Let $(M,g)$ be a compact connected $n$-dimensional Riemannian manifold; let $(X,d)$ denote its associated metric (length) space.  A comment on the original formulation of this post mentioned that $(X,d)$ is [Ahlfors $n$-regular][1], meaning that: there exist constants $L,U>0$ for which 
$$
L  r^n\leq \mu(B(x,r))\leq U r^n$$
where $\mu$ is the $n$-dimensional Hausdorff measure and $B(x,r)$ denotes the ball in $(X,d)$ about $x\in X$ of radius $r>0$
*(i.e. each $\mu(B(x,r))\in \Theta( r^n)$)*.

 1. Where can I find a proof or reference for this fact?
 2. Can the constants $L,U$ be stated using the curvature and dimension of $(M,g)$?



  [1]: https://mathoverflow.net/questions/319955/origin-of-term-ahlfors-david-regular