Estimates of the volume of a region in a Riemannian space

In an $n$-dimensional Riemannian space we consider a compact space with nonnegative curvature and with a strictly convex boundary. We let $V$ be the volume of this region, $S$ the area (the $(n-1)$-dimensional volume) of its boundary, $k_1\geqslant0$ the lower bound of the two-dimensional curvatures and $r$ the radius of an inscribed ball. We prove the estimate $V\geqslant\frac1nSr$. In the case $k_1>0$ we establish that $r<\pi/\sqrt{k_1}$, and that one has the more precise estimate
$$ V\geqslant\frac S{\sin^{n-1}r\sqrt{k_1}}\int_0^r{\sin^{n-1}t\sqrt{k_1}\,dt}. $$

In both cases equality holds if the region considered is a ball in a space of constant curvature $k_1\geqslant0$.
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