] of Milman and Pajor for background information about isotropic convex bodies. It is known that L L B n c > 0 for every convex body in R n (we use the letters c, c 1, C etc. to denote absolute constants). In the opposite direction, let us write L n for the maximum of all isotropic constants of convex bodies in R n, (1.3) L n := max{l : is isotropic in R n }. Bourgain first proved in [4] that L n c 4 n logn and, a few years ago, lartag [8] obtained the estimate L n c 4 n (see also [9] for a second proof of this bound). The purpose of this article is to describe a reduction of the slicing problem (or, equivalently, the question whether L n can be bounded by a quantity independent of the dimension n), to the study of the parameter (1.4) I 1 (, Zq ()) =, x Lq()dx for isotropic convex bodies. Generally, if is a centered convex body of volume 1 in R n, then for every symmetric convex body C in R n and for every q ( n, ), q 0, we define ( (1.5) I q (, C) := x q C dx ) 1/q. The notation I 1 (, Z q ()) is then justified by the fact that, x Lq() is the norm induced on R n by the polar body Z q () of the L q -centroid body of (see the next section for background information on L q -centroid bodies). Our reduction can be viewed as a continuation of Bourgain s approach to the slicing problem in [4]: the bound O( 4 n logn) followed from the inequality (1.6) nl I 1 (, (T ()) ), after obtaining an upper bound for the quantity I 1 (, (T ()) ), where T SL(n) is a symmetric, positive definite matrix such that the mean width of T () satisfies the estimate w(t ()) c n logn (the existence of such a position for is guaranteed by Pisier s estimate on the norm of the Rademacher projection; see [19]). In Section 4 we prove the following statement: Theorem 1.1. There exists an absolute constant ρ (0, 1) with the following property: given κ, τ 1, for every n n 0 (τ) and every isotropic convex body in R n which satisfies the following entropy estimate: (1.7) log N(, tb n ) κn log n t we have that, if q satisfies for all t τ n logn, (1.8) q ρ n and I 1 (, Z q ()) ρnl,

3 then { n (1.9) L Cκ q log n max 1, I 1(, Zq } ()) qnl. Theorem 1.1 can lead to an upper bound for L n, provided that there exist (κ, τ)-regular isotropic convex bodies in R n, i.e. bodies which satisfy the entropy estimate (1.7) for a pair of constants κ, τ, and at the same time have maximal isotropic constant, i.e. L L n. The existence of such bodies is essentially established by [5, Theorem 5.7]. In Section 5 we give a self-contained proof of this fact; see Theorem 5.1. Observe that, for every isotropic convex body in R n, we have that both conditions in (1.8) are satisfied with q =, since I 1 (, Z ()) nl. Therefore, Theorem 1.1 will give us that (1.10) L C 1 n log n for any such body which is regular. Theorem 5.1 then guarantees that, for some absolute constants κ, τ and δ > 0, there exists a (κ, τ)-regular isotropic convex body in R n with L δl n, and hence (1.10) leads us to Bourgain s bound again: L n C 4 n logn. However, the behaviour of I 1 (, Z q ()) may allow us to use much larger values of q. In Section 3 we discuss upper and lower bounds for this quantity. For every isotropic convex body in R n we have some simple general estimates: (i) For every q n, c 1 max { nl, qn, R(Z q ())L } I1 (, Z q ()) c q nl. (ii) If q n, then c 1 max { nl, qnl } I1 (, Z q ()) c q nl. Any improvement of the exponent of q in the upper bound I 1 (, Zq ()) cq nl would lead to an estimate L n Cn α with α < 1 4. It seems plausible that one could even have I 1 (, Zq ()) c qnl, at least when q is small, say q n. Some evidence is given by the following facts: (iii) If is an unconditional isotropic convex body in R n, then for all q n. c 1 qn I1 (, Z q ()) c qn logn (iv) If is an isotropic convex body in R n then, for every q n, there exists a set A q O(n) with ν(a q ) 1 e q such that I 1 (, Z q (U())) c 3 qn L for all U A q. 3

4 The proofs of (i)-(iv) are given in Section 3. We can make a final observation about the reduction of Theorem 1.1 on the basis that there exist (κ, τ)-regular isotropic convex bodies in R n with L δl n (where κ, τ, δ > 0 are absolute constants) which, at the same time, have small diameter : they satisfy γ nl B n, where γ > 0 is an absolute constant (see Theorem 5.9). In Section 6, we show that then it is enough to study the parameter I 1 (, Z q ()) within the class I sd of isotropic convex bodies which are O(γ)- close to the Euclidean ball D n of volume 1 and have uniformly bounded isotropic constant. The precise statement which we prove is the following: if we have an isotropic symmetric convex body in R n satisfying γ nl B n, then we can find an isotropic symmetric convex body C such that L C c 1, c D n C c 3 γd n, and (1.11) I 1 (, Z q ()) qnl c 4 I 1 (C, Z q (C)) qn for all 1 q n, where c 1, c, c 3, c 4 > 0 are absolute constants. Notation and preliminaries We work in R n, which is equipped with a Euclidean structure,. We denote the corresponding Euclidean norm by, and write B n for the Euclidean unit ball, and S n 1 for the unit sphere. Volume is denoted by. We write ω n for the volume of B n and σ for the rotationally invariant probability measure on S n 1. We also denote the Haar measure on O(n) by ν. The Grassmann manifold G n,k of k-dimensional subspaces of R n is equipped with the Haar probability measure µ n,k. Let k n and F G n,k. We will denote the orthogonal projection from R n onto F by P F. We also define B F := B n F and S F := S n 1 F. The letters c, c, c 1, c 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 1, c > 0 such that c 1 a b c a. Also if, L R n we will write L if there exist absolute constants c 1, c > 0 such that c 1 L c. A convex body in R n is a compact convex subset C of R n with nonempty interior. We say that C is symmetric if x C implies that x C. We say that C is centered if it has center of mass at the origin, i.e. x, θ dx = 0 for every C θ S n 1. The support function of a convex body C is defined by (.1) h C (y) := max{ x, y : x C}, and the mean width of C is (.) w(c) := h C (θ)σ(dθ). S n 1 For each < q <, q 0, we define the q-mean width of C by ( ) 1/q (.3) w q (C) := h q C (θ)σ(dθ). S n 1 4

