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 inR d ,i s2.n di1 /. This improves substantially the upper bound of O.n 2di2 / 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).