Volume comparison of balls with two different centers

Question

Let $M$ be complete connected (dim = $n$ ) riemannian manifolds. Assume $Ric \geq - (n-1)$ then it holds

$$ \frac{vol(B(p, R))}{vol(B(q,r))} \leq \frac{V_{-1}(R + s)}{V_{-1}(r)} $$

where $p,q \in M$, $V_1 (r), V_1 (R)$ is volume of ball with radius $r$ in constant curvature -1 space (dim = $n$), and $s = d(p,q)$.

Could you tell me how to derive it.

I know that if $p = q$ it holds

$$ \frac{vol(B(p, R))}{vol(B(p,r))} \leq \frac{V_{-1}(R)}{V_{-1}(r)} $$

(Bishop Gromov volume comparison theorem).

Answer 1

Use $B(p, R)\subset B(q, R+s)$, so

$$ \frac{vol(B(p, R))}{vol(B(q,r))} \leq \frac{vol(B(q, R+s))}{vol(B(q,r))} \le \frac{V_{-1}(R + s)}{V_{-1}(r)}.$$