Fan’s inequality in geodesic spaces

Abstract. Fans minimax inequality is extended to the context of metric spaces with global nonpositive curvature. As a consequence, a much more general result on the existence of a Nash equilibrium is obtained. 1. Preliminaries Suppose that C is a nonempty compact and convex subset of a linear topological space. Fans minimax inequality asserts that any function f: CC! R+ which is quasi-concave in the \u85rst variable and lower semicontinuous in the second variable veri es the minimax inequality, (F) min y2C sup x2C f(x; y) sup z2C f(z; z): As is well known, this result is equivalent to the Brouwer Fixed Point Theorem. See [2], pp. 205-206. The aim of this work is to extend Fans minimax inequality to the framework of global NPC spaces, that is, to the complete metric spaces with global nonpositive curvature. De nition 1. A global NPC space is a complete metric space E = (E; d) for which the following inequality holds true: for each pair of points x0; x1 2 E there exists a point y 2 E such that for all points z 2 E; (NPC) d2(z; y)