DOI: 10.1515/ms-2017-0162 Math. Slovaca 68 (2018), No. 5, 1001–1008 SOME RESULTS ON ABSTRACT CONVEXITY OF FUNCTIONS Hossein Mohebi* — Hasan Barsam** (Communicated by J´an Bors´ ık ) ABSTRACT. In this paper, we study abstract convexity topical and sub-topical functions. We obtain some results on abstract convexity such as support set and subdifferential in view of new elementary functions. Indeed, first we show that the ICR and IR functions are dense in IPH functions then the topical function and sub-topical function on topological vector space by another elementary function is studied. These elementary functions lead to obtain similar results with easier proofs. c 2018 Mathematical Institute Slovak Academy of Sciences 1. Introduction What is abstract convexity? It is well-known that a lower semicontinous convex function is pointwise supremum of the class of affine functions. Indeed, each lower semicontinous convex function can be represented by a class of affine functions. In [7] Rubinov showed that this work can be extended to the other functions. Hence, abstract convexity is representation of a function f as the upper envelope of a class of functions. Moreover, in this case f is so-called abstract convex with respect to this class. In [5], [4], [3] abstract convexity of increasing positively homogeneous IPH functions and topical functions have been studied. In [2–4, 6, 8] abstract convexity of topical function and sub-topical function on topological vector space has been studied. In this paper, first we show increasing co-radiant ICR functions and increasing radiant IR functions are dense in IPH functions, also we study topical functions and sub-topical functions on topological vector space by another elementary functions. These elementary functions lead to obtain similar results with easier proofs. The structure of the paper is as follows. In Section 2, we collect definitions, notations and preliminary results related to topical functions. In Section 3 we study density IPH functions. In Section 4, we obtain some results on support set, subdifferential set and some other notions for topical functions in the framework of abstract convexity. Finally, similar to Section 5 we get some results on abstract convexity sub-topical functions. 2. Preliminaries Let X be a topological vector space. We assume that X is equipped with a closed convex, normal (i.e., if there exists a constant m> 0 such that ‖x‖≤ m‖y‖, whenever 0 ≤ x ≤ y and x, y ∈ X) 2000 Mathematics Subject Classification: Primary 26B25; Secondary 49N15, 06F20. K e y w o r d s: abstract convexity, topical function, sub-topical function, elementary function, support set, subdiffer- ential. 1001 Brought to you by | Université de Strasbourg Authenticated Download Date | 10/27/18 8:48 AM