Convex sets in Hadamard manifolds. (Q1397838)

In the course of the study of some problems of geometric probability in the Lobachevskij plane \(\mathbb H^2\), \textit{L. A. Santalo} and \textit{I. Yañez} proved the following theorem in [J. Appl. Probab. 9, 140--157 (1972; Zbl 0231.60010)]: Let \(\{\Omega(t)\}_{t\in\mathbb R^+}\) be a family of compact \(h\)-convex domains in \(\mathbb H^2\) which expands over the whole plane. Then \[ \lim_{t\to+\infty}\frac{\text{Area\,}(\Omega(t))} {\text{Length\,}(\partial\Omega(t))}=1. \] The paper under review gives wide generalizations of the Santalo-Yañez theorem which are valid for Hadamard manifolds of arbitrary dimension and for any family of \(h\) and \(\lambda\)-convex domains which expand over the whole space and may have even nonsmooth boundaries. The results obtained are new even for Lobachevskij space.