A generalization of a theorem of Fenchel

Abstract

2. We introduce the following notations and definitions. We denote by M a given set in an n-dimensional euclidean space En. We denote the convex hull of any set A CEn by HA. DEFINITION. A point pEEn has the k-point property (k-p.p.) with respect to M if there exist k (or fewer) points ql,q2, , qkEM such that pEzH{qi }. If p has the k-p.p. with respect to M then pzHM. DEFINITION. If each point pEHM has the k-p.p. with respect to M, then M is said to have the k-p.p. In this terminology the lemma in ?1 says that any set MCEn has the (n+1)-p.p. Fenchel has shown2 [2, p. 241, Satz A] that if M is and compact, then M has the n-p.p. As was pointed out by Bunt [3, p. 23, Stelling 15] the compactness condition of this theorem is superfluous and connected can be replaced by having at most n components. Bunt's argument is essentially the following. Suppose that pEHM does not have the n-p.p. with respect to M. Then p is an interior point of an n-simplex with the n +1 vertices in M. Reflect this simplex in the point p and erect with p as vertex the open convex cone on each of the faces of the reflected simplex. It is easy to verify that each such cone contains a point of M, namely a vertex of the original simplex. The n+1 cones are disjoint and the