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Optimal regularity of optimal transport and degenerations of convex cones
Abstract
If T is an optimal transport map between bounded convex domains Ω and Ω', we characterize the regularity/singularity dichotomy for DT at (x,T(x)) ∈ ∂ Ω× ∂ Ω'. This is obtained through a new approach to the regularity theory based on affine degenerations of the tangent cones to $(Ω, Ω')$ at (x,T(x)). We formulate a general principle relating the optimal boundary regularity to a stability criterion and an SL(n) moduli space of pairs of convex cones. Among other results, we establish geometric criteria for existence and non-existence of homogeneous optimal transport maps between convex cones.
Related
- Global regularity and sparsity for regularised optimal transport
- Optimal transport to non-convex polyhedral domains
- Sharp Weighted Sobolev Trace Inequalities on Convex Cones and Rigidity of Extremals
- Monotone invariant valuations on convex cones
- Extremals for Sharp Trace Gagliardo--Nirenberg--Sobolev Inequalities on Convex Cones
BibTeX
@misc{collins2026optimal,
title = {{Optimal regularity of optimal transport and degenerations of convex cones}},
author = {Tristan C. Collins and Benjy Firester and Freid Tong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37847},
eprint = {2609.37847},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37847},
}
RIS
TY - UNPB
TI - Optimal regularity of optimal transport and degenerations of convex cones
AU - Tristan C. Collins
AU - Benjy Firester
AU - Freid Tong
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37847
UR - https://arxiv.org/abs/2609.37847
AB - If T is an optimal transport map between bounded convex domains Ω and Ω', we characterize the regularity/singularity dichotomy for DT at (x,T(x)) ∈ ∂ Ω× ∂ Ω'. This is obtained through a new approach to the regularity theory based on affine degenerations of the tangent cones to $(Ω, Ω')$ at (x,T(x)). We formulate a general principle relating the optimal boundary regularity to a stability criterion and an SL(n) moduli space of pairs of convex cones. Among other results, we establish geometric criteria for existence and non-existence of homogeneous optimal transport maps between convex cones.
ER -