Every closed convex set is the set of minimizers of some $C^{\infty }$-smooth convex function

Abstract

We show that for every closed convex set C in a separable Banach space X there is a C∞-smooth convex function f : X → [0, ∞) so that f-1 (0) = C. We also deduce some interesting consequences concerning smooth approximation of closed convex sets and continuous convex functions.