Convex spaces: Classification by differentiable convex functions

Abstract

The differentiability, of a specified strength, of a convex function at a point, is shown to be characterised by the convergence of subdifferentials in the appropriate topology on the dual space. This is used to prove that if each gauge is densely differentiable then so is each convex function. The generic version of this is equivalent to a conjecture which, for Gateaux differentiability and Banach spaces, is the long standing open question of whether X × ℝ is Weak Asplund whenever X is. Some progress is made towards a resolution.