ON THE ISOTROPY CONSTANT OF PROJECTIONS OF POLYTOPES

Transcription

1 ON THE ISOTROPY CONSTANT OF PROJECTIONS OF POLYTOPES DAVID ALONSO-GUTIÉRREZ, JESÚS BASTERO, JULIO BERNUÉS, AND PAWE L WOLFF Abstract. The isotropy constant of any d-dimensional polytope with n vertices is bounded by C n/d where C > 0 is a numerical constant.. Introduction The boundedness of the isotropy constant (see definition below) is a major conjecture in Asymptotic Geometric Analysis. The answer is known to be positive for many families of convex bodies, see for instance [MP] or [KK] and the references therein. In this paper we focus our attention on the isotropy constant of polytopes or, equivalently, of projections of the unit ball of l n space (in the symmetric case) and of the regular n-dimensional simplex S n (in the non-symmetric case). M. Junge [J] proved that the isotropy constant of all orthogonal projections of B n p, the unit ball of the l n p space, < p, is bounded by Cp an estimate improved to C p in [KM] (p is the conjugate exponent of p and C a numerical constant). Later [J2, Theorem 4] M. Junge showed that the isotropy constant of any symmetric polytope with 2n vertices is bounded by C log n, see also [Mi]. In a recent paper, [KK], B. Klartag and G. Kozma show the boundedness of the isotropy constant of random Gaussian polytopes. The integral over a polytope, which defines its isotropy constant, is computed by passing to an integral over its surface (faces). A consequence of their results is that most (see precise meaning below) d-dimensional projections of B n as well as of S n have bounded isotropy constant. When reading this statement one should have in mind the well-known fact that every symmetric convex body in R d is almost a projection of a B n -ball, with possibly large n. In the same spirit, positive answers for other random d-dimensional polytopes with n ( Cd) vertices were given in [A] and [DGG] Mathematics Subject Classification. 46B20 (Primary), 52A40, 52A39 (Secondary). Key words and phrases. polytopes, projections, isotropy constant. Partially supported by Marie Curie RTN CT The three first named authors partially supported by MCYT Grant(Spain) MTM and DGA E-64. The fourth named author partially supported by National Science Foundation FRG grant DMS (U.S.A.).

2 2 D. ALONSO, J. BASTERO, J. BERNUÉS, AND P. WOLFF Our main theorem (Corollary 3.5) states that for any d-dimensional polytope K with n vertices its isotropy constant L K verifies n L K C d where C > 0 is a numerical constant. We now pass to describe the contents of the paper. The second section introduces the geometric tool (Proposition 2.) necessary to deal with integration on d-dimensional projections of polytopes (Corollary 2.8). Some time ago, one of the authors learned about this tool from Prof. Franck Barthe. The ideas originate from a paper by U. Betke [Be], where a general result was presented, namely a related formula for mixed volumes of two polytopes. However, for the sake of completeness, we provide the proof of the particular result we need. It also seems that the content of the proof is more geometric. In the third Section we use these tools to prove our aforementioned main result (Corollary 3.5) by easily reducing it to the cases K = P E B n or K = P E S n (Theorem 3.4) where E R n is any d-dimensional subspace and P E is the orthogonal projection onto E. Also in this Section we give a proof of the observation that for most subspaces, that is, for a subset A of the Grassmann space G n,d of Haar probability measure c e c 2 max{log n,d}, one has L PE B n < C and L PE S n < C for every E A with numerical constants C, c, c 2 (Proposition 3.3). The next Section studies the isotropy constant of projections of random polytopes with vertices on the sphere S n. Using the techniques from Section 2 and [A] we show that, with high probability, the isotropy constants of all d-dimensional projections of random polytopes are bounded by C n d (Proposition 4.). In the last Section we show that for every isotropic convex body, the isotropy constants of its hyperplane projections are comparable to the isotropy constant of the body itself (Corollary 5.). Recall that the analogous result for hyperplane sections was already proved in [MP]. The proof uses Steiner symmetrization in a similar way as it appears in [BKM], with better numerical constants. In particular, we have L PH Bp n C for any hyperplane H and < p < improving Junge s estimate [J] for the case of hyperplanes. In [ABBW] a different proof of this fact is given with the hope it might be extended to lower dimensional projections. We recall that a convex body K R n is isotropic if it has volume Vol n (K) =, the barycenter of K is at the origin and its inertia matrix is a multiple of the identity. Equivalently, there exists a constant L K > 0 called isotropy constant of K such that L 2 K = K x, θ 2 dx, θ S n.

