Extensions of convex functions with prescribed subdifferentials

Abstract

Let $E$ be an arbitrary subset of a Banach space $X$, $f:E\to \mathbb{R}$ be a function, and $G:E\rightrightarrows X^{\ast }$ be a set-valued mapping. We give necessary and sufficient conditions on $f,G$ for the existence of a continuous convex extension $F:X\to \mathbb{R}$ of $f$ such that the subdifferential $\partial F$ of $F$ coincides with $G$ on $E.$

In this new version we added several examples in the introduction to illustrate the role of our conditions in an infinite-dimensional setting