5 The radius of C is the quantity R(C) := max{ x : x C}. Also, if the origin is an interior point of C, the polar body C of C is defined as follows: (.4) C := {y R n : x, y 1 for all x C}. Finally, we write C for the homothetic image of volume 1 of a convex body C R n, i.e. C := C. C 1/n Recall that if A and B are nonempty sets in R n, then 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. In this paper, B will usually be a multiple of the Euclidean ball: in those cases we also require that the centres of the translates of B are taken from the set A; one can easily check that this additional requirement does not crucially affect our entropy estimates..1 L q -centroid bodies Let be a convex body of volume 1 in R n. For every q 1 and every y R n we set ( 1/q (.5) h Zq()(y) := x, y dx) q. The L q -centroid body Z q () of is the centrally symmetric convex body with support function h Zq(). Note that is isotropic if and only if it is centered and Z () = L B n. Also, if T GL(n) with det T = ±1, then Z p (T ()) = T (Z p ()). From Hölder s inequality it follows that Z 1 () Z p () Z q () Z () for all 1 p q, where Z () = conv{, }. Using Borell s lemma (see [15, Appendix III]), one can check that inverse inclusions also hold: (.6) Z q () β 1 qz 1 (), and more generally, (.7) Z q () β q p Z p() for all 1 p < q. In particular, if is isotropic, then R(Z q ()) β 1 ql. One can also check that if is centered, then Z q () β 3 for all q n (see [16]). All the constants β i, β j that appear in this section are absolute positive constants which may be used again in the arguments of the next sections. Let C be a symmetric convex body in R n and let C denote the norm induced on R n by C. The parameter k (C) is the largest positive integer k n with the property that the measure of F G n,k for which we have 1 w(c)b F P F (C) w(c)b F is greater than n n+k. It is known that (.8) β 4 n w(c) R(C) k (C) β 5 n w(c) R(C). 5

6 The q-mean width w q (C) is equivalent to w(c) as long as q k (C). Litvak, Milman and Schechtman proved in [11] that, for every symmetric convex body C in R n, (i) If 1 q k (C) then w(c) w q (C) β 6 w(c). (ii) If k (C) q n then β 7 q/n R(C) wq (C) β 8 q/n R(C). Let be a centered convex body of volume 1 in R n. Recall that, for every symmetric convex body C in R n and for every q ( n, ), q 0, we define ( (.9) I q (, C) := x q C dx ) 1/q. When C = B n, we write I q () := I q (, B n ) for simplicity. In [17] and [18] it is proved that for every 1 q n/, (.10) I q () n/q w q (Z q ()) and I q () n/q w q (Z q ()). The parameter q () is also defined by (.11) q () := max{q n : k (Z q ()) q}. Then, the main result of [18] states that, for every centered convex body of volume 1 in R n, one has I q () I q () for every 1 q q (). In particular, for all q q () one has I q () β 9 I (). If is isotropic, one can check that q () c n, where c > 0 is an absolute constant (for a proof, see [17]). Therefore, (.1) I q () β 10 nl for every q n. In particular, from (.10) and (.1) we see that w(z q ()) w q (Z q ()) ql for all q n.. The bodies B q (, F ) Another family of convex bodies associated with a centered convex body R n was introduced by Ball in [1] (see also [13]): to define them, let us consider a k- dimensional subspace F of R n and its orthogonal subspace E. For every φ F \{0} we denote by E + (φ) the halfspace {x span{e, φ} : x, φ 0}. Ball proved that, for every q 0, the function ( ) 1 (.13) φ φ 1+ q q+1 q+1 x, φ q dx E + (φ) is the gauge function of a convex body B q (, F ) on F. Several properties of these bodies can be found in [1], [13] and also in [17], [18]. In Section 5, we will make use of only two of those: 6

7 (i) Let R n be isotropic, let 1 k < n and let F G n,k. Then the body B k+1 (, F ) is almost isotropic, namely it has (by definition) volume 1, and we can write B k+1 (, F ) T (B k+1 (, F )) where T (B k+1 (, F )) is an isotropic (in the regular sense) linear image of B k+1 (, F ). In addition, (.14) F 1/k L B k+1 (,F ) L. (ii) Let, F and k < n be as above and consider any p [1, k]. Then (.15) Z p ( Bk+1 (, F ) ) F 1/k P F (Z p ())..3 Two related lemmas We close this section with two lemmas that will be used in the sequel; they reveal some properties of the support function of the L q -centroid bodies of a convex body with respect to subsets or certain integrals of maxima. Lemma.1. Let be a convex body of volume 1 in R n, and consider any points z 1, z,..., z N R n. If q 1 and p max{logn, q}, then (.16) ( ) 1/q max x, z i q dx β 1 max h Z 1 i N 1 i N p()(z i ), where β 1 > 0 is an absolute constant. Proof. Let p max{logn, q} and θ S n 1. Markov s inequality shows that (.17) {x : x, θ e 3 h Zp()(θ)} e 3p. Since x x, θ is a seminorm, from Borell s lemma (see [15, Appendix III]) we get that ( ) e (.18) {x : x, θ e 3 th Zp()(θ)} (1 e 3p 3p t+1 ) 1 e 3p e pt for every t 1. We set S := e 3 max h Z 1 i N p()(z i ). Then, for every t 1 we have that {x : max 1 i N x, z i St} N {x : x, z i e 3 th Zp()(z i )} i=1 Ne pt. 7

