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

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 are exactly $N$. We also prove that every tangent cone at any point on such RCD spaces is a metric cone.