Discrete optimal transport: limits and limitations

We use the finite volume method to discretely approximate the Kantorovich distance W_2 on the space of probability measures in Euclidean space. This method gives the discrete space a Riemannian structure. However, the question of Gromov-Hausdorff convergence was unanswered except for cubic finite volumes on the torus (Gigli-Maas 2013). We show that the limit distance is in general lower than the Kantorovich distance due to cost-decreasing oscillations. However, under a simple geometric condition on the finite volumes, we show Gromov-Hausdorff convergence.