The purpose of this paper is to
prove for all positive integers n and r that if a family of n + 1 + 2r, or more, strongly
convex sets on the n dimensional sphere Sn is such that each intersection of
n + 1 + r of them is empty, then the intersection of some n + 1 of them must be
empty. (Sn is the set of points in n + 1 dimensional Euclidean space satisfying
x12+ x22+⋯+ xn+1l= 1. A set on a sphere is called strongly convex if it does not
contain any pair of diametrically opposite or antipodal points, and if together with
any two of its points it contains the whole of the minor arc of the great circle joining
them.)
