Discontinuities in optimal transport between nonconvex domains

Abstract

We develop a new approach to studying discontinuity sets of measure-preserving convex gradient maps between nonconvex domains. Such maps optimize the Euclidean distance squared. Our idea is to introduce a new `phantom' optimal transport problem, that transports the missing mass from the convex hull of the target to the `excess' Monge--Ampère measure of the minimal convex extension of the Brenier potential. We show that the closure of the discontinuity set of the original map contains the support of this phantom target. Moreover, inside a planar source domain these two sets actually coincide; on $n$-dimensional domains they coincide $\mathcal{H}^{n-1}$-a.e. The Hausdorff exponent $n-1$ is sharp in every dimension $n\geq3$. For planar polygonal targets, we establish a quantitative global length bound for the closure of the discontinuity set depending only on the target geometry, valid for bounded positive densities.