Generic Gâteaux-differentiability of convex functions on small sets

Abstract

The theory of generic differentiability of convex functions on Banach spaces is by now a well-explored part of infinite-dimensional geometry. All the attempts to solve this kind of problem have in common, as a working hypothesis, one special feature of the finite-dimensional case. Namely, convex functions are always considered to be defined on convex sets with nonempty interior. But typically, a convex set in a Banach space does not have interior points even when it is not contained in a closed hyperplane. So this raises the problem of expounding a theory of differentiability of convex functions defined on small sets, i.e., sets without interior points. In our paper [N2] we have made an attempt to solve this problem, discussing questions of generic Frechet-differentiability of convex functions on small sets. In the present paper we deal with problems of generic Gateauxdifferentiability in this context.