Pointwise Estimates for Marginals of Convex Bodies

TL;DR

This paper establishes a pointwise approximation of the marginal densities of isotropic log-concave convex bodies by Gaussian densities in high-dimensional subspaces, extending the central limit theorem to a pointwise setting.

Contribution

It proves a pointwise version of the central limit theorem for convex bodies, showing Gaussian approximation for projections in high-dimensional subspaces.

Findings

Density ratios are close to 1 in large parts of the subspace.

The result applies to typical subspaces of dimension n^c.

It complements previous total-variation results with pointwise estimates.

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 R^n with a log-concave density. For a typical subspace E in R^n 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.