Gravitational allocation on the sphere

Abstract

Given a collection L of n points on a sphere Sn2 of surface area n, a fair allocation is a partition of the sphere into n parts each of area 1, and each is associated with a distinct point of L. We show that, if the n points are chosen uniformly at random and if the partition is defined by a certain “gravitational” potential, then the expected distance between a point on the sphere and the associated point of L is O(log⁡n). We use our result to define a matching between two collections of n independent and uniform points on the sphere and prove that the expected distance between a pair of matched points is O(log⁡n), which is optimal by a result of Ajtai, Komlós, and Tusnády.