Subdifferential representation of convex functions on $X^*$

TL;DR

This paper provides a new subdifferential representation for proper $w^*$-lower semicontinuous convex functions on dual spaces, extending classical results and incorporating the Radon-Nikodym property for refined estimates.

Contribution

It introduces a novel subdifferential representation formula for convex functions on dual spaces, including cases with the Radon-Nikodym property, advancing convex analysis theory.

Findings

Derived a subdifferential representation formula for convex functions on $X^*$.

Extended the representation to cases where $X^*$ has the Radon-Nikodym property.

Provided estimates involving $w^*$-strongly exposed points.

Abstract

In this paper, we obtain subdifferential representation of a proper $w^*$-lower semicontinous convex function on $X^*$ as follows: Let $g$ be a proper convex $w^*$-lower semicontinuous function on $X^*$. Assume that int dom $g$ $\neq\emptyset$ (resp. int (dom ($g^*|_X)$)$\neq\emptyset$). Then given any point $x_0^*$ $\in$ D ($\partial g\cap X$) and $x^*$ $\in$ dom $g$ (resp. $x^*\in X^*$), we have $$g(x^*)=g(x_0^*)+\sup\{\sum_{i=0}^{n-1}\langle x_i,x_{i+1}^*-x_i^*\rangle +\langle x_n,x^*-x_n^*\rangle \},$$ where the above supremum is taken over all integers $n$, all $x_i^*\in X^*$ and all $x_i\in\partial g(x_i^*)\cap X$ for $i=0,1,\cdots,n$. (resp. if, moreover, $X^*$ has the Radon-Nikodym property, then we may estimate the above supremum among the set of $w^*$-strongly exposed points of $g$.)