Helly-type theorems for spheres

Abstract

We prove that (i) a family F of at least n+3 spheres in En has nonempty intersection if each n+1 spheres of F have nonempty intersection, and (ii) if a family F of spheres in En has nonempty intersection, then there exist n+1 or fewer spheres in F whose intersection coincides with the intersection of all spheres of F. © 1989 Springer-Verlag New York Inc.