Helly-type theorems for spheres (Q1119911)

A sphere S of radius r in Euclidean n-space \(E^ n\) consists of all points at distance r from some fixed point called the center of S. Let F be a family of spheres in \(E^ n\). The author proves the following theorems: 1) If the centers of the spheres in F are on a flat of dimension m and every subfamily consisting of at most \(m+2\) spheres in F has nonempty intersection, then F has nonempty intersection. 2) If F has nonempty intersection and the centers of the spheres in F all lie on a flat of dimension m, then there exist \(m+1\) of fewer spheres in F whose intersection is equal to the intersection of all spheres of F. 3) If F has at least \(n+3\) different spheres and any \(n+1\) spheres in F have a common point, then F has nonempty intersection.