Pointwise Estimates for Marginals of Convex Bodies

2007/08/18 by Eldan, Ronen, Klartag, Bo'az · 3 citations
#FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.0708.2513

Abstract

We prove a pointwise version of the multi-dimensional central limit theorem for convex bodies. Namely, let X be an isotropic random vector in Rⁿ with a log-concave density. For a typical subspace E in Rⁿ of dimension n^c, consider the probability density of the projection of X onto E. We show that the ratio between this probability density and the standard gaussian density in E is very close to 1 in large parts of E. Here c > 0 is a universal constant. This complements a recent result by the second named author, where the total-variation metric between the densities was considered.

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