A New Proof of the Bonnice-Klee Theorem

The purpose of this note is to give a new (and much easier) proof for a theorem of Bonnice-Klee [1]. Their result (Theorem 1 below) has proved to be quite useful in establishing a number of results concerning the convex hulls of certain sets (see [4]). It is a generalization of the following two classic results due respectively to Caratheodory and Steinitz. (a) If X is a subset of an n-dimensional real linear space En, and pEcon X (the convex hull of X), then pCcon U for some subset U of X with card U (cardinality of U) at most n + 1. (b) If XCEn, and pC int con X (the interior of con X), then p C int con U for some subset U of X with card U < 2n. The generalization is based on the notion of intermediate interiors. The d-interior of a set X CEn (denoted by intd X) is the set of all points p such that p is in the relative interior of some d-simplex contained in X; equivalently, pCintd X iff there exists a d-dimensional flat F through p such that p is in the interior of XnGF relative to F. The result may be stated as