Certain integral estimates for three-dimensional PM manifolds

In this article we consider three-dimensional PM manifolds with positive curvature which are homeomorphic to a ball and have convex boundary. For these PM manifolds there is defined in a natural way the radius $r$ of the inscribed sphere and the integral mean curvature $H$ of the boundary. The new results consist of a proof of the estimates
$$ V\geqslant\frac13Sr,\quad r\leqslant\frac SH,\quad D<\frac{2S}H+d,\quad V\leqslant Sr,\quad V\leqslant\frac{S^2}H, $$
where $V$ is the volume of the PM manifold, $D$ is the diameter, $S$ is the area of the boundary and $d$ is the intrinsic diameter of the boundary. Incidentally, properties of geodesics and the construction of their boundaries are investigated. The results obtained are completely analogous to the two-dimensional case. In particular, a construction is investigated similar to the special case of cutting out lunes from a two-dimensional PM manifold: it is shown that the union of the geodesics joining an interior point of the PM manifold to a point on the boundary form a finite collection of tetrahedra which are glued together into a “three-dimensional cone” after cutting out from the PM manifold the “remaining material”.
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