A Simple Proof of the Sion Minimax Theorem

For convex subsets X of a topological vector space E, we show that a KKM principle implies a Fan-Browder type fixed point theorem and that this theorem implies generalized forms of the Sion minimax theorem. The von Neumann-Sion minimax theorem is fundamental in convex analysis and in game theory. von Neumann [8] proved his theorem for simplexes by reducing the problem to the 1-dimensional cases. Sion’s generalization [7] was proved by the aid of Helly’s theorem and the KKM theorem due to Knaster, Kuratowski, and Mazurkiewicz [5]. In a recent paper, Kindler [4] proved Sion’s theorem by applying the 1-dimensional KKM theorem (i.e., every interval in R is connected), the 1-dimensional Helly theorem (i.e., any family of pairwise intersecting compact intervals in R has nonempty intersection), and Zorn’s lemma (or other method). In this short note, for convex subsets X of a topological vector space E, we show that a KKM principle implies a Fan-Browder type fixed point theorem and that this theorem implies a generalized form of the Sion minimax theorem. Definition. If a multimap G : X ( X satisfies