Topical and sub-topical functions, downward sets and abstract convexity

Abstract

We study topical and sub-topical functions (i.e., functions which are increasing in the natural partial ordering of ℝn and additively homogeneous, respectively additively sub-homogeneous), and downward sets (i.e., subsets of ℝn which contain, along with each element, all smaller elements), in the framework of abstract convex analysis, with the aid of the additive min-type coupling function . We study primal and dual link between topical functions and closed downward sets, via plus-Minkowski gauges and ϕ-support functions of sets and level sets and ϕ-support sets of functions. Also, we give characterizations of topical functions in terms of their Fenchel-Moreau conjugates and biconjugates with respect to the above coupling function ϕ