A central limit theorem for convex sets

Abstract

We show that there exists a sequence ε n ց 0 for which the following holds: Let K ⊂ R n be a compact, convex set with a non-empty interior. Let X be a random vector that is distributed uniformly in K. Then there exist a unit vector θ in R n , t 0 ∈ R and σ > 0 such thatwhere the supremum runs over all measurable sets A ⊂ R, and where ·, · denotes the usual scalar product in R n . Furthermore, under the additional assumptions that the expectation of X is zero and that the covariance matrix of X is the identity matrix, we may assert that most unit vectors θ satisfy ( * ), with t 0 = 0 and σ = 1. Corresponding principles also hold for multidimensional marginal distributions of convex sets.

On nearly radial marginals of high-dimensional probability measures

“…There is no hope for approximate Gaussians. Theorem 1.3 bears a strong relation to the proof of the central limit theorem for convex bodies presented in [14,15] (see [16] for another proof, which at present works only for a subclass of convex bodies). That proof begins by showing that marginals of the uniform measure on a convex body are approximately spherically-symmetric.…”

Isoperimetry for spherically symmetric log-concave probability measures

“…Otherwise, since we are aiming at results which do not depend on n and because of the Central-Limit Theorem ( [13]), we cannot expect better isoperimetric profile than the one of the Gaussian measure, proportional to…”

“…The first lemma is due to Klartag [13]. The balls are centered at the maximum of density in order to capture a large fraction of the mass.…”

“…Lemma 8 (Klartag [13]). Let ν(dr) = r n−1 ρ(r) dr be a probability measure on R + with ρ a log-concave function of class C 2 .…”

“…When φ satisfies (H1'), inequalities of Theorem 4 are optimal in a for n = 1, according to Lemma 14. In the supergaussian case, the central limit theorem for convex bodies of Klartag (see [13]), in the simpler case of spherically symmetric distributions, ensures that we cannot find a profile bounding from below Is µ n,φ for all n, better than the Gaussian one (times a constant depending possibly on n). Else we should have concentration properties stronger than Gaussian.…”