Revisiting the Bobkov-Chistyakov-Götze theorem on approximately Gaussian marginals

We study one-dimensional marginals of an isotropic, log-concave probability measure in ${\mathbb{R}}^n$. We show that the Kolmogorov distance between a typical marginal and the standard Gaussian distribution is at most $C\mathrm{/}n$, for a universal constant $C>0$. This bound is optimal in general, up to the value of the universal constant. It improves the Gaussian approximation bound of Bobkov, Chistyakov and Götze by a logarithmic factor in the log-concave setting.