Comparison theorems on convex hypersurfaces in Hadamard manifolds (Q1599390)

Let \(M\) be a Hadamard manifold with sectional curvature \(K\) satisfying \(0\geq K\geq -k_{2}^{2}\). Let \(\Omega\) be a compact \(k_2\)-convex domain in \(M\). The authors give sharp upper estimates for the term \(\lim\text{max} d( o,\partial\Omega) -r\), where \(o\) is the centre of an inball, and \(r\) is the inradius of \(\Omega\). Estimates for the term \(\frac{M_d (\partial\Omega)}{\lim\text{volume}(\partial\Omega)}\) (with \(M_d\) being the total \(d\)-mean curvature) are also given.