An inequality of the isoperimetric type for a~domain in a~Riemannian space

B. V. Dekster

Sbornik. Mathematics, Tome 19 (1973) no. 2, pp. 257-274

Résumé

We consider in the $n$-dimensional Riemannian space a domain with compact closure $T$ bounded by a regular hypersurface $\Gamma$. We assume that the sectional curvatures in $T$ are positive and the boundary $\Gamma$ is strictly convex. We let $V$ denote the volume of $T$, $S$ the $(n-1)$-dimensional volume of $\Gamma$, $H$ the integral mean curvature of $\Gamma$, and $r$ the radius of the inscribed ball. The basic result is the inequality $V\leqslant\frac{S^2}H$, which is implied by the two estimates $V\leqslant Sr$ and $r\leqslant\frac SH$. Both these bounds are exact. Bibliography: 6 titles.