A new ellipsoid associated with convex bodies

Abstract

It is shown that corresponding to each convex body there is an ellipsoid that is in a sense dual to the Legendre ellipsoid of classical mechanics. Sharp affine isoperimetric inequalities are obtained between the volume of the convex body and that of its corresponding new ellipsoid. These inequalities provide exact bounds for the isotropic constant associated with the new ellipsoid. Among other things, this leads to a new approach to establishing Ball’s maximal shadows conjecture (for symmetric convex bodies). Corresponding to each origin–symmetric convex (or more general) subset of Euclidean n-space, R, there is a unique ellipsoid with the following property: The moment of inertia of the ellipsoid and the moment of inertia of the convex set is the same about every 1-dimensional subspace of R. This ellipsoid is called the Legendre ellipsoid of the convex set. The Legendre ellipsoid and its polar (the Binet ellipsoid) are well-known concepts from classical mechanics. See Milman and Pajor [MPa1, MPa2], Lindenstrauss and Milman [LiM] and Leichtweis [Le] for some historical references. It has slowly come to be recognized that along side the Brunn-Minkowski theory there is a dual theory. The nature of the duality between this dual Brunn-Minkowski theory and the Brunn-Minkowski theory is subtle and not yet understood. It is easily seen that the Legendre (and Binet) ellipsoid is an object of this dual BrunnMinkowski theory. This observation leads immediately to the natural question regarding the possible existence of a dual analog of the classical Legendre ellipsoid in the Brunn-Minkowski theory. It is the aim of this paper to demonstrate the existence of precisely this dual object. In retrospect, one may well wonder why Research supported, in part, by NSF Grant DMS–9803261 Typeset by AMS-TEX 1 2 A NEW ELLIPSOID ASSOCIATED WITH CONVEX BODIES the new ellipsoid presented in this note was not discovered long ago. The simple answer is that the definition of the new ellipsoid becomes obvious only with the notion of L2-curvature in hand. However, the Brunn-Minkowski theory was only recently extended to incorporate the new notion of Lp-curvature (see [L2], [L3]). A positive definite n × n real symmetric matrix A generates an ellipsoid, (A), in R, defined by (A) = {x ∈ R : x·Ax ≤ 1}, where x·Ax denotes the standard inner product of x and Ax in R. Associated with a star-shaped (about the origin) set K ⊂ R is its Legedre ellipsoid, Γ2K, which is generated by the matrix [mij(K)] where mij(K) = n+ 2 V (K) ∫ K (ei ·x)(ej ·x) dx, with e1, . . . , en denoting the standard basis for R and V (K) denoting the ndimensional volume of K. We will associate a new ellipsoid Γ−2K with each convex body K ⊂ R. One approach to defining Γ−2K without introducing new notation is to first define it for polytopes and then use approximation (with respect to the Hausdorff metric) to extend the definition to all convex bodies. Suppose P ⊂ R is a polytope that contains the origin in its interior. Let u1, . . . , uN denote the outer unit normals to the faces of P , let a1, . . . , aN denote the areas (i.e., (n − 1)-dimensional volumes) of the corresponding faces and let h1, . . . , hN denote the distances from the origin to the corresponding faces. The ellipsoid Γ−2P is generated by the matrix [mij(P )] where mij(P ) = 1 V (P ) N ∑ l=1 al hl (ei ·ul)(ej ·ul). An alternate definition of the operator Γ−2 will be given after additional notation is introduced. The easily established affine nature of the operator Γ2 is formally stated in: Lemma 1. If K ⊂ R is star shaped about the origin, then for each φ ∈ GL(n),