Shape recognition via Wasserstein distance

The Kantorovich-Rubinstein-Wasserstein metric defines the distance between two probability measures mu and nu on Rd+1 by computing the cheapest way to transport the mass of mu onto nu, where the cost per unit mass transported is a given function c(x, y) on R2d+2. Motivated by applications to shape r ecognition, we analyze this transportation problem with the cost c(x, y) = \x - y\(2) and measures supported on two curves in the plane. or more gener ally on the boundaries of two domains Omega, Lambda subset of Rd+1. Unlike the theory for measures that are absolutely continuous with respect to Lebe sgue, it turns out not to be the case that mu -a.e. x is an element of part ial derivative Omega is transported to a single image y is an element of pa rtial derivative Lambda; however, we show that the images of x are almost s urely collinear and parallel the normal to partial derivative Omega at x. I f either domain is strictly convex, we deduce that the solution to the opti mization problem is unique. When both domains are uniformly convex, we prov e a regularity result showing that the images of x is an element of partial derivative Omega are always collinear, and both images depend on x in a co ntinuous and (continuously) invertible way. This produces some unusual extr emal doubly stochastic measures.