Restricted Weak Upper Semi-continuity of Subdifferentials of Convex Functions on Banach Spaces

Abstract

Let X be a Banach space and f a continuous convex function on X. Suppose that for each x ∈ X and each weak neighborhood V of zero in X* there exists δ > 0 such that $$\partial f(y)\subset\partial f(x)+V\;\;{\rm for\;all}\;y\in X\;{\rm with}\;\|y-x\|<\delta. $$ Then every continuous convex function g with \(g \leqslant f\) on X is generically Frechet differentiable. If, in addition, \(\lim\limits_{\|x\|\rightarrow\infty}f(x)=\infty\), then X is an Asplund space.