On the Representation of Approximate Subdifferentials for a Class of Generalized Convex Functions

Abstract

Given a family of real-valued functions Φ defined in a normed vector space X, we study a class of Φ-convex functions having a simpler representation for the ɛ-Φ-subdifferential. The case Φ=X* with X being a Banach space (the Fenchel case) is particularly analysed, and we find that the sublinear lower semicontinuous functions satisfy the simpler representation with respect to X*. As a side result, we provide various new subdifferential-type charaterizations of positively homogeneous functions among those which are lower semicontinuous and convex. In addition, we also discuss that family Φ related to the the so-called prox-bounded functions. In this more general framework our simpler representation may give rise to a new notion of enlargement of the subdifferential.