A noncompact minimax theorem
Chung Ha · Pacific Journal of Mathematics · 1981
This paper contains an extension of Ky Fan's theorem on sets with convex sections (for the case two sets are involved) by relaxing the compactness condition.It is then applied to obtain a generalization of Sion's minimax theorem in which neither underlying set is assumed to be compact.Ky Fan gave his theorem on sets with convex sections for a family of n sets (n ^ 2) and its various interesting consequences in [1,2].It was recently extended in [4] by removing the compactness condition on the underlying sets.Our first result is the following Theorem 1, which is also an extension of the Fan's theorem for the case n = 2 in the same direction, but under a much weaker condition.The proof of Theorem 1 relies on the minimax inequality of Fan in its geometric formulation ([3], Theorem 2).It says that if Z is a nonempty compact convex set in a Hausdorίf topological vector space and if S is a subset of Z x Z such that the set {z e Z: (x, z) e S} is open in Z for each xeZ and the set {xeZ: (x, z) eS) is nonempty and convex for each ze Z, then there exists x o e Z such that (x Of x Q ) e S.