Convexity beyond Vector Spaces, Alternative Theorems and Minimax Equality

The concept of convexlike (concavelike) functions was introduced by Ky Fan (1953), who has proved the rst minimax theorem without linear structure of the underlying spaces. Further extensions or generalizations of this concept have been used later in optimization and decision theory, but the most signiicant applications are in the framework of the game theory, where the strategy spaces are not endowed with natural algebraic structures. In the present paper one introduces new convexity and connect-edness conditions and establishes the relationships with other known convexlike type properties. The main results concern the minimax equality in a topological framework. They generalize classical mini-max theorems of Fan and KK onig, and are independent of most similar results known in the literature. The proofs make use of some special alternative theorems which also hold in a pure topological framework, without any vector space structure.