  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.
