Optimal regularity of optimal transport and degenerations of convex cones

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