Some properties of convex sets related to fixed point theorems

Abstract

This paper is a study of certain geometric properties of convex sets in topological vector spaces (which are always assumed to be separated). These properties are closely related to fixed point theorems. While Theorems 7, 9, and 10 are explicitly fixed point or coincidence theorems for non-compact convex sets, Theorem 8 includes Theorem 9 (a coincidence theorem) as a special case. Theorems 1, 2, and 3 may be called "matching theorems", as the conclusions in these theorems assert the existence of a certain "matching". Consider for example, the conclusion of Theorem 3: "then there exists a non-empty finite subset {xl, x2 . . . . , x,} of X such that the convex hull of {x~, xz ..... x,} contains a point of the corresponding intersection (~ A(xi)". Notice that when n = 1, this becomes i=1 x~eA(x 0, a fixed point. The property required of the finite subset {Xl, x2 . . . . . x,} is a "matching" property involving two intuitively opposite conditions: a larger subset {xl, x2 . . . . . x,} of X would make its convex hull larger (and thus make it easier to have the required property), but would make the corresponding intersection