On Gaussian Marginals of Uniformly Convex Bodies

Abstract

Recently, Bo&#39;az Klartag showed that arbitrary convex bodies have Gaussian marginals in most directions. We show that Klartag&#39;s quantitative estimates may be improved for many uniformly convex bodies. These include uniformly convex bodies with power type 2, and power type $p>2$ with some additional type condition. In particular, our results apply to all unit-balls of subspaces of quotients of $L_p$ for $1<p<\infty$. The same is true when $L_p$ is replaced by $S_p^m$, the $l_p$-Schatten class space. We also extend our results to arbitrary uniformly convex bodies with power type $p$, for $2 \leq p < 4$. These results are obtained by putting the bodies in (surprisingly) non-isotropic positions and by a new concentration of volume observation for uniformly convex bodies.