Regularly abstract convex functions with respect to the set of Lipschitz continuous concave functions. (2nd January 2023) - Gorokhovik, Valentin V.
Abstract : Given a set H of functions defined on a set X, á function f : X ↦ 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 C ˆ ( 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 ¯ to be L C ˆ -convex it is necessary and sufficient that f be lower semicontinuous and bounded from below by a Lipschitz continuous function; moreover, each L C ˆ -convex function is regularly L C ˆ -convex as well. We focus on L C ˆ -subdifferentiability of functions at a given point. We prove that the set of points at which an L C ˆ -convex function is L C ˆ -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 C ˆ θ of the set L C ˆ consisting of such Lipschitz continuous concave functions that vanish at the origin we introduce the notions of L C ˆ θ -subgradient and L C ˆ 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 C ˆ θ -subdifferentials as well as L C ˆ θ -subdifferential conditions for global extremum points are established. Symmetric notions of abstract L C ˇ -concavity and L C ˇ -superdifferentiability of functions where L C ˇ := L C ˇ ( X, R ) is the set of Lipschitz continuous convex functions are also considered.