3 ON THE ISOTROPY CONSTANT OF PROJECTIONS OF POLYTOPES 3 It is well known [MP], that every convex body K R n has an affine transformation K isotropic, so we can write L K := L K. This is well defined and moreover, (.) nl 2 K = inf { Vol n (K) 2 n + a+t K x 2 dx; a R n, T SL(n) For a convex body K R n, R(K) = max{ x : x K} and r(k) = min{ x : x K} are the circumradius and the inradius of K respectively. We will think of S n as an n-dimensional regular simplex in R n with center of mass at the origin. We will write n = conv{e,..., e n+ } for the natural position of an n-dimensional regular simplex in R n+. The Lebesgue measure on an affine subspace E will be denoted by λ E. For a measurable set A E, if d is a dimension of E, Vol d (A) will stand for λ E (A). The notation a b means a c b a c 2 for some numerical constants c, c 2 > 0. } 2. Projections of polytopes Throughout this section, K R n is a polytope (non-empty but possibly of empty interior), E R n is a linear subspace of dimension d ( d n ) and P E is the orthogonal projection onto E. Let us fix some notation and recall necessary definitions (we follow the book by Schneider [S, Ch., 2]). For a subset A R n, aff A denotes the minimal affine subspace which contains A. The dimension of a convex set A is dim aff A. When writing relint A we mean the relative interior of A w.r.t. the topology of aff A. If G R n is an affine subspace then G 0 denotes the linear subspace parallel to G. A convex subset F K of a polytope K is called a face if for any x, y K, (x + y)/2 F implies x, y F (see also [S, Sec..4, pp. 8]). The set of j-dimensional faces (j-faces, in short) of K will be denoted as F j (K) (j = 0,,..., n), and F(K) = n j=0 F j(k) { } is the set of all faces of K ( is also a face). K can be decomposed into a disjoint union of {relint F ; F F(K)} (see [S, Thm. 2..2]). For that reason for any x K the unique face F F(K) such that x relint F will be denoted by F (K, x). For x K, a normal cone of K at x is N(K, x) = {u R n ; z K z x, u 0}. N(K, x) is a closed convex cone. convex cone, namely We shall also consider another closed S(K, x) = λ>0 λ(k x).

4 4 D. ALONSO, J. BASTERO, J. BERNUÉS, AND P. WOLFF (In general, i.e. when K is a convex body, S(K, x) does not have to be closed.) By [S, (2.2.)], (2.2) N(K, x) = S(K, x), where the polarity used here is the polarity of convex cones, namely, if C R n is a convex cone, C = {y R n ; x C x, y 0} (see also [S, Sec..6, pp. 34]). We shall also need to consider normal cones taken w.r.t. an affine subspace. If G is an affine subspace of R n and L G is a convex body, then for x L we define a normal cone for L at x taken w.r.t. G: N G (L, x) = {u G 0 ; z L z x, u 0}. Note that N G (L, x) G 0. The similar duality relation to (2.2) holds: (2.3) N G (L, x) G 0 = S(L, x), where the polarity is taken w.r.t. G 0. For any face F F(K), define N(K, F ) := N(K, x), where x relint F. This definition does not depend on the choice of x (see [S, Sec. 2.2., pp. 72]). N G (L, F ) is analogously defined. For a given polytope K R n and a linear subspace E R n of dimension d, let us fix any u E \ {0} which satisfy (2.4) u / {P E N(K, F ) ; F F(K)\{ }, dim P E N(K, F ) n d }. Clearly, such u exists, since (2.4) excludes only a finite union of sets of dimension < n d from E which is of dimension n d. Consider the following subsets of F(K): F(K, E, u) := {F F(K) ; u P E N(K, F )}, F d (K, E, u) := F(K, E, u) F d (K). Proposition 2.. Let K R n be a polytope, E a d-dimensional subspace, u E verifying (2.4) and F(K, E, u) as described above. Then (a) {P E (relint F ) ; F F(K, E, u)} is a family of pair-wise disjoint sets, (b) {P E F ; F F(K, E, u)} = P E K. Moreover, F(K, E, u) 0 j d F j(k) and for each F F d (K, E, u), P E F : F P E F is an affine isomorphism. In the proof of the proposition we shall use several lemmas. Lemma 2.2. Let L be a polytope in R n and G R n be an affine subspace. If x L G then P G0 N(L, x) = N G (L G, x).

