Examples of Convex Functions and Classifications of Normed Spaces

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.