Regularly abstract convex functions with respect to the set of Lipschitz continuous concave functions

Abstract

Given a set H of functions defined on a set X, à function f : X Þ Ñ R is called abstract H-convex if it is the upper envelope of its H-minorants, i.e., such its minorants which belong to the set H; and f is called regularly abstract H-convex if it is the upper envelope of its maximal (with respect to the pointwise ordering) H-minorants. In the paper we first present the basic notions of (regular) H-convexity for the case when H is an abstract set of functions. For this abstract case a general sufficient condition based on Zorn's lemma for a H-convex function to be regularly H-convex is formulated.The goal of the paper is to study the particular class of regularly H-convex functions, when H is the set L p CpX, Rq of real-valued Lipschitz continuous classically concave functions defined on a real normed space X. For an extended-real-valued function f : X Þ Ñ R to be L p C-convex it is necessary and sufficient that f be lower semicontinuous and bounded from below by a Lipschitz continuous function; moreover, each L p C-convex function is regularly L p C-convex as well. We focus on L p Csubdifferentiability of functions at a given point. We prove that the set of points at which an L p C-convex function is L p C-subdifferentiable is dense in its effective domain. This result extends the well-known classical Brøndsted-Rockafellar theorem on the existence of the subdifferential for convex lower semicontinuous functions to the more wide class of lower semicontinuous functions. Using the subset L p C θ of the set L p C consisting of such Lipschitz continuous concave functions that vanish at the origin we introduce the notions of L p C θ -subgradient and L p C θ -subdifferential of a function at a point which generalize the corresponding notions of the classical convex analysis. Some properties and simple calculus rules for L p C θ -subdifferentials as well as L p C θ -subdifferential conditions for global extremum points are established. Symmetric notions of abstract L q C-concavity and L q C-superdifferentiability of functions where L q C :" L q CpX, Rq is the set of Lipschitz continuous convex functions are also considered.