ESTIMATES OF THE VOLUME OF A REGION IN A RIEMANNIAN SPACE

Abstract

In an -dimensional Riemannian space we consider a compact space with nonnegative curvature and with a strictly convex boundary. We let be the volume of this region, the area (the -dimensional volume) of its boundary, the lower bound of the two-dimensional curvatures and the radius of an inscribed ball. We prove the estimate . In the case we establish that , and that one has the more precise estimate In both cases equality holds if the region considered is a ball in a space of constant curvature .Figures: 5. Bibliography: 12 items.