Geodesic
An inequality of the isoperimetric type for a domain in a Riemannian space
Sbornik. Mathematics, Tome 19 (1973) no. 2, pp. 257-274 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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B. V. Dekster. An inequality of the isoperimetric type for a domain in a Riemannian space. Sbornik. Mathematics, Tome 19 (1973) no. 2, pp. 257-274. http://geodesic.mathdoc.fr/item/SM_1973_19_2_a9/

[1] B. V. Dekster, “Otsenki ob'ema oblasti v rimanovom prostranstve”, Matem. sb., 88 (130) (1972), 61–87 | MR | Zbl

[2] P. K. Rashevskii, Rimanova geometriya i tenzornyi analiz, izd-vo «Nauka», Moskva, 1964 | MR

[3] B. V. Dekster, “Nekotorye integralnye otsenki dlya trekhmernykh razvertok”, Matem. sb., 81 (123) (1970), 256–278 | MR | Zbl

[4] D. Gromol, V. Klingenberg, V. Meier, Rimanova geometriya v tselom, izd-vo «Mir», Moskva, 1971

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[6] J. Simons, “Minimal varieties”, Ann. Math., 88:1 (1968), 62–105 | DOI | MR | Zbl