Approximate convexity and submonotonicity in locally convex spaces (original) (raw)

We introduce some new concepts of local Lipschitz mappings, Clarke’s subdifferential, approximate convexity and sub-monotonocity 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’s 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.