for every $x \in M$ and every $v \in T_xM$. For $p \in M$ and $r>0$, let $B_g(p,r)$ denote the geodesic ball of radius $r$ centered at $p$, and let $\operatorname{vol}_g$ denote the Riemannian volume measure.
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Let $V_{-K}^n(r)$ denote the volume of the radius-$r$ geodesic ball in the simply connected $n$-dimensional space form of constant sectional curvature $-K$. Then, for every $p \in M$ and every $R>0$,