A constructive solution to Talagrand's Gaussian convexification problem

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 ${\gamma }_n(A)\ge 7\mathrm{/}8$. We prove that for every $\lambda >4$ and every $4\le t<\lambda$, one can construct a finite polyhedron $C\subset {\mathbb{R}}^n$ such that $$C\subset \lambda (A+A+A)\text{and}{\gamma }_n(C)\ge 1-\frac{1}{t}\mathrm{.}$$ The construction is finite-dimensional: a separation argument produces polyhedral convex barriers, Gaussian integrability yields a uniform slope bound, and a finite net reduces the global problem to a finite convex minimization problem. As an application, we give a finite semidefinite construction of an optimal-scale ellipsoid. There is a universal constant $c>0$ such that, for $n\ge 2$, one can construct an ellipsoid $E$ satisfying $${\gamma }_n(E)\ge \frac{1}{2}\text{and}c\sqrt{\frac{\log n}{n}}E\subset \lambda (A+A+A)\mathrm{.}$$ The same finite-barrier method also yields a constructive bounded-step convexification result for balanced Gaussian sets.