Generic Fréchet Differentiability of Convex Functions Dominated by a Lower Semicontinuous Convex Function

In this paper, an extended real-valued proper lower semicontinuous convex functionfon a Banach space is said to have the Fréchet differentiability property (FDP) if every proper lower semicontinuous convex functiongwithg≤fis Fréchet differentiable on a denseGδsubset of intdomg, the interior of the effective domain ofg. We show thatfhas the FDP if and only if thew*-closed convex hull of the image of the subdifferential map offhas the Radon–Nikodým 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.