On Talagrand's Convexity Conjecture

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.