A View on Optimal Transport from Noncommutative Geometry

We discuss the relation between the Wasserstein distance of order 1 between probability distributions on a metric space, arising in the study of MongeKantorovich transport problem, and the spectral distance of noncommutative geometry. We first show that on Riemannian spin manifolds the two distances coincide. Then, on convex manifolds in the sense of Nash embedding, we provide some natural upper and lower bounds to the distance between any two probability distributions. Finally, specializing to the Euclidean space R, we explicitly compute the distance for a particular class of distributions.