Generic Fr´echet Differentiability of Convex Functions Dominated by a Lower Semicontinuous Convex Function

In this paper, an extended real-valued proper lower semicontinuous convex function f on a Banach space is said to have the Fr´echet differentiability property FDP.if every proper lower semicontinuous convex function g with gFf is Fr´echet differentiable on a dense G subset of int dom g, the interior of the d effective domain of g. We show that f has the FDP if and only if the wU-closed convex hull of the image of the subdifferential map of f has the Radon]Nikod´ym property. This is a generalization of the main theorem in a paper by Lixin and Shuzhong to appear.. According to this result, it also gives several new criteria of Asplund spaces.