8 It follows that max x, z i q dx = q 1 i N 0 s q 1 {x : max x, z i s} ds 1 i N S q + q s q 1 {x : max x, z i s} ds S 1 i N ) = S (1 q + q t q 1 {x : max x, z i St} dt 1 1 i N ) S (1 q + qn t q 1 e pt dt 1 ( = S q 1 + qn ) p q t q 1 e t dt p ( S q 1 + qn ) p q e p p q (3S) q, where we have also used the fact that, for every p q 1, (.19) p t q 1 e t dt e p p q. This finishes the proof (with β 1 = 3e 3 ). Remark.. It is a well-known fact (see e.g. [6, Proposition.5.1]) that (.0) max x, z i dx C 1 logn max h Z 1 i N 1 i N 1()(z i ). Through a variant of the argument in [6], and using (.19) as well, one can also show that for q logn, ( ) 1/q (.1) max x, z i q dx C logn max h Z 1 i N 1 i N 1()(z i ). Now, both inequalities can be directly deduced from Lemma.1 combined with (.6), however the lemma provides additional information on how well the quantities ( max i x, z i q dx ) 1/q ( and maxi x, z i q dx ) 1/q maxi h Zq()(z i ) can be compared: for q logn, using also (.7), we have that (.) ( ) 1/q max x, z i q dx β 1 max h Z logn ()(z i ) C logn 1 i N 1 i N q whereas for q > logn, ( ) 1/q ( 1/q (.3) max x, z i q dx max x, z i dx) q. 1 i N 1 i N max h Z 1 i N q()(z i ), 8

9 We now turn our attention to the L q -centroid bodies of subsets of. Lemma.3. Let be a convex body of volume 1 in R n and let 1 q, r n. There exists an absolute constant β > 0 such that if A is a subset of with A 1 e β q, then (.4) Z p () Z p (A) for all 1 p q. Also, for the opposite inclusion, it suffices to have A r conclude that to (.5) Z p (A) Z p () for all r p n. Proof. Let θ S n 1. Note that ( 1/p (.6) h (θ) = Zp(A) x, θ dx) p 1 = A A 1 p + 1 n ( 1/p x, θ dx) p. A We first prove (.5): since A and assuming that A r, we have ( h Zp()(θ) = r p r n 1/p ( ) 1/p x, θ dx) p x, θ p dx A ( ) 1/p x, θ p dx 1 A h Z (θ) p(a) for all r p n. On the other hand, assuming that A 1 e β q and using the fact that, θ p β, θ p, we have x, θ p dx = x, θ p dx + x, θ p dx A \A ( ) 1/ A 1+ p n x, θ p dx + \ A 1/ x, θ p dx A x, θ p dx + e β q/ (β ) p x, θ p dx A x, θ p dx + 1 x, θ p dx A for every p q, if β > 0 is chosen large enough. This proves (.4). 3 Simple estimates for I 1 (, Z q ()) In this section we give some upper and lower bounds for I 1 (, Z q ()) which hold true for every isotropic convex body in R n and any 1 q n. In fact, our 9

10 arguments are quite direct and make use of estimates for simple parameters of the bodies Z q (), such as their radius or their volume, so that it is straightforward to reach analogous upper and lower bounds for I 1 (, Z q (M)) in the more general case when and M are not necessarily the same isotropic convex body. Since h Zq()(x) R(Z q ()) x, we have that (3.1) I 1 (, Zq ()) R(Z q ()) x dx R(Z q ()) nl, which, in combination with the fact that R(Z q ()) β 1 ql (a direct consequence of (.6)), leads to the bound (3.) I 1 (, Z q ()) β 1 q nl. More generally, we have that (3.3) I 1 (, Zq (M)) R(Z q (M)) x dx β 1 q nl L M. However, in the case that M is an orthogonal transformation of, the next lemma shows that the average of the quantity I 1 (, Z q (M)) can be bounded much more effectively than in (3.3). Lemma 3.1. Let be a centered convex body of volume 1 in R n. q n, (3.4) ( I1( q, Z q (U()) ) ) 1/q dν(u) C q/niq (), O(n) For every where C > 0 is an absolute constant. Proof. We write I1( q, Z q (U()) ) dν(u) O(n) = = = O(n) O(n) = c q n,q I q q (, Z q (U()) ) dν(u) = c q n,qi q q O(n) U() x, y q dy dx dν(u) x, Uy q dν(u) dy dx y q x, θ q dσ(θ) dy dx S n 1 y q x q dy dx (), where c n,q q/n. 10

