Given a collection L of n points on a sphere [Formula: see text] 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 [Formula: see text] 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 [Formula: see text], which is optimal by a result of Ajtai, Komlós, and Tusnády.
Keywords: allocation; bipartite matching; gravity; transportation.
Copyright © 2018 the Author(s). Published by PNAS.