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

We study topical and sub-topical functions (i.e., functions f : Rn → R = [-∞, +∞] 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 functio