On nearly radial marginals of high-dimensional probability measures

Abstract

Abstract. Suppose that µ is an absolutely continuous probability measure on R n , for large n. Then µ has low-dimensional marginals that are approximately spherically-symmetric. More precisely, if n ≥ (C/ε) Cd , then there exist d-dimensional marginals of µ that are ε-far from being sphericallysymmetric, in an appropriate sense. Here C > 0 is a universal constant.

Cone-volume measure of general centered convex bodies

“…Carlen and D. Cordero-Erausquin [14], and J. Bennett, A. Carbery, M. Christ and T. Tao [7] in their study of the Brascamp-Lieb inequality. Moreover, the case when strict inequality holds for all subspaces in (1.3) for a measure μ is due to B. Klartag [27]. Isotropic measures on S n−1 are also discussed, e.g.…”

Affine images of isotropic measures

“…One of the aims of this paper is to extend work of Carlen and CorderoErausquin [11] for discrete measures, and Klartag [37] for arbitrary measures, and provide an answer to Problem 1.1.…”

“…Klartag [37] established that if a general measure satisfies the strict subspace concentration inequality, then it has an affine isotropic image.…”

Approximations of convex bodies by measure-generated sets

“…We remark that the proof of Corollary 1.7 (or, equivalently, of Theorem 1.5) is based on high-dimensional phenomena, rather than concentration of measure (which is used to obtain the same result in the case of log-concave measures). Other results in the spirit of Corollary 1.7, where the log-concavity assumption on the measure may be relaxed, can be found in [15,24,16,17,18]. This paper is organized as follows.…”