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 ℋn−1-a.e. The Hausdorff exponent n−1 is sharp in every dimension n≥3. 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.
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BibTeX
@misc{mccann2026discontinuities,
title = {{Discontinuities in optimal transport between nonconvex domains}},
author = {Robert McCann and Arghya Rakshit},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03950},
eprint = {2611.03950},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03950},
}
RIS
TY - UNPB
TI - Discontinuities in optimal transport between nonconvex domains
AU - Robert McCann
AU - Arghya Rakshit
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03950
UR - https://arxiv.org/abs/2611.03950
AB - 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 ℋ^n−1-a.e. The Hausdorff exponent n−1 is sharp in every dimension n≥3. 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.
ER -