A Bishop-type inequality on metric measure spaces with Ricci curvature bounded below
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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.References
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Additional Information
- Yu Kitabeppu
- Affiliation: Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
- MR Author ID: 1055218
- Email: y.kitabeppu@gmail.com
- 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
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 3137-3151
- MSC (2010): Primary 51F99; Secondary 53C20
- DOI: https://doi.org/10.1090/proc/13517
- MathSciNet review: 3637960