Optimal regularity of optimal transport and degenerations of convex cones

Abstract

If $T$ is an optimal transport map between bounded convex domains $\Omega$ and ${\Omega }^{\prime }$, we characterize the regularity/singularity dichotomy for $DT$ at $(x,T(x))\in \partial \Omega \times \partial {\Omega }^{\prime }$. This is obtained through a new approach to the regularity theory based on affine degenerations of the tangent cones to $(\Omega ,{\Omega }^{\prime })$ 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.