Extensions of convex functions with prescribed subdifferentials

Daniel Azagra; Juan Ferrera; Javier Gómez-Gil; Carlos Mudarra

Extensions of convex functions with prescribed subdifferentials

Functional Analysis 2019-04-18 v2

Authors: Daniel Azagra , Juan Ferrera , Javier Gómez-Gil , Carlos Mudarra

Abstract

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

Keywords

convex analysis function theory banach spaces

Cite

@article{arxiv.1811.05538,
  title  = {Extensions of convex functions with prescribed subdifferentials},
  author = {Daniel Azagra and Juan Ferrera and Javier Gómez-Gil and Carlos Mudarra},
  journal= {arXiv preprint arXiv:1811.05538},
  year   = {2019}
}

Comments

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

Extensions of convex functions with prescribed subdifferentials

Abstract

Keywords

Cite

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