On Evenly Convex Functions

A subset of Rn is said to be evenly convex if it is the intersection of some family (possibly empty) of open halfspaces. This class of convex sets was introduced by Fenchel in 1952 in order to extend the polarity theory to nonclosed convex sets. This paper deals with functions with evenly convex epigraphs, the so-called evenly convex functions. We study the main properties of this class of convex functions that contains the important class of lower semicontinuous convex functions. In particular, a characterization of even convexity in terms of lower semicontinuity is given. We also show that the class of evenly convex functions is closed under the main operations.