Sum of Gaussian vectors and large sets

Abstract

We prove that the convexity problem of M. Talagrand is equivalent to the subgaussian vector problem: can any centered $1$-subgaussian random vector in $\mathbb{R}^n$ be realized as the sum of a universal number of standard Gaussian vectors? We introduce methods to study this problem and, using elementary arguments, we settle it for $1$-subgaussian random variables and random vectors with good norm and covariance bounds. These results already confirm the permutation invariant case of the convexity problem, and give optimal estimates on the largest ellipsoid contained in a sum of large sets in Gaussian spaces. We also propose a Riemannian version of the convexity problem for spaces with nonnegative Ricci curvature.