11 Recall that in the case that is isotropic, one has from [17] that I q () max{ nl, R(Z q ())}. Then, Lemma 3.1 shows that, for every q n, ( (3.5) I1( q, Z q (U()) ) dν(u) O(n) ) 1/q C 1 max{ qn, q q/n} L, where C 1 > 0 is an absolute constant. Therefore, for every q n, there exists a set A q O(n) with ν(a q ) 1 e q such that I 1 (, Z q (U()) ) C qn L for all U A q. It is thus conceivable that there are properties of the bodies Z q () which we can exploit to also bound I 1 (, Z q ()) more effectively than in (3.1) and (3.). We now pass to lower bounds; we will present three simple arguments. For the first one we do not have to assume that or M are in the isotropic position, only that they are centered and have volume 1: from [13, Corollary..a] we have that (3.6) I 1 (, Zq (M)) = h Zq(M)(x) dx n n + 1 Then, by the Blaschke Santaló inequality, we get that 1 Z q (M) 1/n. (3.7) I 1 (, Z q (M)) c 1 n Z q (M) 1/n c qnlm for all q n, because Z q (M) 1/n c 3 q/n LM for this range of values of q by a recent result of lartag and E. Milman (see [9]). When n q n, we have the weaker lower bound Z q (M) 1/n c 4 q/n, which is due to Lutwak, Yang and Zhang (see [1]). It follows that I 1 (, Zq (M)) c 5 qn for this range of values of q. For the second argument, we require that is isotropic and we write (3.8) I 1 (, Zq (M)) = h Zq(M)(x) dx = max x, z dx z Z q(m) max z Z q(m) = c R(Z q (M))L. x, z dx c max z L z Z q(m) Finally, if M is isotropic as well, we can use Hölder s inequality to get (3.9) I 1 (, Zq (M)) = h Zq(M)(x) dx h Z(M)(x) dx = x L M dx c nl L M. All the estimates presented above are gathered in the next proposition. Proposition 3.. Let and M be isotropic convex bodies in R n. q n, For every (3.10) c 1 max { nl L M, qn, R(Z q (M))L } I1 (, Z q (M)) c q nl L M. 11

12 In addition, if q n then (3.11) c 1 max { nl L M, qnl M } I1 (, Z q (M)) c q nl L M. The situation is more or less clear in the unconditional case. Recall that a convex body in R n is called unconditional if it is symmetric with respect to all coordinate hyperplanes (for some orthonormal basis of R n ). Then, it is easy to check that one can bring to the isotropic position by applying an operator which is diagonal with respect to this basis. It is also not hard to prove that the isotropic constant of satisfies L 1. The upper bound follows from the Loomis Whitney inequality; see also []. It is known (from [3]) that, for every q, one has h Zq()(y) c qn y, where c > 0 is an absolute constant. This leads us to the estimates (3.1) c 1 qn I1 (, Zq ()) c qn x dx c qn logn for all q n (the same estimates hold true for the quantity I 1 (, Z q (M)) when M is too an unconditional isotropic convex body). 4 The reduction Let κ, τ > 0. Throughout this paper, we say that an isotropic convex body in R n is (κ, τ)-regular if (4.1) log N(, tb n ) κn log n t for all t τ n logn. The purpose of this section is to present a reduction of the slicing problem to the study of the quantity I 1 (, Zq ()) for (κ, τ)-regular isotropic convex bodies: we show that any upper bound for I 1 (, Zq ()) immediately leads to an upper bound for the isotropic constant L of a regular convex body. Note that the dependence seems to be nontrivial, in the sense that using the simple estimates of Section 3 we can already recover the currently known bound for L with a loss of a logarithmic factor, while a small (although not necessarily easy) improvement to those estimates will also result in new bounds for L. In a sense, we will have fully presented our reduction by the end of the next section, where we provide a self-contained proof of the fact that there exist regular isotropic convex bodies in R n with L L n. First, let us see how the quantity I 1 (, Zq ()) and the isotropic constant of a regular convex body are connected. Theorem 4.1. There exists an absolute constant ρ (0, 1) with the following property: given κ, τ 1, for every n n 0 (τ) and every (κ, τ)-regular isotropic convex body in R n we have that, if q satisfies (4.) q ρ n and I 1 (, Z q ()) ρnl, 1

13 then { n (4.3) L Cκ q log n max 1, I 1(, Zq } ()) qnl. Proof. We define a convex body W in R n, setting (4.4) W := {x : h Zq()(x) C 1 I 1 (, Z q ())}, where C 1 = e β and β > 0 is the constant which appears in Lemma.3. From Markov s inequality we have that W 1 e β and also trivially that W 1 q (as long as β 1). Then we set (4.5) 1 := W. Applying both cases of Lemma.3 to the set W with p =, we see that (4.6) This implies that (4.7) 1 4 L = 1 4 for every θ S n 1, and hence (4.8) We also have nl 4 1 Z ( 1 ) Z () Z ( 1 ). x, θ dx x, θ dx 4 1 i=1 x, θ dx = 4L n x, e i dx = 1 x dx 4nL. 1 (4.9) 1 = W 1/n W W, thus for every x 1 we have x/ W, and using (.5) of Lemma.3 again, we can write (4.10) h Zq( 1)(x) h Zq()(x) = 4h Zq()(x/) 4C 1 I 1 (, Z q ()). Finally, (4.11) log N( 1, tb n ) log N(, tb n ) 4κn log n t, for all t τ n logn. We now write (4.1) nl 4 x dx 4 1 max x, z dx. z

14 (4.11) tells us that for every t τ n logn, we can find z 1,..., z Nt 1 such that 1 Nt ( (z i + tb n ), and N t exp 4κn log n t ). It follows that i=1 (4.13) max z 1 x, z max x, z i + max x, w = 1 i N t w tb n max x, z i + t x, 1 i N t and hence (4.14) nl max x, z i dx + 4t x dx 1 i N t 1 max x, z i dx + 8t nl. 1 i N t We choose (4.15) t 0 = 16C κ max { 1, I 1(, Z q ()) qnl } n 3/ q log n, where C = 16C 1 β β 1 with β the constant appearing in (.7) and β 1 the constant from Lemma.1. With this choice of t 0, we have n (4.16) t 0 16C κ q n log n 16C κ n log n, ρ as long as q satisfies (4.), and (4.17) t 0 16C κ I 1(, Z q ()) ql From (4.16) it is clear that (4.18) t 0 16C κ n log n ρ n log n. 4τ n logn, provided that n n 0 (τ, κ, ρ), so the argument above, leading up to (4.14), remains valid for t = t 0. We also set p 0 := 4κn log n. Observe that p t 0 q (as 0 long { as q is assumed } to satisfy (4.)), if ρ is chosen properly: indeed, we have max 1, I1(,Z q ()) qnl ρ n/q, and hence (4.19) t 0 16C κρ n log n. q If we choose ρ < 1/(4C ), then we have (4.0) p 0 = 4κn log n t 0 4κn q log n 16C κρn log n = q q. 4C ρ 14

