Essentially Smooth Lipschitz Functions

Abstract

In this paper we address some of the most fundamental questions regarding the differentiability structure of locally Lipschitz functions defined on separable Banach spaces. For example, we examine the relationship between integrability, D-representability, and strict differentiability. In addition to this, we show that on any separable Banach space there is a significant family of locally Lipschitz functions that are integrable, D-representable and possess desirable differentiability properties. We also present some striking applications of our results to distance functions. Academic Press

Integration of multivalued operators and cyclic submonotonicity

“…Since ∂ C f is a w * -cusco mapping ( [3]), combining Proposition 1 with formulas (6) and (7), we obtain in view of [18,Theorem 2.4] the following corollary.…”

“…Case 1: Suppose that U is convex. Then the set U ∩ Z is connected; so relation (29) yields g 1 = g 2 + c for some c ∈ R (see [5,Proposition 4.12] or [3,Proposition 5.9]). Since…”

“…Let us note that subdifferentiability properties of marginal functions have been studied by many authors; see for instance [3], [7], [8] and [11].…”

“…In a different context, a concept of cyclicity has also been used by Qi in [19], where the author characterizes operators that coincide with a subdifferential of some locally Lipschitz function, using the Lebesgue measure and (implicitly) the Rademacher theorem. Elaborating upon these ideas, Borwein and Moors [3] introduce and study the class S e (X) of essentially smooth (locally Lipschitz) functions, that is, functions f whose Clarke subdifferential ∂ C f is single-valued in the complement of a Haar null set. One of the main features of this class stems from the fact that for every f ∈ S e (X), the problem of finding a locally Lipschitz function g such that ∂ C g ⊆ ∂ C f has a unique solution modulo a constant (i.e., g = f + c).…”

Lipschitz functions with maximal Clarke subdifferentials are generic

“…Finally we note that in [1] a careful study was made of those Lipschitz functions whose Clarke's subdifferentials are minimal with respect to set inclusion, viewed as norm to w* upper semicontinuous multifunctions with nonempty convex compact images. These functions capture most of the concrete classes of Lipschitz functions occurring in practice -convex functions, smooth functions, distance functions in appropriately smooth norms, etc.…”