A support theorem for generalized convexity and its applications

In the present paper we introduce a notion of (ω,t)-convexity as a natural generalization of the notion of usual t-convexity, t-strongly convexity, approximate t-convexity, delta t-convexity and many other. The main result of this paper establishes the necessary and sufficient conditions under which an (ω,t)-convex map can be supported at a given point by an (ω,t)-affine support function. Several applications of this support theorem are presented. For instance, new characterizations of inner product spaces among normed spaces involving the notion of (ω,t)-convexity are given.