Restricted weak upper semi-continuity of subdifferentials of convex functions on Banach spaces

摘要

Let X be a Banach space and f a continuous convex function on X. Suppose that for each X E X and each weak neighborhood V of zero in X* there exists delta > 0 such that partial derivative f(y) subset of partial derivative f (x) + V for all y is an element of X with parallel to y - x parallel to < delta. Then every continuous convex function g with g <= f on X is generically Frechet differentiable. If, in addition, [GRAPHICS] f (x) = infinity, then X is an Asplund space.