Lipschitz Continuity of Convex Functions

TL;DR

This paper establishes necessary and sufficient conditions for convex functions in Banach spaces to be Lipschitz continuous, using subdifferential and normal cone characterizations, with applications to extensions and Moreau envelopes.

Contribution

It introduces new criteria for Lipschitz continuity of convex functions based on subdifferential selections and normal cones, extending previous results.

Findings

Characterization of Lipschitz continuity via subdifferential boundedness

Conditions for Lipschitz continuity on bounded sets using boundary subdifferentials

Extension of Lipschitz convex functions and analysis of Moreau envelopes

Abstract

We provide some necessary and sufficient conditions for a proper lower semicontinuous convex function, defined on a real Banach space, to be locally or globally Lipschitz continuous. Our criteria rely on the existence of a bounded selection of the subdifferential mapping and the intersections of the subdifferential mapping and the normal cone operator to the domain of the given function. Moreover, we also point out that the Lipschitz continuity of the given function on an open and bounded (not necessarily convex) set can be characterized via the existence of a bounded selection of the subdifferential mapping on the boundary of the given set and as a consequence it is equivalent to the local Lipschitz continuity at every point on the boundary of that set. Our results are applied to extend a Lipschitz and convex function to the whole space and to study the Lipschitz continuity of its Moreau envelope functions.