On Generalized Convexity for Set and Vector-Valued Functions

It is well-known that a real valued function is convex if and only if its epigraph is a convex set; moreover, the class of quasiconvex functions, introduced by De Finetti in 1949 [9], and later developed by Mangasarian [19], is defined as the set of all functions with convex lower level sets, even if their epigraph is not convex. Hence, starting from these basic definitions, in this paper we discuss and compare different classes of generalized convex functions, already introduced in the literature, and we give new definitions based on the generalization of the concept of quasiconvexity, weakening the convexity of the lower level sets of a vector-valued function. More precisely, we present the generalizations of convex functions following a scheme concerning the generalized convex sets. We prove also some results about the weakened convexity of sets useful to understand the relationships among functions. These relationships involve the definition of epigraph, image set and level set of a vector-valued function.