15 We therefore see that, using Lemma.1 with q = 1, we can write (4.1) p 0 max x, z i dx β 1 max h Zp0 ( 1 i N t0 1 i N t0 1)(z i ) β 1 β max h Zq( q 1 i N t0 1)(z i ). 1 Combining the above with (4.14), (4.10) and the definition of C, we get (4.) nl C p 0 q I 1(, Z q ()) + 8t 0 nl. Also, from (4.17) and the definition of p 0, we have p 0 (4.3) C q I 1(, Zq ()) = 4C κi 1 (, Zq ()) qt n log n nl. Therefore, (4.) becomes (4.4) nl C 3 t 0 nl. This gives us that (4.5) L C 4 t 0 n { = Cκ max 1, I 1(, Zq } ()) n qnl q log n, as we desired. 5 Regular convex bodies with maximal isotropic constant Recall that L n := max{l : is isotropic in R n }. In order to be able to use the argument of the previous section to bound L n, we need to establish the existence of (κ, τ)-regular convex bodies, namely bodies satisfying (4.1), whose isotropic constant is as close to L n as possible. The following theorem, formulated in the more general setting of log-concave measures, was proven in [5]. Theorem 5.1. There exist absolute constants κ, τ and δ > 0 such that, for every n N, there exists an isotropic convex body in R n with the following properties: (i) L δl n. (ii) log N(, tb n ) κn log n t, for all t τ n logn. For the reader s convenience, we will give an outline of the proof in the setting of convex bodies. First, we recall the following theorem by Pisier which will be used in several steps of the argument (see [19] for a proof in the symmetric case; this can easily be extended to the general case): 15

16 Theorem 5.. Let be a centered convex body of volume 1 in R n. For every α (0, ) there exists an ellipsoid E α with E α = 1 such that, for every t 1, (5.1) log N(, te α ) κ(α) t α n, where κ(α) > 0 is a constant depending only on α. Remark 5.3. One can take κ(α) κ1 α, where κ 1 > 0 is an absolute constant. An ellipsoid E α which satisfies (5.1) is called an α-regular M-ellipsoid for. Secondly, let us gather some useful facts about ellipsoids in R n that we are going to need for the proof of Theorem 5.1 (proofs for these facts can be found in [5], [10] and []). Lemma 5.4. Let E be an ellipsoid in R n, then E = T (B n ) for some T GL(n). We denote the eigenvalues of the matrix T T by λ 1 λ n > 0 (recall that T T is a symmetric, positive definite matrix). Then, for all 1 k n 1, (5.) max F G n,k E F = and max P F (E) = ω k F G n,k (5.3) min E F = min P F (E) = ω k F G n,k F G n,k k i=1 n λ i i=n k+1 Also, if the dimension n is even, we can find a subspace F G n,n/ such that P F (E) = λ n/ B F (= λ n/ B n F ). In view of the last part of Lemma 5.4, we choose to restrict ourselves to the cases that the dimension n is even, n = m for some m 1, and prove Theorem 5.1 for those. However, as we will see in Remark 5.7, it is not hard to then extend the theorem to all dimensions. Proof of Theorem 5.1. We start with an isotropic convex body 1 with L 1 δ 1 L m, where δ 1 (0, 1). Then, one has the following upper bound for the volume of sections of 1. Lemma 5.5. For every k-codimensional subspace E of R m, 1 E 1/k c 1 (δ 1 ), where c 1 (δ 1 ) > 0 depends only on δ 1. Proof. Let E be a k-codimensional subspace of R m, and denote its orthogonal subspace by F. We consider the body B k+1 ( 1, F ), a convex body in the subspace F defined as in Subsection., and we recall that (5.4) c 1 L Bk+1 ( 1,F ) L 1 1 E 1/k c L Bk+1( 1,F ) L 1 16 λ i.

17 for some absolute constants c 1, c independent of m or k. On the other hand, it is not hard to check that if k j then L k c 3 L j (see e.g. [6, Theorem 4..]). Thus, (5.5) L Bk+1 ( 1,F ) L k c 3 L m = (c 3 /δ 1 )L 1, and the lemma follows with c 1 (δ 1 ) = c c 3 /δ 1. We will now invoke Pisier s theorem to also give a lower bound for the volume of m-dimensional sections of 1 that contain its barycenter. Lemma 5.6. For every F G m,m we have 1 F 1/m c (δ 1 ), where c (δ 1 ) > 0 depends only on δ 1. Proof. We consider an α-regular M ellipsoid E α for 1 (for the proof of this lemma we could have fixed α = 1; however, some steps of this more general argument will be needed again later). Set t α = max{1, [κ(α)] 1/α }. Then, (5.6) P F ( 1 ) N( 1, t α E α ) P F (t α E α ) e m P F (t α E α ) for every F G m,m. We also need the Rogers-Shephard inequality (see [0]) for both 1 and E α : since 1 = E α = 1, we know that (5.7) 1 = c 1 1 F 1/m P F ( 1 ) 1/m c, and similar estimates hold true for E α (see [1] or [14] for the left hand side inequality). The idea of the argument is the following: inequality (5.7) helps us relate the volume of m-dimensional sections of 1 (or E α ) to that of m-dimensional projections of 1 (or E α respectively); an upper bound for the former will give us a lower bound for the latter and vice versa. Also, inequality (5.6) allows us to compare the maximum (or minimum) volume of the m-dimensional projections of 1 to the maximum (or minimum) volume of the corresponding projections of E α. However, as we recalled in Lemma 5.4, the maximum volume of the m-dimensional projections of an ellipsoid is the same as the maximum volume of its m-dimensional sections, so we can use inequalities (5.6) and (5.7) once more to get from upper bounds for the volume of sections of 1 to lower bounds. We now give the precise argument: combining (5.7) with the conclusion of Lemma 5.5, we see that min P F ( 1 ) 1/m c 3 (δ 1 ). We then get from (5.6) F G m,m that min P F (t α E α ) 1/m c 4 (δ 1 ). Now, using (5.7) for E α we get E α F 1/m F G m,m c 5 (δ 1 )t α for every F G m,m. But from (5.) we have that (5.8) max P F (E α ) 1/m = max E α F 1/m c 5 (δ 1 )t α. F G m,m F G m,m Using (5.6) once again, we get P F ( 1 ) 1/m c 6 (δ 1 )t α for every F G m,m. Inserting this estimate into (5.7), we see that 1 F 1/m c 7 (δ 1 )/t α for every F G m,m. We may choose α = 1 now, and complete the proof with c (δ 1 ) = c 7 (δ 1 )/t 1. 17

18 Conclusion of the proof of Theorem 5.1. Let α (1, ) and let E α be an α-regular M ellipsoid for. Recall that E α = 1. Also, if E α = T (B n ) = T (B m ), let λ 1 λ m > 0 be the eigenvalues of the matrix T T ; observe from Lemma 5.4 that (5.9) B m m i=m+1 λ i = min P F (E α ) max P F (E α ) = B m F G m,m F G m,m Using (5.6) and the conclusion of Lemma 5.6, we get m λ i. i=1 (5.10) B m 1/m λ m min P F (E α ) 1/m e min P F ( 1 ) 1/m F G m,m t α F G m,m e t α min 1 F 1/m c 8(δ 1 ), F G m,m t α and hence (5.11) λ m c 9(δ 1 ) t α n. In a similar way, using (5.8), we see that B m 1/m λ m max P F (E α ) 1/m F G m,m c 5 (δ 1 )t α, and hence λ m c 10 (δ 1 )t α n. But from the last part of Lemma 5.4 we know that there exists a subspace F 0 G m,m such that P F0 (E α ) = λ m B F0, therefore, (5.1) c 9 (δ 1 ) t α nbf0 P F0 (E α ) c 10 (δ 1 )t α nbf0. Let W := B m+1 ( 1, F 0 ) and := W U(W ), where U O(m) satisfies U(F 0 ) = F 0. Since W is almost isotropic and L U(W ) = L W, from [6, Lemma 1.6.6] we see that = W U(W ) is an almost isotropic convex body in R n R m with L = L W. We will show that satisfies (i) and (ii); the same conclusion will then immediately follow (perhaps with slightly different constants for property (ii)) for any isotropic linear image T () of satisfying T (). Proof of (i): Since L = L W, from (5.4) we get (5.13) L = L W c 1 L 1 1 F 0 1/m c 1 c (δ 1 )L 1 δl n, where δ = δ 1 c (δ 1 )/c. For the last two inequalities we have used Lemma 5.6 and the fact that L 1 δ 1 L n. Proof of (ii): Using the fact that N(A A, B B) N(A, B) for any two nonempty sets A, B, and also the fact that B m B m B m, we may write for any s > 0, (5.14) N (, s nb n ) N ( W U(W ), s n(bf0 B F 0 )) N ( W, s nb F0 ). 18

19 From (.15) we know that (5.15) Z m (B m+1 ( 1, F 0 )) 1 F 0 1/m P F0 (Z m ( 1 )), therefore, using Lemmas 5.5, 5.6 and the fact that conv(c, C) Z m (C) for every centered convex body C of volume 1 in F 0 or in R n, we get (5.16) conv(w, W ) Z m (B m+1 ( 1, F 0 )) 1 F 0 1/m P F0 (Z m ( 1 )) δ1 P F0 (conv( 1, 1 )). But then, recalling also (5.1), we have for every r > 0, (5.17) N(W, c 10 (δ 1 )t α r nb F0 ) N ( conv(w, W ), c 10 (δ 1 )t α r nb F0 ) N ( conv(w, W ), rp F0 (E α ) ) N ( c 11 (δ 1 )P F0 (conv( 1, 1 )), rp F0 (E α ) ) N ( c 11 (δ 1 )conv( 1, 1 ), re α ) N ( 1 1, c 1 (δ 1 )r(e α E α ) ) N( 1, c 13 (δ 1 )re α ) (note that for the last two inequalities we have also used that E α is convex and symmetric, so E α E α = E α, that 1 is convex and contains the origin, so conv( 1, 1 ) 1 1, as well as the fact that N(A A, B B) N(A, B) ). It follows that (5.18) N(, t ( ) 4 n B n c 13 (δ 1 )t ) N 1, E α c10 (δ 1 )t α for every t > 0. Since E α is an α-regular M ellipsoid for 1, it remains to consider large enough t τ(δ 1, α), where (5.19) τ(δ 1, α) := c 10 (δ 1 )t α /c 13 (δ 1 ) = t α /c 14 (δ 1 ), to deduce from (5.1) and (5.18) that (5.0) log N(, t n B n ) 4 log N ( 1, c ) 14(δ 1 )t E α 4κ(α)tα α n t α c α 14 (δ 1) t α. Choosing α = 1 logn, we have κ(α) κ 1 logn and t α t as long as, say, t n. This completes the proof. Remark 5.7. Now that we have proven the existence of an isotropic body in R m which has properties (i) and (ii) of Theorem 5.1, we can easily prove the existence of such bodies in R m 1 as well: just note that for every subspace F G m,m 1 we have that L F R(). Combining this with the properties 19

