H\"older continuity for optimal multivalued mappings

TL;DR

This paper investigates the regularity of multivalued optimal transportation maps between measures on spheres, establishing H"older continuity for these maps in the context of average squared distance optimization.

Contribution

It provides a quantitative analysis of H"older continuity for multivalued optimal maps between measures on spheres, extending previous theoretical results.

Findings

Established H"older continuity for bivalent optimal maps

Quantified regularity of multivalued maps on Euclidean spheres

Extended regularity results to average distance squared optimization

Abstract

Gangbo and McCann showed that optimal transportation between hypersurfaces generally leads to multivalued optimal maps - bivalent when the target surface is strictly convex. In this paper we quantify H\"older continuity of the bivalent map optimizing average distance squared between arbitrary measures supported on Euclidean spheres.