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 R C D 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 R C D ∗ ( K , N ) RCD^*(K,N) spaces is exactly N N . We also prove that every tangent cone at any point on such R C D RCD spaces is a metric cone.