same in different appearances. The symbols $C, C^{\prime},
\bar{C}, \tilde{C}$ etc. denote universal constants that are assumed to be sufficiently large, while $c, c^{\prime}, \bar{c},
\tilde{c}$ etc. denote sufficiently small universal constants. We abbreviate $\log$ for the natural logarithm, $\EE$ for expectation, $Prob$ for probability and $Vol$ for volume.

*Acknowledgements.* I would like to thank Charles Fefferman, Emanuel Milman and Vitali Milman for interesting discussions on related subjects, and to Boris Tsirelson for mentioning the central limit problem for convex sets in his graduate course at Tel-Aviv University.

Some background on log-concave functions {#section2}
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Here we gather some useful facts pertaining mostly to log-concave densities. For more information about log-concave functions, the reader is referred to, e.g., [@ball_log_concave], [@dedicata] and [@lovasz]. The raison d’être of log-concave densities on $\RR^n$ stems from the classical Brunn-Minkowski inequality and its generalizations. Let $E \subset \RR^n$ be a subspace, and let $f: \RR^n \rightarrow [0, \infty)$ be an integrable function. We denote the marginal of $f$ with respect to the subspace $E$ by $$\pi_E(f)(x) = \int_{x + E^{\perp}} f(y) dy \ \ \ (x \in E)$$ where $x + E^{\perp}$ is the affine subspace in $\RR^n$ that is orthogonal to $E$ and passes 