A generalization of Helly’s theorem

Abstract

1. The purpose of this note is to present a generalization, stated as Theorem 2, of Helly's theorem on the intersection of convex sets. Throughout this paper, En will denote a euclidean space of n dimensions, and for any set A CEn, H(A) will denote the convex hull of A; i.e., the smallest convex set containing A. Furthermore, by d(A) =r we will mean that H(A) contains a linear manifold of dimension r, but no higher dimension. Finally, for any linear manifold P, P* will denote the homogeneous linear manifold of dim d(P) parallel to P. It is well known, [1; 2], that Helly's theorem is essentially a consequence of the following fact: Any set of n+2 points in En can be partitioned into two nonempty disjoint sets A and B in such a way that H(A)fnH(B) $ 0. In proving Theorem 2, we shall first establish Theorem 1, which is a generalization of the above proposition.