Stability of Supporting and Exposing Elements of Convex Sets in Banach Spaces

Abstract

To a convex set in a Banach space we associate a convex function (the separating function), whose subdifferential provides useful information on the nature of the supporting and exposed points of the convex set. These points are shown to be also connected to the solutions of a minimization problem involving the separating function. We investigate some relevant properties of this function and of its conjugate in the sense of Legendre-Fenchel. Then we highlight the connections between set convergence, with respect to the slice and Attouch-Wets topologies, and convergence, in the same sense, of the associated functions. Finally, by using known results on the behaviour of the subdifferential of a convex function under the former epigraphical perturbations, we are able to derive stability results for the set of supported points and of supporting and exposing functionals of a closed convex subset of a Banach space.