Examples of Convex Functions and Classifications of Normed Spaces

Abstract

. We study various properties of convex functions and their connections to the structure of the spaces on which they are defined. In particular, it is shown boundedness properties of convex functions on various bornologies are related to sequential convergence in dual topologies. Convex functions whose subdifferentials have range with nonconvex interior are constructed on nonreflexive spaces, and we exhibit examples of convex functions on infinite dimensional spaces whose subdifferentials have sparse domains. 1. Research supported by an NSERC Research Grant and by the Shrum Endowment. 2. Research supported by an NSERC Scientific International Exchange Grant. 3. NSERC postdoctoral fellow. 1991 AMS Classification. 46B20, 52A41. Key Words: Convex function, bounded subdifferential, domain, range, Mackey convergence, support points. 1. Introduction. It is often important to know when convex functions are bounded on a certain class of sets, for example, to show the Moreau-Yosida approxim...