I am reading the proof of Cheng's maximal diameter theorem in Chavel's Riemannian Geometry, and having a question.
Given a complete Riemannian manifold $M$, with all Ricci curvatures bounded from below by $(n −1)\kappa$, $\kappa > 0$, if $\text{diam}(M) = \pi/\sqrt{\kappa}$, then $M$ is isometric to $\mathbb{S}^n_\kappa$, the standard sphere of constant sectional curvature equal to $\kappa$.
In the proof, first pick $x$ and $y$ such that $d(x,y) = \pi/\sqrt{\kappa}$. Then use the Gromov's lemma to show that
$$ \frac{V(x;\frac{\pi}{2\sqrt{\kappa}})}{V(\mathbb{S}^n_\kappa)/2} \ge \frac{V(x;\frac{\pi}{\sqrt{\kappa}})}{V(\mathbb{S}^n_\kappa)} = \frac{V(M)}{V(\mathbb{S}^n_\kappa)}$$
The same lemma applies to $y$. Then using, $B(x;\frac{\pi}{2\sqrt{\kappa}}) \cap B(y;\frac{\pi}{2\sqrt{\kappa}}) = \emptyset$, we have $V(x;\frac{\pi}{2\sqrt{\kappa}}) = V(y;\frac{\pi}{2\sqrt{\kappa}}) = \frac{V(M)}{2}$. Then, it immediately concludes that "$V(M) = V(\mathbb{S}^n_\kappa)$" and thus they are isometric.
My question is, how to get "$V(M) = V(\mathbb{S}^n_\kappa)$"?
Thanks in advance for any help.