Sum of Gaussian vectors and large sets

We show that for some constant $\kappa>0$, any centered $\kappa$-subgaussian random variable is equal to the sum of three standard Gaussian random variables, confirming a conjecture of M. Talagrand. We also prove that given $\Lambda\geq 1$, any centered random vector $X$ in $\mathbb{R}^n$ such that $\|X\|\leq \Lambda$ almost surely and $\|\mathrm{Cov}(X)\|\leq {\Lambda^2 }{e^{-\Lambda^2}}$ is equal to the sum of a universal number of standard Gaussian random vectors. In particular, a centered random vector is subgaussian if and only if it is a finite sum of Gaussian random vectors. We apply these results to settle the permutation invariant case of M. Talagrand's convexity problem, and to give optimal estimates on the largest ellipsoid contained in a sum of large sets in Gaussian spaces.