significant contributions regarding the central limit problem for convex sets were made by Bastero and Bernués [@bb], Bobkov [@bobkov], Bobkov and Koldobsky [@bobkov_koldobsky], Brehm and Voigt [@brehm_voigt], Brehm, Hinow, Vogt and Voigt [@BHVV], Koldobsky and Lifshits [@koldobsky_lifshits], E. and M. Meckes [@meckes], E. Milman [@emanuel], Naor and Romik [@NR], Paouris [@pa_clt], Romik [@romik_phd], S. Sodin [@sodin], Wojtaszczyk [@woj] and others.

Let us explain a few ideas from our proof. We begin with a general principle that goes back to Sudakov [@sudakov] and to Diaconis and Freedman [@dia_thin_shell] (see also the expositions of Bobkov [@bobkov] and von Weizsäcker [@weizs]. A sharpening for the case of convex bodies was obtained by Anttila, Ball and Perissinaki [@ABP]). This principle reads as follows: Suppose $X$ is any random vector in $\RR^n$ with zero mean and identity covariance matrix. Then most of the marginals of $X$ are approximately gaussian, if and only if the random variable $|X| / \sqrt{n}$ is concentrated around the value one. In other words, typical marginals are approximately gaussian if and only if most of the mass is concentrated on a “thin spherical shell” of radius $\sqrt{n}$ and width much smaller than $\sqrt{n}$. Therefore, to a certain extent, our task is 