Gromov-Hausdorff convergence of discrete transportation metrics

This paper continues the investigation of discrete transportation distances initiated in [13] and further studied in [8] (see also [5] and [14]). We prove that the discrete transportation metrics on the d-dimensional discrete torus with mesh size 1/N converge, when N goes to infinity to the standard 2-Wasserstein distance on the continuous torus. This is the first result of a passage to the limit from a discrete transportation problem to a continuous one, and proves that the theory built by the second author is fully compatible with the continuous case. © 2013 Society for Industrial and Applied Mathematics