Regularity of optimal maps on the sphere: the quadratic cost and the reflector antenna

TL;DR

This paper investigates optimal transportation problems on the sphere with quadratic and reflector antenna cost functions, demonstrating positive curvature conditions that ensure smooth solutions and regularity of optimal maps.

Contribution

It establishes uniform positivity of cost-sectional curvature for these problems, enabling application of existing regularity results to obtain smooth solutions and map regularity.

Findings

Cost-sectional curvature is uniformly positive for both problems.

Global smooth solutions exist for smooth positive data.

Optimal maps are Hölder continuous under weak assumptions.

Abstract

Building on the results of Ma, Trudinger and Wang \cite{MTW}, and of the author \cite{L5}, we study two problems of optimal transportation on the sphere: the first corresponds to the cost function $d^2(x,y)$, where $d(\cdot,\cdot)$ is the Riemannian distance of the round sphere; the second corresponds to the cost function $-\log|x-y|$, it is known as the reflector antenna problem. We show that in both cases, the {\em cost-sectional curvature} is uniformly positive, and establish the geometrical properties so that the results of \cite{L5} and \cite{MTW} can apply: global smooth solutions exist for arbitrary smooth positive data and optimal maps are H\"older continuous under weak assumptions on the data.