We investigate barycenters of probability measures on proper Alexandrov spaces of curvature bounded below, and show that they enjoy several properties relevant to or different from those in metricâ€¦ (More)

We establish the existence of Euclidean tangent cones on Wasserstein spaces over compact Alexandrov spaces of curvature bounded below. By using this Riemannian structure, we formulate and constructâ€¦ (More)

We extend Cordero-Erausquin, McCann and SchmuckenschlÃ¤gerâ€™s Riemannian Borell-Brascamp-Lieb inequality to Finsler manifolds. Among applications, we establish the equivalence between Sturm, Lott andâ€¦ (More)

This paper studies the heat flow on Finsler manifolds. A Finsler manifold is a smooth manifold M equipped with a Minkowski norm F (x, Â·) : TxM â†’ R+ on each tangent space. Mostly, we will require thatâ€¦ (More)

(RBC and UKQCD Collaborations) 1Physics Department, University of Connecticut, Storrs, CT 06269-3046, USA 2RIKEN-BNL Research Center, Brookhaven National Laborator y, Upton, NY 11973, USA 3SUPA,â€¦ (More)

We prove that Alexandrov spaces of nonnegative curvature have Markov type 2 in the sense of Ball. As a corollary, any Lipschitz continuous map from a subset of an Alexandrov space of nonnegativeâ€¦ (More)

We introduce a class of generalized relative entropies (inspired by the Bregman divergence in information theory) on the Wasserstein space over a weighted Riemannian or Finsler manifold. We proveâ€¦ (More)

We extend results proven by the second author ([Oh]) for nonnegatively curved Alexandrov spaces to general compact Alexandrov spaces X with curvature bounded below: the gradient flow of aâ€¦ (More)

We give a necessary and sufficient condition on a Randers space for the existence of a measure for which Shenâ€™s S-curvature vanishes everywhere. Moreover, such a measure coincides with theâ€¦ (More)

We investigate the m-relative entropy, which stems from the Bregman divergence, on weighted Riemannian and Finsler manifolds. We prove that the displacement convexity of the m-relative entropy isâ€¦ (More)