Gravitational Allocation For Uniform Points On The Sphere

Given a collection L of n points on a sphere S-n(2) of surface area n, a fair allocation is a partition of the sphere into n cells each of area 1, and each associated with a distinct point of L. We show that if the n points are chosen uniformly at random and the partition is defined by considering a "gravitational" potential defined by the n points, then the expected distance between a point on the sphere and the associated point of L is O (root log n) which is optimal by a result of Ajtai, Komlos and Tusnady. Furthermore, we prove that the expected number of maxima for the gravitational potential is Theta(n/log n). We also study gravitational allocation on the sphere to the zero set L of a particular Gaussian polynomial, and we quantify the repulsion between the points of L by proving that the expected distance between a point on the sphere and the associated point of L is O(1).