Improved geodesics for the reduced curvature-dimension condition in branching metric spaces | Zendy

Tapio Rajala | Zendy

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Improved geodesics for the reduced curvature-dimension condition in branching metric spaces

Author(s) -

Tapio Rajala

Publication year - 2013

Publication title -

discrete and continuous dynamical systems

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.289

H-Index - 70

eISSN - 1553-5231

pISSN - 1078-0947

DOI - 10.3934/dcds.2013.33.3043

Subject(s) - geodesic , upper and lower bounds , mathematics , ricci curvature , curvature , metric space , dimension (graph theory) , convexity , space (punctuation) , mathematical analysis , measure (data warehouse) , metric (unit) , combinatorics , pure mathematics , geometry , operations management , economics , linguistics , philosophy , database , computer science , financial economics

In this note we show that in metric measure spaces satisfying the reduced curvature-dimension condition CD*(K,N) we always have geodesics in the Wasserstein space of probability measures that satisfy the critical convexity inequality of CD*(K,N) also for intermediate times and in addition the measures along these geodesics have an upper-bound on their densities. This upper-bound depends on the bounds for the densities of the end-point measures, the lower-bound K for the Ricci-curvature, the upper-bound N for the dimension, and on the diameter of the union of the supports of the end-point measures.

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