20 (.14), (.15) for the almost isotropic convex body B m (, F ) in the (m 1) dimensional subspace F, we get that (5.1) L Bm(,F ) F m 1 L L δl m δ L m 1, c 3 1 and also that (5.) ( B m (, F ) Z m 1 Bm (, F ) ) F 1 m 1 PF (Z m 1 ()) P F (). Since for every t > 0, N(P F (), tb F ) = N ( P F (), tp F (B m ) ) N(, tb m ), we conclude that the body B m (, F ) will also satisfy properties (i) and (ii) of Theorem 5.1 with perhaps slightly different, but still independent of the dimension, constants κ, τ and δ. In the statement of Theorem 5.1, we can add one more property about the radius of the body that we look for: we can require that R() γ nl where γ > 0 is an absolute constant. The first step towards this is to use Bourgain s argument [4] which reduces the slicing problem to the study of bodies of small diameter; one can prove the following fact (see e.g. [6, Proposition.3.1]). Lemma 5.8. There exists an isotropic convex body 1 in R n with L 1 and R( 1 ) γ 1 nl1, where δ 1, γ 1 > 0 are absolute constants. δ 1 L n Then, we can repeat the proof of Theorem 5.1 starting with the body 1 R n = R m given by Lemma 5.8. One has now that R(W ) c(δ 1 )γ 1 nl1 : to see this, write R(W ) = R ( B m+1 ( 1, F 0 ) ) c 1 1 F0 1/m R ( ( (5.3) P F0 Zm ( 1 ) )) c (δ 1 )R ( conv( 1, 1 ) ) = c (δ 1 )R( 1 ) c (δ 1 )γ 1 nl1. It is also easy to check that R() = R(W U(W )) R(W ), hence R() γ nl for some absolute constant γ > 0. Similarly for the odd dimensions, we see that for every F G m,m 1, (5.4) R(B m (, F )) R(P F ()) R() cγ m 1 L Bm(,F ), where we have made use of (5.1), (5.). Thus, we can state the following version of Theorem 5.1. Theorem 5.9. There exist absolute constants κ, τ, γ and δ > 0 such that, for every n N, there can be found an isotropic convex body in R n with R() γ nl, L δl n, and the property that log N(, tb n ) κn log n t for all t τ n logn. 0

21 Definition Let I(κ, τ, γ, δ) denote the class of isotropic convex bodies whose existence is established in Theorem 5.9. Let ρ > 0 be the absolute constant in Theorem 4.1. Then, we define A(n, κ, τ, γ, δ) to be the set of all q [, ρ n] for which there exists I(κ, τ, γ, δ) such that I 1 (, Zq ()) ρnl. Observe that already, by (3.), A(n, κ, τ, γ, δ) can be shown to contain an interval of the form [, c n] where c > 0 is an absolute constant. Clearly, any improvement to the upper bound in (3.) will automatically give us that A(n, κ, τ, γ, δ) contains an even larger part of [, ρ n]. For those q we set { I1 (, Z } q ()) (5.5) B(q) = inf qnl : I(κ, τ, γ, δ). Then, Theorem 4.1 implies the following: for every q A(n, κ, τ, γ, δ), (5.6) δ L n Cκ n/q log n max{1, B(q)}. In other words, we have: Theorem There exist absolute constants κ, τ, γ and δ > 0 such that, for every n N, { } Cκ (5.7) L n min n/q log δ n max{1, B(q)} : q A(n, κ, τ, γ, δ). The estimate L n c 4 n logn is a direct consequence of Theorem 5.11: observe that B() 1. 6 Isotropic convex bodies of small diameter In [7, Section 3] it is proven that for every isotropic convex body there exists a second isotropic convex body C with bounded isotropic constant and the same behaviour as with respect to linear functionals. Theorem 6.1. Let be an isotropic convex body in R n. There exists an isotropic convex body C in R n with the following properties: (i) L C c 1. (ii) c Z q (C) Zq() L + qb n c 3 Z q (C) for all 1 q n. (iii) c 4 I q (C, W ) Iq(,W ) L + I q (D n, W ) c 5 I q (C, W ) for all 1 q n and every symmetric convex body W in R n. The constants c i, i = 1,..., 5 are absolute positive constants. The body C is defined as the convolution of with a multiple of B n. If we also assume that is symmetric, then using the fact that Z n (C) C and Z n (), we see that (6.1) C L + D n. 1

