Sharp Bounds on Geometric Permutations of Pairwise Disjoint Balls in R d

Abstract

We prove that the maximum number of geometric permutations, induced by line transversals to a collection of n pairwise disjoint balls in \R d , is Θ (n d-1 ) . This improves substantially the upper bound of O(n 2d-2 ) known for general convex sets [9]. We show that the maximum number of geometric permutations of a sufficiently large collection of pairwise disjoint unit disks in the plane is two, improving the previous upper bound of three given in [5].