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A Bishop-type inequality on metric measure spaces with Ricci curvature bounded below


Author: Yu Kitabeppu
Journal: Proc. Amer. Math. Soc. 145 (2017), 3137-3151
MSC (2010): Primary 51F99; Secondary 53C20
DOI: https://doi.org/10.1090/proc/13517
Published electronically: January 6, 2017
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Abstract: We define a Bishop-type inequality on metric measure spaces with Riemannian curvature-dimension condition. The main result in this short article is that any $ RCD$ spaces with the Bishop-type inequalities possess only one regular set in not only the measure theoretical sense but also the set theoretical one. As a corollary, the Hausdorff dimension of such $ RCD^*(K,N)$ spaces is exactly $ N$. We also prove that every tangent cone at any point on such $ RCD$ spaces is a metric cone.


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Additional Information

Yu Kitabeppu
Affiliation: Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
Email: y.kitabeppu@gmail.com

DOI: https://doi.org/10.1090/proc/13517
Keywords: RCD spaces, regular sets
Received by editor(s): April 6, 2016
Received by editor(s) in revised form: June 28, 2016, and August 14, 2016
Published electronically: January 6, 2017
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2017 American Mathematical Society