22 From the previous section, we know that for our purposes it is enough to study the quantity I 1 (, Z q ()) in the cases that is an isotropic symmetric convex body of small diameter; that is, we can assume that R() γ nl for some γ 1. The next proposition, which makes use of Theorem 6.1, shows us that it even suffices to consider isotropic convex bodies which are c(γ)-isomorphic to a ball. Proposition 6.. Let be an isotropic symmetric convex body in R n with R() γ nl. Then, there exists an isotropic symmetric convex body C such that: (i) L C c 6, (ii) c 7 D n C c 8 γd n, and (iii) I 1 (, Z q ()) c 9 I 1 (C, Z q (C))L where c 6, c 7, c 8, c 9 > 0 are absolute constants. for all 1 q n, Proof. We will use the fact that w q (Z q ()) q/ni q (), and hence (6.) c 1 ql w q (Z q ()) γ ql for all 1 q n. We consider the body C defined by Theorem 6.1. It is clear that 1 L C c 6 for some absolute constant c 6 > 0. Since L Z q () c 3 Z q (C), we have c 3 L Zq () Zq (C), and hence (6.3) I 1 (C, Z q (C)) I 1 (C, c 3 L Z q ()) = 1 c 3 L I 1 (C, Z q ()). Applying the inequality c 5 I 1 (C, W ) I1(,W ) L with W = Z q (), we get (6.4) I 1 (C, Zq I 1 (, Zq ()) (C)) c 9 L, with c 9 = (c 3 c 5 ) 1 Finally, from (6.1) and the fact that L γ nb n, we see that c 7 D n C c 8 γd n. In view of this result, we can give one more version of the reduction theorem of Section 4. Definition 6.3. Let I sd (γ) denote the class of isotropic convex bodies that satisfy (i) L C c 6 and (ii) c 7 D n C c 8 γd n, where c i > 0 are absolute constants (e.g. the ones in Proposition 6.). For every q n, set { I1 (, Z } q ()) Γ(q) = sup : I sd (γ). qn Then, Theorem 5.11 and Proposition 6. imply the following:

23 Theorem 6.4. There exist absolute constants κ, τ, γ and δ > 0 such that, for every n N, { } Cκ (6.5) L n min n/q log δ nγ(q) : q A(n, κ, τ, γ, δ). In other words, studying the behaviour of I1(,Z q ()) qn within the class I sd (γ) is enough in order to understand the behaviour of the parameter B(q) as well as whether that behaviour can lead to improved upper bounds for L n. Acknowledgements. The second named author is partially supported by an NSF grant (DMS ). The third named author is supported by a scholarship of the University of Athens. Part of this work was carried out while the second and the third named authors were visiting the Isaac Newton Institute for Mathematical Sciences in Cambridge; they would like to thank the Institute for the hospitality and the organisers of the Discrete Analysis Programme for their invitation. References [1]. M. Ball, Logarithmically concave functions and sections of convex sets in R n, Studia Math. 88 (1988), [] S. G. Bobkov and F. L. Nazarov, On convex bodies and log-concave probability measures with unconditional basis, Geom. Aspects of Funct. Analysis (Milman- Schechtman eds.), Lecture Notes in Math (003), [3] S. G. Bobkov and F. L. Nazarov, Large deviations of typical linear functionals on a convex body with unconditional basis, Stochastic Inequalities and Applications, Progr. Probab. 56, Birkhauser, Basel, 003, [4] J. Bourgain, On the distribution of polynomials on high dimensional convex sets, Geom. Aspects of Funct. Analysis (Lindenstrauss-Milman eds.), Lecture Notes in Math (1991), [5] N. Dafnis, G. Paouris, Small ball probability estimates, ψ -behavior and the hyperplane conjecture, Journal of Functional Analysis 58 (010), [6] A. Giannopoulos, Notes on isotropic convex bodies, Warsaw University Notes (003). [7] A. Giannopoulos, G. Paouris and P. Valettas, On the existence of subgaussian directions for log-concave measures, Contemporary Mathematics 545 (011), [8] B. lartag, On convex perturbations with a bounded isotropic constant, Geom. Funct. Anal. 16 (006), [9] B. lartag and E. Milman, Centroid Bodies and the Logarithmic Laplace Transform - A Unified Approach, arxiv: v1. [10] B. lartag and V. D. Milman, Rapid Steiner Symmetrization of most of a convex body and the slicing problem, Combin. Probab. Comput. 14, no. 5-6 (005) [11] A. Litvak, V. D. Milman and G. Schechtman, Averages of norms and quasi-norms, Math. Ann. 31 (1998),

24 [1] E. Lutwak, D. Yang and G. Zhang, L p affine isoperimetric inequalities, J. Differential Geom. 56 (000), [13] V. D. Milman and A. Pajor, Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space, Geom. Aspects of Funct. Analysis (Lindenstrauss-Milman eds.), Lecture Notes in Math (1989), [14] V.D. Milman, A. Pajor, Entropy and Asymptotic Geometry of Non-Symmetric Convex Bodies, Advances in Mathematics, 15 (000), [15] V.D. Milman and G. Schechtman, Asymptotic Theory of Finite Dimensional Normed Spaces, Lecture Notes in Math. 100 (1986), Springer, Berlin. [16] G. Paouris, ψ -estimates for linear functionals on zonoids, Lecture Notes in Mathematics 1807 (003), 11. [17] G. Paouris, Concentration of mass in convex bodies, Geometric and Functional Analysis 16 (006), [18] G. Paouris, Small ball probability estimates for log-concave measures, Trans. Amer. Math. Soc. (011), DOI: /S [19] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Mathematics 94 (1989). [0] C. A. Rogers and G. C. Shephard, Convex bodies associated with a given convex body, J. London Soc. 33 (1958), [1] J. Spingarn, An inequality for sections and projections of a convex set, Proc. Amer. Math. Soc. 118 (1993), [] C. Zong, Strange phenomena in convex and discrete geometry, Universitext, Springer (003). Apostolos Giannopoulos: Department of Mathematics, University of Athens, Panepistimioupolis , Athens, Greece. apgiannop@math.uoa.gr Grigoris Paouris: Department of Mathematics, Texas A & M University, College Station, TX U.S.A. grigoris paouris@yahoo.co.uk Beatrice-Helen Vritsiou: Department of Mathematics, University of Athens, Panepistimioupolis , Athens, Greece. bevritsi@math.uoa.gr 4