Gateaux Differentiability of Convex Functions and Weak Dentable Set in Nonseparable Banach Spaces

摘要

In this paper, we prove that if C~(??) is a ε-separable bounded subset of X~(??), then every convex function g ≤ σ_C is Gateaux differentiable at a dense G_δ subset G of X~? if and only if every subset of ?σ_C(0) ∩ X is weakly dentable. Moreover, we also prove that if C is a closed convex set, then dσ_C(x?) = x if and only if x is a weakly exposed point of C exposed by x~?. Finally, we prove that X is an Asplund space if and only if, for every bounded closed convex set C~* of X~*, there exists a dense subset G of X~(**) such that σ_(C?) is Gateaux differentiable on G and dσ_(C~?)(G) ? C~?. We also prove that X is an Asplund space if and only if, for every w~?-lower semicontinuous convex function f, there exists a dense subset G of X~(??) such that f is Gateaux differentiable on G and df(G) ? X~?.