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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Strambach, Karl

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Barycenters in Alexandrov spaces of curvature bounded below

1Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan

Supported in part by the Grant-in-Aid for Young Scientists (B) 20740036

Citation Information: . Volume 14, Issue 4, Pages 571–587, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: 10.1515/advgeom-2011-058, December 2012

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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 metric spaces of curvature bounded above. We prove the reverse variance inequality, and show that the push forward of a measure to the tangent cone at its barycenter has the flat support.

Keywords: Barycenter; Alexandrov space; variance inequality; Wasserstein space.

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