On the compactness of the hyperspace of faces

Abstract

By a flat we mean a translate of a subspace of R and by a hyperplane, a flat of dimension d — 1. If X is a compact convex subset of R the symbols dim (X), rel int (X), and rel bd (X) denote respectively, the dimension of the flat generated by X, the interior, and the boundary of X with respect to the flat X generates. A hyperplane H is called a supporting hyperplane of X if H Π X Φ 0 and H n relint (X) = 0 . A set A is called a face of X if A = X, A = 0 or if there exists a supporting hyperplane H of X such that i = J ϊ n X The set of proper faces (those not X or 0) is denoted by ^(X). A set J3 is called a poonem of X if there exists sets j?o, Blf , £ m such that £ m = X and B^eJ^iB,) for i = 1, , m. The set of poonems of X is denoted by &*(X). A point x in X is called a ^-exposed [A -extreme] point if for some j <^k, x belongs to a i-dimensional face [i-dimensional poonem] of X. The symbols expfe (X) and ext^ (X) denote the set of fc-exposed and fc-extreme points of X respectively. A point x of X is called an exposed point of X if {x} G ^(X) and x is called an extreme point if whenever x 6 [a, b] c X, we have x = a or x = b, where [a, b] denotes the closed line segment from a to 6. The symbols ext (X) and exp (X) denote the extreme points and exposed points of X, respectively. Note that exp (X) — exp0 (X) and ext (X) = ext0 (X). Also, let q denote the Hausdorff metric. Finally, if DaR, the symbols CI(JD) and conv (D) denote the closure of D and the convex hull of D respectively. We require the following results