A constructive solution to Talagrand's Gaussian convexification problem

RESUMO

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 $\gamma_n(A)\ge 7/8$. 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 \[ C\subset \lambda(A+A+A) \qquad\text{and}\qquad \gamma_n(C)\ge 1-\frac1t. \] 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 $n\ge2$, one can construct an ellipsoid $E$ satisfying \[ \gamma_n(E)\ge\frac12 \qquad\text{and}\qquad c\sqrt{\frac{\log n}{n}}\,E\subset \lambda(A+A+A). \] The same finite-barrier method also yields a constructive bounded-step convexification result for balanced Gaussian sets.