Inserted: 25 sep 2012
Last Updated: 16 jul 2014
Journal: Publ. Math. Inst. Hautes Études Sci.
We prove that, for general cost functions on $\mathbb R^n$, or for the cost $d^2/2$ on a Riemannian manifold, optimal transport maps between smooth densities are always smooth outside a closed singular set of measure zero.