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Simplices in the Euclidean Ball

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Simplices in the Euclidean Ball Simplices in the Euclidean ball Matthieu Fradelizi, Grigoris Paouris, Carsten Sch¨utt Abstract We establish some inequalities for the second moment Z 1 2 jxj2dx jKj K of a convex body K under various assumptions on the position of K. 1 Introduction The starting point of this paper is the article [3], 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 Z 2 1 2 r jxj2dx > (1) jKj K 9n where jxj2 stands for the Euclidean norm of x and jKj 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 Z 1 2 C2(K) := jxj2dx ; (2) jKj K 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). n n Let n > 2. We denote by K the set of all convex bodies in R , i.e. the set of compact convex sets with non empty interior and by ∆n the regular simplex in Rn n−1 n with vertices in S , the Euclidean unit sphere. For K 2 K , we denote by gK , its centroid, 1 Z gK = xdx: jKj K Under these notations we prove the following theorem. 1 Theorem 1.1. Let r > 0, K 2 Kn such that all its extreme points have Euclidean norm greater than r. Then Z 2 2 1 2 n n + 1 2 r + (n + 1)jgK j2 C2(K) := jxj2dx > C2(r∆ ) + jgK j2 = : jKj K n + 2 n + 2 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 x2 = ∆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 2 Kn. We say that K is in John position if the ellipsoid of maximal volume contained 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 [6] (see also [7]) that if K 2 K satisfies gK = 0 and 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 + 3 1 + (n + 3)jg j2 = C (Bn) ≥ C (K) ≥ C (∆n) + jg j2 = K : 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 1 (n + 1)(n + 2) Let K be a convex body in John position. Then n n2 n2 + (n2 + n + 2)jg j2 = C (Bn) ≤ C (K) ≤ C (n∆n)+ 1 + jg j2 = 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 1 3 2 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. n n n 1. If K ⊂ B2 and 0 2 K then C2(K) ≤ C2(B2 ) = n+2 . n n n 2. If K ⊃ B2 then C2(K) ≥ C2(B2 ) = n+2 . We do not have uniqueness in the first case of this proposition. Take e.g. one half of an Euclidean unit ball. 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 = f(x; y) 2 L; jyj ≤ 1=2g 1+a2 5+15a2 with a large enough. Indeed, C2(L) = 6 and C2(K) = 72 . The paper is organized as follows. In x2, we gather some background material needed in the rest of the paper. We prove Theorem 1.1 in x3, Theorem 1.2 in x4 and Proposition 1.3 in x5. 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 2 Rn, and K 2 Kn, one has Z 1 2 2 C2(K − a) = jx − aj2dx = C2(K) − 2hgK ; ai + jaj2: jKj K Hence 2 C2(K − gK ) = C2(K) − jgK j2 (3) n minimizes C2(K − a) among translation a 2 R . Let T be a non-singular linear transform, then Z Z 1 2 1 2 C2(TK) = jxj2dx = jT xj2dx: jTKj TK jKj K The preceding quantity may be computed in terms of C2(K) if K is in isotropic position (see below). 3 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 X X I = ciui ⊗ ui and ciui = 0: i=1 i=1 Notice that in this case, one has, for x 2 Rn N N N X 2 X 2 X x = cihx; uiiui; jxj2 = cihx; uii and ci = n: (4) i=1 i=1 i=1 Moreover, for any linear map T on Rn, its Hilbert-Schmidt norm is given by N N 2 ? X ? X 2 kT kHS := tr(T T ) = cihui;T T uii = cijT uij2: i=1 i=1 If A is an affine transformation and T its linear part, i.e. A(x) = T (x) + A(0), then N X 2 2 2 cijAuij2 = kT kHS + njA(0)j2 : (5) i=1 Indeed, N N N N X 2 X 2 2 X X 2 cijAuij2 = cijT ui + A(0)j2 = kT kHS + 2 cihT ui;A(0)i + cijA(0)j2 i=1 i=1 i=1 i=1 2 2 = kT kHS + njA(0)j2 : 2.2 John, L¨ownerand isotropic positions Let K 2 Kn. Recall that K is in John position if the ellipsoid of maximal volume 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 ([8], see also [1]) characterizes these positions. n Theorem 2.1. Let K 2 Kn. Then K is in John position if and only if B2 ⊆ K n−1 and there exist u1; : : : ; uN 2 @K \ S that form a representation of identity. n Also K is in L¨owner position if and only if B2 ⊇ K and there exist u1; : : : ; uN 2 @K \ Sn−1 that form a representation of identity. 4 Let K 2 Kn. We say that K is in isotropic position if jKj = 1, gK = 0 and Z 2 2 n−1 hx; θi dx = LK ; 8 θ 2 S : (6) K 2 If K is in isotropic position then C2(K) = nLK . Note that the isotropic position is unique up to orthogonal transformations and that for any convex body K 2 Kn there exist an affine transformation A such that AK is in isotropic position (see [10] or [5]). The quantity LK is called the isotropic constant of K. For any non singular linear transform T on Rn, Z 2 hx; T xidx = LK trT: K In particular if K is isotropic and T 2 GLn then Z 2 1 2 2 ∗ kT kHS C2(TK) = jT xj2dx = LK trT T = C2(K) : (7) jKj K n 2 From the arithmetic geometric inequality, it implies that C2(TK) ≥ j det(T )j n C2(K). With (3), it gives, as it is well known, that the isotropic position minimizes C2(AK) among affine transforms A that preserve volume.
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