Pointwise estimates for marginals of convex bodies

We prove a pointwise version of the multi-dimensional central limit theorem for convex bodies. Namely, let mu be an isotropic, log-concave probability measure on R-n. For a typical subspace E subset of R-n of dimension n(c), consider the probability density of the projection of mu 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. (c) 2007 Elsevier Inc. All rights reserved.