Extensions of convex functions with prescribed subdifferentials

Abstract

Let E be an arbitrary subset of a Banach space X, f : E → R be a function, and G : E ⇒ X * be a set-valued mapping. We give necessary and sufficient conditions on f, G for the existence of a continuous convex extension F : X → R of f such that the subdifferential ∂F of F coincides with G on E.