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- W2079577078 abstract "We give some separation theorems to extend the intersection theorem of von Neumann, Fan, and others, omitting hypotheses of convexity and local convexity for one of the coordinate spaces. Let X, Y be two nonempty compact convex sets, each in a Euclidean space. Let E, F be closed subsets of X x Y. Von Neumann's intersection theorem [6] asserts that, if for each x E X and y E Y, both E(y) = {x' E XI(x', y) E E} and F(x) = {y' E YI(x, y ) E F} are nonempty and convex, then E n F :A 0 . For the case where Y = X and F = {(x, x) x E X}, this theorem reduces to Kakutani's fixed-point theorem [4]. Ky Fan extended the theorem to locally convex spaces in 1952 [1]. As a further extension, we prove the theorem allowing X or Y to be nonconvex in a nonlocally convex topological vector space. Our main result is Theorem 3, a generalization of von Neumann's intersection theorem [6]. Its proof relies on some conclusions about separating disjoint graphs, which are stated below as Theorems 1 and 2. These elementary theorems may find additional uses of interest. Theorem 4 is a generalization of von Neumann's minimax theorem [5]. Let X, Y be topological spaces. A correspondence U: X -* Y is a function from X to the family of subsets of Y. A correspondence U: X -* Y is said to be with open graph (closed graph) if the set Gr U = {(x, y) E X x Y ly E U(x)} is open(closed) in X x Y. We identify U and Gr U. Let U l(y) = {x E X ly E U(x)}. Let coS denote the convex hull of the set S and ConvS denote the convex closure of the set S where S is a subset of a topological vector space, conventionally Hausdorff. The empty set is both convex and compact. The following is a separation theorem, which is a stronger form of Hausdorff separation principle. Theorem 1. Let X be a compact subset of a topological vector space E1 and Y a compact subset of a locally convex space E2. Let U and D be two correspondences from X into Y of closed graphs such that for each x E X, Received by the editors February 3, 1989 and, in revised form, June 6, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 54H25. ? 1991 American Mathematical Society 0002-9939/91 $1.00 + $.25 per page" @default.
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- W2079577078 date "1991-04-01" @default.
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- W2079577078 title "Separation and von Neumann intersection theorems" @default.
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