Partial regularity of optimal transport with Coulomb cost

We prove that for two-marginal optimal transport with Coulomb cost on Rᵈ R d , the optimal map is a C1,α C 1 , α diffeomorphism outside a closed set of Lebesgue measure zero provided the marginals are α α -Hölder continuous, bounded, and strictly positive. Excluding a set of measure zero is necessary as optimal maps for the Coulomb cost have long been known to exhibit jump singularities across codimension 1 surfaces (even for smooth marginals on convex domains).