Approximate convexity and submonotonicity in locally convex spaces

Abstract

We introduce some new concepts of locally Lipschitz mappings, Clarke subdifferential, approximate convexity and submonotonocity in locally convex spaces. We show that, if f is approximately convex and bounded above, then f is locally Lipschitz. We also prove that a Lipschitz function is approximately convex if and only if its Clarke subdifferential is a submonotone operator. Several properties of approximate convexity are discussed. Our results can be viewed as extensions and refinements of the previously known results from Banach spaces to locally convex spaces. © 2010 Iranian Mathematical Society.