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Simplices in the Euclidean Ball Matthieu Fradelizi, Grigoris Paouris, Carsten Schütt

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Simplices in the Euclidean Ball Matthieu Fradelizi, Grigoris Paouris, Carsten Schütt Simplices in the Euclidean ball Matthieu Fradelizi, Grigoris Paouris, Carsten Schütt To cite this version: Matthieu Fradelizi, Grigoris Paouris, Carsten Schütt. Simplices in the Euclidean ball. Canadian Mathematical Bulletin, 2011, 55 (3), pp.498-508. 10.4153/CMB-2011-142-1. hal-00731269 HAL Id: hal-00731269 https://hal-upec-upem.archives-ouvertes.fr/hal-00731269 Submitted on 12 Sep 2012 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Simplices in the Euclidean ball Matthieu Fradelizi, Grigoris Paouris,∗ Carsten Sch¨utt To appear in Canad. Math. Bull. Abstract We establish some inequalities for the second moment 1 2 |x|2dx |K| ZK of a convex body K under various assumptions on the position of K. 1 Introduction The starting point of this paper is the article [2], where it was shown that if all the extreme points of a convex body K in Rn have Euclidean norm greater than r > 0, then 1 r2 x 2dx > (1) K | |2 9n | | ZK where x 2 stands for the Euclidean norm of x and K for the volume of K. | | | | r2 We improve here this inequality showing that the optimal constant is , n + 2 with equality for the regular simplex, with vertices on the Euclidean sphere of radius r. We also prove the same inequality under the different condition that K is in L¨ownerposition. More generally, we investigate upper and lower bounds on the quantity 1 C (K) := x 2dx , (2) 2 K | |2 | | ZK under various assumptions on the position of K. Some hypotheses on K are neces- 2 sary because C2(K) is not homogeneous, one has C2(λK) = λ C2(K). Let n > 2. We denote by n the set of all convex bodies in Rn, i.e. the set of compact convex sets with nonK empty interior and by ∆n the regular simplex in Rn ∗supported by an NSF grant 1 n−1 n with vertices in S , the Euclidean unit sphere. For K , we denote by gK , its centroid, ∈ K 1 g = xdx. K K | | ZK Under these notations we prove the following theorem. Theorem 1.1. Let r > 0, K n such that all its extreme points have Euclidean norm greater than r. Then ∈K 1 n + 1 r2 + (n + 1) g 2 C (K) = x 2dx > C (r∆n) + g 2 = | K |2 . 2 K | |2 2 n + 2 | K |2 n + 2 | | ZK Moreover, if K is a polytope there is equality if and only if K is a simplex with its vertices on the Euclidean sphere of radius r. In Theorem 1.1, for a general K, we don’t have a characterization of the equal- ity case because we deduce it by approximation from the case of polytopes. We conjecture that the equality case is still the same. Notice that the condition imposed on K that all its extreme points have Eu- clidean norm greater than r is unusual. For example, if K has positive curvature, n n it is equivalent to either K rB2 or K rB2 = . Moreover, this hypothe- sis is not continuous with respect⊃ to the Hausdorff∩ distance.∅ Indeed, if we define P = conv(∆n, x), where x / ∆n is a point very close to the centroid of a facet of ∆n then the distance of ∆n∈and P is very small but the point x will be an extreme point of P of Euclidean norm close to 1/n, i.e. much smaller than 1, the Euclidean norm of the vertices of ∆n. Other conditions on the position of K may be imposed. To state it, let us first recall the classical definitions of John and L¨ownerposition. Let K n. We say that K is in John position if the ellipsoid of maximal volume contained∈ K in K is n B2 . We say that K is in L¨ownerposition if the ellipsoid of minimal volume that n contains K is B2 . n It was proved by Gu´edonin [5] (see also [6]) that if K satisfies gK = 0 and ∈K n if K ( K) is in L¨ownerposition (which is equivalent to say that B2 is the ellipsoid ∩ − n of minimal volume containing K and centered at the origin) then C2(K) C2(∆ ). Using the same ideas, we prove the following theorem. ≥ Theorem 1.2. Let K be a convex body in L¨ownerposition. Then n (n + 1)2 n + (n + 1)2 g 2 = C (Bn) C (K) C (∆n) + g 2 = | K |2 . n + 2 2 2 ≥ 2 ≥ 2 n(n + 2)| K |2 n(n + 2) Moreover, if K is symmetric, then n 2n = C (Bn) C (K) C (Bn) = . n + 2 2 2 ≥ 2 ≥ 2 1 (n + 1)(n + 2) 2 Let K be a convex body in John position. Then n n + 1 n2 + 2(n + 1) g 2 = C (Bn) C (K) C (n∆n) + 2 g 2 = | K |2 . n + 2 2 2 ≤ 2 ≤ 2 n + 2 | K |2 n + 2 Moreover, if K is symmetric, then n n = C (Bn) C (K) C (Bn ) = . n + 2 2 2 ≤ 2 ≤ 2 ∞ 3 The inequalities involving the Euclidean ball in Theorem 1.2 are deduced from the following proposition. Proposition 1.3. Let K be a convex body. 1. If K Bn and 0 K then C (K) C (Bn) = n with equality if and only ⊂ 2 ∈ 2 ≤ 2 2 n+2 if K = tx; 0 t 1, x S , where S Sn−1. { ≤ ≤ ∈ } ⊂ 2. If K Bn then C (K) C (Bn) = n , with equality if and only K = Bn. ⊃ 2 2 ≥ 2 2 n+2 2 In view of Proposition 1.3, it could be conjectured that for every centrally symmetric convex bodies K, L such that K L one has C2(K) C2(L). But this is not the case. It can be seen already in dimension⊂ 2, by taking≤ L = conv((a, 0), ( a, 0), (0, 1), (0, 1)) − − K = (x, y) L; y 1/2 { ∈ | | ≤ } 2 2 1+a 5+15a with a large enough. Indeed, C2(L) = 6 and C2(K) = 72 . The paper is organized as follows. In 2, we gather some background material needed in the rest of the paper. We prove§ Theorem 1.1 in 3, Theorem 1.2 in 4 and Proposition 1.3 in 5. § § § Acknowledgment. We would like to thank B. Maurey and O. Gu´edonfor dis- cussions. 2 Preliminaries As mentioned before the quantity C2(K) is not affine invariant. Let us investigate the behaviour of C2(K) under affine transform. We start with translations. For a Rn, and K n, one has ∈ ∈K 1 C (K a) = x a 2dx = C (K) 2 g , a + a 2. 2 − K | − |2 2 − K | |2 | | ZK Hence C (K g ) = C (K) g 2 (3) 2 − K 2 − | K |2 3 n minimizes C2(K a) among translation a R . Let T be a non-singular linear transform, then − ∈ 1 1 C (TK) = x 2dx = T x 2dx. 2 TK | |2 K | |2 | | ZTK | | ZK The preceding quantity may be computed in terms of C2(K) if K is in isotropic position (see below). 2.1 Decomposition of identity n−1 Let u1, . , uN be N points in the unit sphere S . We say that they form a representation of the identity if there exist c1, . cN positive integers such that N N I = c u u and c u = 0. i i ⊗ i i i i=1 i=1 X X Notice that in this case, one has, for x Rn ∈ N N N x = c x, u u , x 2 = c x, u 2 and c = n. (4) i i i | |2 i i i i=1 i=1 i=1 X X X Moreover, for any linear map T on Rn, its Hilbert-Schmidt norm is given by N N T 2 := tr(T ⋆T ) = c u ,T ⋆T u = c T u 2. HS i i i i| i|2 i=1 i=1 X X If A is an affine transformation and T its linear part, i.e. A(x) = T (x) + A(0), then N c Au 2 = T 2 + n A(0) 2 . (5) i| i|2 HS | |2 i=1 X Indeed, N N N N c Au 2 = c T u + A(0) 2 = T 2 + 2 c T u ,A(0) + c A(0) 2 i| i|2 i| i |2 HS i i i| |2 i=1 i=1 i=1 i=1 X X X X = T 2 + n A(0) 2 . HS | |2 2.2 John, L¨ownerand isotropic positions Let K n. Recall that K is in John position if the ellipsoid of maximal volume ∈K n contained in K is B2 and that K is in L¨ownerposition if the ellipsoid of minimal n volume that contains K is B2 . The following theorem ([7], see also [1]) characterizes these positions. 4 n Theorem 2.1. Let K n. Then K is in John position if and only if B2 K and there exist u , . , u∈ K ∂K Sn−1 that form a representation of identity.⊆ 1 N ∈ ∩ n Also K is in L¨owner position if and only if B2 K and there exist u1, .
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