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arXiv · 2609.37847

Optimal regularity of optimal transport and degenerations of convex cones

Tristan C. Collins·Benjy Firester·Freid Tong
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Abstract

If $T$ is an optimal transport map between bounded convex domains $Ω$ and $Ω'$, we characterize the regularity/singularity dichotomy for $DT$ at $(x,T(x)) \in \partial Ω\times \partial Ω'$. 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 $\mathrm{SL}(n)$ 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.

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Tristan C. Collins, Benjy Firester, Freid Tong. 2026-09-29. Optimal regularity of optimal transport and degenerations of convex cones. https://arxiv.org/abs/2609.37847

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