5 ON THE ISOTROPY CONSTANT OF PROJECTIONS OF POLYTOPES 5 Proof. By taking polars w.r.t. G 0 we see that the assertion is equivalent to (2.5) N(L, x) G 0 = N G (L G, x) G 0 (for the l.h.s. we used the fact that for a convex cone C, (P G0 C) G 0 = C G 0 ). Since G 0 = G x by (2.2) we get N(L, x) G 0 = S(L, x) (G x) = λ>0 λ(l x) (G x) = S(L G, x). Applying (2.3) we see that the r.h.s. of (2.5) is also equal to S(L G, x). Lemma 2.3. [S, Sec. 2.2] Let L be a polytope contained in an affine subspace G R n. Then N G (L, F ) = G 0. F F 0 (L) Lemma 2.4. With the hypothesis as in the previous lemma, for x, y L, N G (L, x) N G (L, y) = N G (L, (x + y)/2). Proof. The inclusion is immediate from the definition of a normal cone. For the converse inclusion take u N G (L, (x+y)/2). Then x x+y 2, u 0, y x+y 2, u 0, so x y, u = 0. Now, for all z L, z x, u = z x + y, u + y x, u 0, 2 2 so u N G (L, x). Similarly u N G (L, y). Lemma 2.5. [S, Sec. 2.4] With the hypothesis as in Lemma 2.3, for F F(L), dim N G (L, F ) = dim G dim F Remark 2.6. Actually we shall use only the inequality dim N G (L, F ) dim G dim F which simply follows from the fact N G (L, F ) ( (aff F ) 0 ) G0. Lemma 2.7. Let K R n be a polytope, E R n be a linear subspace of dimension d and u E satisfies (2.4). Let y K, x = P E y E, K x = K (x + E ) (K x is a polytope in x + E ). If one of the equivalent condition holds: (i) u P E N(K, y), (ii) u N x+e (K x, y), then {y} F 0 (K x ) and dim F (K, y) d. Proof. The conditions (i) and (ii) are equivalent by Lemma 2.2. Consider F = F (K, y) and F = F (K x, y). By the condition (2.4) on u, dim P E N(K, F ) n d, so dim N x+e (K x, F ) n d and also dim N(K, F ) n d. Therefore Lemma 2.5 applied to K x and F implies dim F = 0, so {y} = F F 0 (K x ). Eventually, applying the same lemma to K and F yields dim F d.

6 6 D. ALONSO, J. BASTERO, J. BERNUÉS, AND P. WOLFF Proof of Proposition 2.. (a). Take F, F 2 F(K, E, u) such that for some x E, x P E (relint F ) P E (relint F 2 ) which means that for i =, 2 one can find y i x + E that y i relint F i and then u P E N(K, F i ) = P E N(K, y i ). Consider a convex polytope K x = K (x + E ). Lemma 2.7 implies that {y }, {y 2 } F 0 (K x ) and also u N x+e (K x, y ) N x+e (K x, y 2 ) = N x+e (K x, (y + y 2 )/2), where the last equality is due to Lemma 2.4. But again, Lemma 2.7 implies that also {(y + y 2 )/2} F 0 (K x ), hence y = y 2 (see definition of a face) and consequently, F = F (K, y ) = F 2. (b). The inclusion is obvious. For the inclusion take arbitrary x P E K. Put K x = K (x + E ). K x is a non-empty polytope in x + E. By Lemma 2.3 one can find y x + E such that {y} F 0 (K x ) and u N x+e (K x, y). Lemma 2.7 (or just Lemma 2.2) implies u P E N(K, F ) where F = F (K, y). Consequently, F F(K, E, u). By the definition of F(K, E, u) and Lemma 2.7, any face F F(K, E, u) has dimension d. Finally we show that for F F d (K, E, u), P E F : F P E F is an isomorphism. The condition (2.4) and Lemma 2.5 implies n d dim P E N(K, F ) dim N(K, F ) n dim F = n d, so dim P E N(K, F ) = dim N(K, F ) = n d. This means ( span N(K, F ) ) E = {0} and N(K, F ) E = {0}. The definition of a normal cone yields (aff F ) 0 N(K, F ), which finally gives (aff F ) 0 E = {0}. The following corollary is an immediate consequence of Proposition 2.. Corollary 2.8. Let K R n be a convex polytope, E R n a d-dimensional linear subspace ( d n ). Then there exists a subset F of F d (K) such that for any integrable function f : E R, (2.6) P E K In particular (for f ), f(x) λ E (dx) = F F Vol d (P E F ) Vol d (F ) Vol d (P E K) = F F Vol d (P E F ). F f(p E y) λ afff (dy).

