A constructive solution to Talagrand's Gaussian convexification problem

Source abstract

Talagrand asked for an explicit construction of a large convex subset of a bounded Minkowski sum of a large Gaussian set. Let $A\subset\mathbb{R}^n$ be closed with $γ_n(A)\ge 7/8$. We prove that for every $λ>4$ and every $4\le t 0$ such that, for $n\ge2$, one can construct an ellipsoid $E$ satisfying $$γ_n(E)\ge\frac12 \qquad\text{and}\qquad c\sqrt{\frac{\log n}{n}}\,E\subset λ(A+A+A).$$ The same finite-barrier method also yields a constructive bounded-step convexification result for balanced Gaussian sets.