Constructing optimal maps for Monge's transport problem as a limit of strictly convex costs

Given two densities on $\mathbf{R}^n$ with the same total mass, the Monge transport problem is to find a Borel map $s:\mathbf{R}^n \to\mathbf{R}^n$rearranging the first distribution of mass onto the second, while minimizing the average distance transported. Here distance is measured by a norm with a uniformly smooth and convex unit ball. This paper gives a complete proof of the existence of optimal maps under the technical hypothesis that the distributions of mass be compactly supported. The maps are not generally unique. The approach developed here is new, and based on a geometrical change-of-variables technique offering considerably more flexibility than existing approaches.