Equal area partitions of the sphere with diameter bounds, via optimal transport

TL;DR

This paper proves the existence of equal area partitions of the sphere with diameter bounds using optimal transport, providing tools for analyzing partitions and distances on the sphere.

Contribution

It introduces a novel approach to partition the sphere with equal areas and diameter bounds via optimal transport methods, with applications to Monge--Kantorovich distances.

Findings

Existence of equal area partitions with diameter bounds

Bounds on maximum diameter of partitions when sampling points uniformly

Application to sliced Monge--Kantorovich distances

Abstract

We prove existence of equal area partitions of the unit sphere via optimal transport methods, accompanied by diameter bounds written in terms of Monge--Kantorovich distances. This can be used to obtain bounds on the expectation of the maximum diameter of partition sets, when points are uniformly sampled from the sphere. An application to the computation of sliced Monge--Kantorovich distances is also presented.