7 ON THE ISOTROPY CONSTANT OF PROJECTIONS OF POLYTOPES 7 Proof. Choose any u E satisfying (2.4) for K and E and put F = F d (K, E, u). By Proposition 2., f(x) λ E (dx) = f(x) λ E (dx) P E K F F P E F = Vol d (P E F ) f(p E y) λ afff (dy). Vol F F d (F ) F For our purposes we shall use above corollary with f(x) = x 2. In such case, the obvious inequality P E y y and the identity = Vol d (P E F ) Vol F F d (P E K) lead to the following estimate: if P E K is a body of dimension d (i.e. is nondegenerated) then (2.7) x 2 λ E (dx) max y 2 λ afff (dy). Vol d (P E K) P E K F F Vol d (F ) F 3. Projections of the l n -ball and the regular simplex First of all, we are going to see that most projections of B n on d- dimensional subspaces (d n) have the isotropy constant bounded. It is well known that any symmetric convex polytope in R d with 2n vertices is linearly equivalent to P E B n for some E G n,d. Indeed, if T : R n R d is a linear transformation of full rank, then taking the d-dimensional subspace E = (ker T ) R n, T can be represented as T E P E where T E : E R d is a linear isomorphism being a restriction of T to the subspace E. As an immediate consequence we obtain the following Lemma 3.. Let K = conv{±v,..., ±v n } R d be a symmetric convex polytope with non-empty interior and let T : R n R d be the linear map such that T e i = v i. Then for E = (ker T ) G n,d, P E B n and K are linearly equivalent. One may also prove a similar lemma in the non-symmetric case. Recall that n = conv{e,..., e n+ } H R n+ where H, as in the whole of this section, denotes the hyperplane orthogonal to the vector (,..., ) R n+. Lemma 3.2. Let K = conv{v,..., v n+ } R d (n d) be a convex polytope with non-empty interior. Let T : R n+ R d be the linear map that T e i = v i v 0 where v 0 = n+ n+ i= v i and E = (ker T ) R n+. Then E H is a subspace of dimension d and K v 0 is linearly equivalent to P E n. Consequently, K is affinely equivalent to some orthogonal projection of the n-dimensional regular simplex S n onto a d-dimensional subspace.

8 8 D. ALONSO, J. BASTERO, J. BERNUÉS, AND P. WOLFF Proof. Clearly (,..., ) ker T, so E H. Since K has non-empty interior, vectors v i v 0 span the whole of R d, so T is of full rank. Therefore dim E = d and the argument given above applies. Now we can prove the following result concerning the isotropy constant of random projections of B n and S n. Proposition 3.3. There exist absolute constants C, c, c 2 > 0 such that the Haar probability measure of the set of subspaces E G n,d verifying is greater than c e c 2 max{log n,d}. L PE B n < C and L P E S n < C Proof. For small values of d, namely d c log n, the isotropy constant of a random projection is bounded by an absolute constant with probability greater than c n c 2 as a consequence of Dvoretzky s theorem. Let G = (g ij ) be a d n Gaussian random matrix, i.e. the g ij s are i.i.d N (0, ) Gaussian random variables. Since (ker G) = Im (G t ) R n, G t being the transpose matrix of G, and the columns of G t are independent and rotationally invariant random vectors in R n, then a random subspace E = (ker G) has dimension d a.s. and is distributed according to the Haar probability measure µ on G n,d. Therefore for any constant C > 0, µ{e G n,d ; L PE B n < C} = P{L P E B n < C}. Lemma 3. and the affine invariance of the isotropy constant imply L PE B n = L conv (±Ge,...,±Ge n) a.s. Klartag and Kozma proved in [KK] that if C is a sufficiently large absolute constant, P{L conv (±Ge,...,±Ge n) < C} > c e c 2d which completes the proof in the symmetric case. For the non-symmetric case, we proceed analogously. For a d (n + ) Gaussian random matrix G = (g ij ), take Ḡ = (g ij n+ n+ k= g ik) i d,j n+. Since the sum of the columns of Ḡ is zero, (ker Ḡ) = Im (Ḡt ) H R n+. Moreover, since rows of Ḡ (equivalently, columns of Ḡ t ) are independent canonical Gaussian random vectors in H, the random subspace E = (ker Ḡ) H is distributed according to the Haar probability measure on G H,d (Grassmann manifold of d-dimensional subspaces of H). Lemma 3.2 and the affine invariance of the isotropy constant imply L PE n = L conv (Ge,...,Ge n+ ) a.s. Since P E n = P E (P H n ) and P H n is an n-dimensional regular simplex (in H), a non-symmetric counterpart of the result of Klartag and Kozma [KK], finishes the proof. P{L conv (Ge,...,Ge n+ ) < C} > c e c 2d,

