Partial Regularity for optimal transport maps

Guido De Philippis, Alessio Figalli

Published 2012-09-25, updated 2018-06-05Version 2

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.