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

TL;DR

This paper introduces a Bishop-type inequality for metric measure spaces with Ricci curvature bounds, showing that such spaces have a unique regular set, Hausdorff dimension N, and tangent cones are metric cones.

Contribution

It establishes a Bishop-type inequality for RCD spaces, proving the uniqueness of the regular set and characterizing tangent cones as metric cones.

Findings

Spaces satisfying the Bishop-type inequality have a unique regular set.

The Hausdorff dimension of these RCD spaces is exactly N.

Tangent cones at any point are metric cones.

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.