9 ON THE ISOTROPY CONSTANT OF PROJECTIONS OF POLYTOPES 9 In the final part of the section we will use the tools from Section 2 to prove the main result. In particular, whenever d cn the boundedness of the isotropy constant holds not only for most projections of B n and S n but deterministically for all of them. Theorem 3.4. Let E R n be a subspace of dimension d n and K = P E B n, T = P ES n. Then where C > 0 is a universal constant. L K, L T C n/d Proof. As an immediate consequence of (.), (3.8) L 2 K d Vol d (K) 2/d Vol d (K) Applying (2.7), we obtain the bound (3.9) x 2 λ E (dx) Vol d (K) Vol d ( d ) K K x 2 λ E (dx). d x 2 λ aff d (dy) = 2 d + 2 (for the last equality see e.g. [KK, Lemma 2.3]). To estimate Vol d (K) note that n /2 B n 2 Bn, so n /2 (B n 2 E) P EB n. Therefore (3.0) Vol d (K) /d c nd. Combining these two, we get L 2 K d nd 2 c 2 d + 2 n C d. In the case of the simplex it is convenient to embed E and S n into H. More precisely, we take S n = conv{p H e i : i =,..., n+} H R n+ and assume E H. Now observe that T = P E S n = P E n so (2.7) again yields Vol d (T ) T x 2 λ E (dx) 2 d + 2. To bound the volume radius of T from below, we use the Rogers-Shephard inequality [RS]: ( ) 2d Vol d (T T ) Vol d (T ). d Note that T T conv(t T ) = conv(p E n P E n ) = P E ( conv( n n ) ) = P E B n+.

10 0 D. ALONSO, J. BASTERO, J. BERNUÉS, AND P. WOLFF Combining with the estimate (3.0), ( ) 2d /d Vol d (T ) /d Vol d (P E B n+ ) /d d ( ) 2d /d c c. d (n + )d nd Due to Lemma 3.2, we immediately get the following: Corollary 3.5. Let K R d be a non-degenerated (dim K = d) convex polytope with n vertices. Then n L K C d. 4. Isotropy constant of projections of random polytopes In this section we consider polytopes generated by the convex hull of vertices randomly chosen on the S n. The main result is Proposition 4.. There exist absolute constants C, c and c 2, such that if m n, {P i } m i=0 are independent random vectors on Sn and K = conv{±p,..., ±P m } or K = conv{p 0,..., P m }, then n P{L PE K C d E G n,d d n } c e c2n. The proof follows [A]. We shall only sketch the main ideas as the technical computations can be found in that reference. Sketch of the proof. Let E R n denotes an d-dimensional subspace. The ideas in what follows will give us the proof for m cn with an absolute constant c. If m < cn, Corollary 3.5 gives deterministically L PE K C m d C n d. Apply once again (.). Writing r(k) the inradius of K and using the inequality (2.7), the main consequence of Proposition 2., we obtain that for any polytope K R n and any d-dimensional subspace E, L 2 P E K C r(k) 2 max x 2 λ afff (dx). F F d (K) Vol d (F ) When K is the symmetric convex hull of m independent random points in S n, it was proved in [A, Lemma 3.], that for some constant c such that cn m ne n 2, P { r(k) < 2 2 F } log m n e n. n The same proof gives the statement in the non-symmetric case.

