Geometric Permutations of Disjoint Unit Spheres

http://www.elsevier.com/locate/comgeoWe show that a set of $n$ disjoint unit spheres in $R^d$ admits at most two distinct geometric permutations if $n \geq 9$, and at most three if $3 \leq n \leq 8$. This result improves a Helly-type theorem on line transversals for disjoint unit spheres in $R^3$: if any subset of size $18$ of a family of such spheres admits a line transversal, then there is a line transversal for the entire family