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This paper continues the study of general convexity and general concavity which are described in an abstract form on arbitrary sets. The main feature is the systematic use of a very versatile technique introduced in this paper via ATM-maps and MTM-maps. In this sense we give simplest applications of ATM-maps and MTM-maps to fixed point theory, geometry, variational inequalities, and minimax theory. 1. General convex topological spaces Fundamental elements of general convexity. In this section we shall consider general convexity introduced in 1997 by Tasković [42] which is described in an abstract form on sets and topological spaces. Also, we formulate a fixed point theorem for general nonexpansive mappings in the general convex topological spaces. In this sense, let X be a nonempty set or topological space and let A : X× X → R+ := [0,+∞) be a function. In this part we considered a topological space X (or a nonempty set X) with a general convex structure denoted by G(x, y, λ) iff there exist a mapping G : X ×X × I → X (for the closed unit interval I := [0, 1]) and a bisection function g : (R+) → R+ satisfying the following inequality A ( z, G(x, y, λ) ) ≤ max { A(z, x), A(z, y), g ( A(z, x), A(z, y) )} (S) for all x, y, z ∈ X and arbitrary λ ∈ [0, 1]. Similarly, a topological space (or a nonempty set) X is with a general affine structure iff there exists a mapping G : X ×X × I → X such that (S) for all x, y, z ∈ X and arbitrary λ ∈ R. 2000 Mathematics Subject Classification. Primary: 47H10, 05A15. Secondary: 54H25.

General convexity, general concavity, fixed points, geometry, and min-max points