Regularity of optimal transportation between spaces with different dimensions

TL;DR

This paper investigates the regularity of optimal transportation solutions when the source space has higher dimension than the target, revealing conditions under which the problem simplifies or solutions remain continuous.

Contribution

It introduces a canonical foliation for the source when the target is c-convex, reducing the problem to equal dimension transportation, and identifies conditions for continuity in low-dimensional cases.

Findings

Source admits a canonical foliation reducing the problem to equal dimensions.

Regularity depends on c-convexity of the target.

Continuity of optimal maps is ensured under specific conditions in low-dimensional cases.

Abstract

We study the regularity of solutions to an optimal transportation problem where the dimension of the source is larger than that of the target. We demonstrate that if the target is $c$-convex, then the source has a canonical foliation whose co-dimension is equal to the dimension of the target and the problem reduces to an optimal transportation problem between spaces with equal dimensions. If the $c$-convexity condition fails, we do not expect regularity for arbitrary smooth marginals, but, in the case where the source is 2-dimensional and the target is 1 dimensional, we identify sufficient conditions on the marginals and cost to ensure that the optimal map is continuous.