# Gravitational allocation for uniform points on the sphere

## TL;DR

This paper introduces a gravitational allocation method for partitioning a sphere among uniformly random points, achieving an expected matching distance of order ((
log n)), which is proven to be optimal.

## Contribution

It presents a novel gravitational allocation scheme for fair partitioning and matching on the sphere with optimal expected distance bounds.

## Key findings

- Expected distance between matched points is O(((
log n)))
- Partitioning scheme is fair and based on gravitational fields
- Optimality of the matching distance is established

## Abstract

Given a collection $\mathcal L$ of $n$ points on a sphere $\mathbf{S}^2_n$ of surface area $n$, a fair allocation is a partition of the sphere into $n$ parts each of area $1$, and each associated with a distinct point of $\mathcal L$. We show that if the $n$ points are chosen uniformly at random and the partition is defined by considering the gravitational field defined by the $n$ points, then the expected distance between a point on the sphere and the associated point of $\mathcal L$ is $O(\sqrt{\log n})$. 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 $O(\sqrt{\log n})$, which is optimal by a result of Ajtai, Koml\'os, and Tusn\'ady.