11 ON THE ISOTROPY CONSTANT OF PROJECTIONS OF POLYTOPES On the other hand, with probability, each d-dimensional face of K is a simplex F = conv{q,..., Q d+ } with Q i = ε i P ji (or just Q i = P ji in the non-symmetric case) where j < < j d+ m and ε i {, }. The same proof as in [KK] and [A] shows that with probability we have (4.) Vol d (F ) F x 2 λ afff (dx) = 2 d (d + )(d + 2) d+ i i 2 Q i, Q i2. In order to give a bound for this quantity for a fixed F F d (K) we proceed in the same way as in [A, Theorem 3.], by using a version of Bernstein s inequality as stated in [BLM]. We thus obtain d+ (4.2) P Q i, Q i2 > ɛ(d + ) 2e cɛn i i 2 for every ɛ > ɛ 0, where ɛ 0 is an absolute constant. Now, for each F F d (K) let Q F,..., QF d+ be vertices of F. Applying (4.2) and the union bound over F d (K) (whose cardinality is clearly bounded by ( ) 2m d+ ), we obtain for ɛ log m n > ɛ 0, P max d+ Q F i, Q F i 2 > ɛ(d + ) log m F F d (K) n i i 2 ( ) 2m 2e cɛn log m n 2e cɛn log m 2em +(d+) log n d+ d + 2e cɛn log m 2em +n log n n, since the function x log C x is increasing when C x > e. Consequently, by the union bound over d, { P d n s.t. max F F d (K) d+ i i 2 Q F i, Q F i 2 > ɛ(d + ) log m n 2e cɛn log m n } 2em +n log n +log n Since m cn, considering the complement set and using (4.), we can fix ɛ > 0 a large enough numerical constant to obtain { P d n max F F d (K) x 2 λ afff (dx) C Vol d (F ) F d log m } n 2e cn log m n Thus, there exist constants c, C > 0 such that if cn m ne n 2 then the set of points (P,..., P m ) for which the inequality L PE K C n d holds for

12 2 D. ALONSO, J. BASTERO, J. BERNUÉS, AND P. WOLFF every d-dimensional subspace E and for every d n has probability greater than 2e cn log m n e n > c e c 2n. In case m > ne n 2, for n large enough, r(k) 4 with probability greater than e n so with this probability dl 2 P E K x 2 dx Vol d (P E K) 2 d Vol d (P E K) P E K Vol d ( 4 Bd 2 ) cd 2 d and the proof is complete. 5. A general result In this section we prove a general relation between the isotropy constant of the hyperplane projections of an isotropic convex body and of the body itself. Corollary 5.. Let K be an isotropic convex body and let H be a hyperplane. Then L PH K L K Its proof relies on the next Proposition which improves the numerical constants appearing in a more general statement in [BKM] for the case of projections onto hyperplanes. Proposition 5.2. Let K R n be an isotropic convex body and let H = ν be a hyperplane. If S(K) is the Steiner symmetrization of K with respect to H then, ( c log n ) L K L n S(K) L K for some numerical constant c > 0. Proof. Without loss of generality, we may assume that ν = e n = (0,..., 0, ). Write E = e n for the -dimensional subspace generated by e n. The Steiner symmetrization of K is defined by { S(K) := (y, t) R n R ; y P H K, t } 2 Vol (K (x + E)). Clearly P H K = P H S(K) = S(K) H. Now we study the inertia matrix of S(K). First notice that for x P H K, Vol (K (x+e)) = Vol (S(K) (x+e)). For every θ S n H, Fubini s theorem yields x, θ 2 dx = y + te n, θ 2 dt dy S(K) = P H K P H K S(K) (x+e) y, θ 2 Vol (K (x + E)) dy = L 2 K.

13 ON THE ISOTROPY CONSTANT OF PROJECTIONS OF POLYTOPES 3 Using the fact that S(K) (x+e) t dt = 0 for x P HK, in the similar fashion we show that for every θ S n H, x, θ x, e n dx = 0. S(K) Also S(K) (x+e) t2 dt K (x+e) t2 dt for x P H K, thus x, e n 2 dx x, e n 2 dx = L 2 K. S(K) Taking σ > 0 such that σ 2 := S(K) x, e n 2 dx/l 2 K, we obtain that the inertia matrix of S(K) is M = L 2... K. The volume of S(K) is, so L S(K) = (detm) /2n (see [MP]), which means (5.3) L S(K) = σ /n L K = K σ 2 ( /2 S(K) x, e n dx) 2 Since σ, we obtain L S(K) L K. L K /n L K. A well-known fact due to Hensley [H] states that Vol n (K H) ( ) /2 x, e K n 2 dx for any convex body K with volume and center of mass at the origin. Using this fact for S(K) in (5.3) we obtain that for some absolute constant c > 0, ( ) c /n ( ) c /n L S(K) L K = L K. Vol n (S(K) H)L K Vol n (P H K)L K Now we use the following inequality: n Vol n (P H K)Vol (K E) Vol n (K) =. (For the proof, see for instance [P, Lemma 8.8]. The proof of this inequality given in [P] works in the non-symmetric case.) It yields Vol n (P H K) where r(k) is the inradius of K. n Vol (K E) n 2r(K)

