Subdifferential representation of convex functions on $X^{\ast }$

Abstract

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

14 pages