On the Regularity of Optimal Transportation Potentials on Round Spheres

TL;DR

This paper investigates the regularity of optimal transportation potentials on round spheres, extending previous results to broader cost-functions via stereographic projection and verifying the (A3) condition.

Contribution

It generalizes Loeper's calculation of the (A3) condition on spheres to a wider class of cost-functions using stereographic projection techniques.

Findings

The (A3) condition is satisfied for cost-functions expressed as functions of geodesic distance.

The stereographic projection approach effectively verifies the (A3) condition on the sphere.

Several examples of cost-functions are analyzed for (A3) satisfaction.

Abstract

In this paper the regularity of optimal transportation potentials defined on round spheres is investigated. Specifically, this research generalises the calculations done by Loeper, where he showed that the strong (A3) condition of Trudinger and Wang is satisfied on the round sphere, when the cost-function is the geodesic distance squared. In order to generalise Loeper's calculation to a broader class of cost-functions, the (A3) condition is reformulated via a stereographic projection that maps charts of the sphere into Euclidean space. This reformulation subsequently allows one to verify the (A3) condition for any case where the cost-fuction of the associated optimal transportation problem can be expressed as a function of the geodesic distance between points on a round sphere. With this, several examples of such cost-functions are then analysed to see whether or not they satisfy this (A3) condition.