Extensions of convex functions with prescribed subdifferentials
Volume 253 / 2020
Studia Mathematica 253 (2020), 199-213
MSC: 26B05, 26B25, 49J52, 54C20, 54C60.
DOI: 10.4064/sm181212-17-5
Published online: 21 January 2020
Abstract
Let $E$ be an arbitrary subset of a Banach space $X$, $f: E \rightarrow \mathbb {R}$ a function, and $G:E \rightrightarrows X^*$ a set-valued mapping. We give necessary and sufficient conditions on $f, G$ for the existence of a continuous convex extension $F: X \rightarrow \mathbb {R} $ of $f$ such that the subdifferential $\partial F$ of $F$ coincides with $G$ on $E.$
