We examine the entropic measure, recently constructed by von Renesse and Sturm, a measure over the metric space of probability measures on the unit interval equipped with the 2-Wasserstein distance. We show that equipped with this measure, Wasserstein space over the interval does not admit generalized Ricci lower bounds in the entropic displacement convexity sense of Lott-Villani-Sturm. We discuss why this is contrary to what one might expect from heuristic considerations.
All Science Journal Classification (ASJC) codes
- Geometry of spaces of measures
- Lower bounds on curvature
- Optimal transport
- Wasserstein space