Geometric permutations of balls with bounded size disparity

Abstract

We study combinatorial bounds for geometric permutations of balls with bounded size disparity in d-space. Our main contribution is the following theorem: given a set S of n disjoint balls in R d , if n is sufficiently large and the radius ratio between the largest and smallest balls of S is γ, then the maximum number of geometric permutations of S is O( γ log γ ). When d=2, we are able to prove the tight bound of 2 on the number of geometric permutations for S, which is the best possible bound because it holds even when γ=1. Our theorem shows how the number of permutations varies as a function of the size disparity among balls, thus gracefully bridging the gap between two extreme bounds that were known before: the O(1) bound for congruent balls, and the Θ( n d−1 ) bound for arbitrary balls.