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

Given a set H of functions defined on a set X, a function f : X (sic) (R) over bar 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 regularlyH-convex functions, when H is the set L (C) over cap (X, R) of real-valued Lipschitz continuous classically concave functions defined on a real normed space X. For an extended-real-valued function f : X (R) over bar to be L (C) over cap -convex it is necessary and sufficient that f be lower semicontinuous and bounded from below by a Lipschitz continuous function; moreover, each L (C) over cap -convex function is regularly L (C) over cap -convex as well. We focus on L (C) over cap -subdifferentiability of functions at a given point. We prove that the set of points at which an L (C) over cap -convex function is L (C) over cap -subdifferentiable is dense in its effective domain. This result extends the well-known classical Brondsted-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 (C) over cap (theta) of the set L (C) over cap consisting of such Lipschitz continuous concave functions that vanish at the origin we introduce the notions of L (C) over cap (theta)-subgradient and L (C) over cap (theta)-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 (C) over cap (theta)-subdifferentials as well as L (C) over cap (theta)-subdifferential conditions for global extremum points are established. Symmetric notions of abstract L(sic)C-concavity and LC(sic)-superdifferentiability of functions where LC(sic) := LC(sic) (X, R) is the set of Lipschitz continuous convex functions are also considered.