On Talagrand's Convexity Conjecture

Abstract

We prove that any random vector in ${\mathbb{R}}^n$ which is dominated in convex order by a standard Gaussian vector can be written as the sum of three standard Gaussian vectors. This implies that any $1$-subgaussian random vector in ${\mathbb{R}}^n$ is the sum of a universal number of Gaussian vectors. It also solves M. Talagrand's convexity problem, which in turn implies a weak version of a combinatorial analogue to the problem.

22 pages, no figures. Comments welcomed! Major changes in v2: Supplied second half of the if and only if in Proposition 3.6. New proof of the main theorem in Section 5 gives explicit constants. Connections with the existing literature in Appendix B expanded on in more detail