\, Proj_E(X) \, , \, Z_E \, \right) \leq \eps,$$ where $Z_E$ is a standard gaussian random vector in the subspace $E$. \[thm\_multi\]

That is, most $k$-dimensional marginals of an isotropic, log-concave function, are approximately gaussian with respect to the total-variation metric, provided that $k <<  \frac{\log
n}{\log \log n}$. Note the clear analogy between Theorem \[thm\_multi\] and Milman’s precise quantitative theory of Dvoretzky’s theorem, an analogy that dates back to Gromov [@gromov Section 1.2]. Readers that are not familiar with Dvoretzky’s theorem are referred to, e.g., [@GM Section 4.2], to [@dvo_30_years] or to [@linden]. Dvoretzky’s theorem shows that $k$-dimensional geometric projections of an $n$-dimensional convex body are $\eps$-close to a Euclidean ball, provided that $k <c
\eps^2 \log n$. Theorem \[thm\_multi\] states that $k$-dimensional marginals, or measure-projections, of an $n$-dimensional convex body are $\eps$-close to gaussian when $k <
c \eps^2 \log n / (\log \log n)$. Thus, according to Dvoretzky’s theorem, the geometric shape of the support of the marginal distribution may be approximated by a very regular body – a Euclidean ball, or an ellipsoid – whereas Theorem \[thm\_multi\] demonstrates that the marginal distribution itself is very regular; it is approximately normal.

More parallels between Theorem \[thm\_multi\] 