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

Kitabeppu, Yu

doi:10.1090/proc/13517

A Bishop-type inequality on metric measure spaces with Ricci curvature bounded below
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Proc. Amer. Math. Soc. 145 (2017), 3137-3151 Request permission

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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