14 4 D. ALONSO, J. BASTERO, J. BERNUÉS, AND P. WOLFF Since every isotropic convex body verifies r(k) L K (see [G] or [KLS], for instance) we obtain ( ) 2c /n L S(K) L K. n Proof of Corollary 5.. Since P H K = S(K) H we have L PH K = L S(K) H L S(K) L K where the first equivalence is the corresponding one for sections of convex bodies as proved in [MP]. Acknowledgements. The fourth named author would like to thank Prof. Stanis law Szarek for some useful discussions. Part of this work has been done while the fourth named author enjoyed the hospitality of University of Zaragoza staying there as Experienced Researcher of the European Network PHD. References [A] D. Alonso-Gutiérrez, On the isotropy constant of random convex sets, Proc. Amer. Math. Soc. 36 (2008), no. 9, pp [ABBW] D. Alonso-Gutiérrez, J. Bastero, J. Bernués, P. Wolff The slicing problem for hyperplane sections of Bp n. Preprint available in [Be] U. Betke, Mixed volumes of polytopes, Arch. Math. 58 (992), pp [BLM] J. Bourgain, J. Lindenstrauss and V.D. Milman, Minkowski sums and symmetrizations, in: Geometric aspects of functional analysis (986/87), Lecture Notes in Math. 37, pp , Springer, Berlin, 988. [BKM] J. Bourgain, B. Klartag, V.D. Milman, Symmetrization and isotropic constants of convex bodies, in: Geometric aspects of functional analysis, Lecture Notes in Math. 850, pp. 0 5, Springer, Berlin, [DGG] N. Dafnis, A. Giannopoulos and O. Guédon, On the isotropic constant of random polytopes, Advances in Geometry (to appear). [G] A. Giannopoulos, Notes on isotropic convex bodies, Warsaw, October [H] D. Hensley, Slicing convex bodies bounds for slice area in terms of the body s covariance, Proc. Amer. Math. Soc. 79 (980), no. 4, pp [J] M. Junge, Hyperplane conjecture for quotient spaces of L p, Forum Math. 6 (994), pp [J2] M. Junge, Proportional subspaces of spaces with unconditional basis have good volume properties, in: Geometric aspects of functional analysis, Oper. Theory Adv. Appl. 77, pp. 2 29, Birkhäuser, Basel, 995. [KLS] R. Kannan, L. Lovász and M. Simonovits, Isoperimetric problems for convex bodies and a localization lemma, Discrete and Comput. Geom. 3 (995), pp [KK] B. Klartag and G. Kozma, On the hyperplane conjecture for random convex sets, to appear in Israel J. Math. [KM] B. Klartag and E. Milman, On volume distribution in 2-convex bodies, Israel J. Math. 64 (2008), pp [Mi] E. Milman, Dual mixed volumes and the slicing problem, Adv. Math. 207 (2006), pp

15 ON THE ISOTROPY CONSTANT OF PROJECTIONS OF POLYTOPES 5 [MP] V. Milman and A. Pajor, Isotropic positions and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space, in: Geometric aspects of functional analysis (987 88), Lecture Notes in Math. 376, pp , Springer, Berlin, 989. [P] G. Pisier, The volume of convex bodies and Banach space geometry, Cambridge Univ. Press, Cambridge 989. [RS] C.A. Rogers and G.C. Shephard, The difference body of a convex body, Arch. Math. (Basel) 8 (957), pp [S] R. Schneider, Convex bodies: the Brunn-Minkowski theory, Cambridge Univ. Press, Cambridge 993. Dpto. de Matemáticas, Facultad de Ciencias, Universidad de Zaragoza, Zaragoza, Spain address, (David Alonso): daalonso@unizar.es address, (Jesús Bastero): bastero@unizar.es address, (Julio Bernués): bernues@unizar.es address, (Pawe l Wolff): pawel.wolff@case.edu Department of Mathematics, Case Western Reserve University, Cleveland, OH , U.S.A.