preprint

Inserted: 30 sep 2026

Year: 2026

Abstract:

If $T$ is an optimal transport map between bounded convex domains $Ω$ and $Ω'$, we characterize the regularitysingularity 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.