Discoveries / probability-statistics

Talagrand's Convexity Problem

Talagrand's convexity problem asks whether a universal number of Minkowski sum operations turns any set of large Gaussian measure into one containing a convex body of comparable measure. It is equivalent to a question about subgaussian vectors: is every centered $1$-subgaussian random vector in $\mathbb{R}^n$ the sum of a universal number of standard Gaussian vectors? Both are answered affirmatively, via the sharper statement that any random vector dominated in convex order by a standard Gaussian is the sum of three standard Gaussian vectors.