# Gromov-Hausdorff convergence of discrete transportation metrics

created by gigli on 27 Jul 2012

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Submitted Paper

Inserted: 27 jul 2012
Last Updated: 27 jul 2012

Year: 2012

Abstract:

This paper continues the investigation of ‘Wasserstein-like’ transportation distances for probability measures on discrete sets. We prove that the discrete transportation metrics on the d-dimensional discrete torus $T^d_N$ with mesh size $1/N$ converge, when $N$ goes to infinity, to the standard 2-Wasserstein distance on the continuous torus in the sense of Gromov– Hausdorff. This is the first result of a passage to the limit from a discrete transportation problem to a continuous one, and proves compatibility of the recently developed discrete metrics and the well-established 2-Wasserstein metric.