A lack of Ricci bounds for the entropic measure on Wasserstein space over the interval

2011/11/30 by Otis Chodosh
Mathematics · #Combinatorics #Computer science #Convexity #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Interval (graph theory) #Mathematical analysis #Mathematics #Measure (data warehouse) #Metric (unit) #Metric space #Point processes and geometric inequalities #Probability measure #Pure mathematics #Space (punctuation) #Unit (ring theory) #Unit interval #Upper and lower bounds #Wasserstein metric #math.MG #math.PR

paper · pdf · doi:10.1016/j.jfa.2012.03.007

published as Journal of Functional Analysis, 262, 10, 4570-4581 (2012)

arxiv created 2012/03/02 · openalex publication_date 2012/03/14 · arxiv updated 2012/09/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

This is a condensed form of the author's essay, which can be found at [arXiv:1105.2883]. We prove that the entropic measure constructed by von Renesse-Sturm over Wasserstein space on the unit interval (probability measures on the unit interval equipped with the 2-Wasserstein metric) does not admit generalized Ricci lower bounds in the sense of Lott-Villani-Sturm. We discuss why this is surprising, considering various heuristic arguments.

Citations

• Optimal Transport and Ricci Curvature: Wasserstein Space Over the Interval
• Optimal Transport
• Topics in Optimal Transportation
• Ricci curvature for metric-measure spaces via optimal transport
• Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality