Geometric permutations of disjoint unit spheres

Abstract

We show that a set of n disjoint unit spheres in R(d) admits at most two distinct geometric permutations if n >=, 9, and at most three if 3 <=, n <= 8. This result improves a Helly-type theorem on line transversals for disjoint unit spheres in R 3 : if any subset of size at most 18 of a family of such spheres admits a line transversal, then there is a line transversal for the entire family. (c) 2004 Elsevier B.V. All rights reserved.