Links between functions and subdifferentials

TL;DR

This paper explores the relationship between functions and their subdifferentials, identifying classes of lower semicontinuous functions that are uniquely determined or recoverable from their subdifferentials.

Contribution

It characterizes large classes of lower semicontinuous functions that are either subdifferentially determined or representable, advancing understanding of subdifferential calculus.

Findings

Identification of classes of functions with subdifferential determination

Conditions under which functions are subdifferentially representable

Extension of subdifferential theory to broader function classes

Abstract

A function in a class $\mathcal{F}(X)$ is said to be subdifferentially determined in $\mathcal{F}(X)$ if it is equal up to an additive constant to any function in $\mathcal{F}(X)$ with the same subdifferential. A function is said to be subdifferentially representable if it can be recovered from a subdifferential. We identify large classes of lower semicontinuous functions that possess these properties.