Some analytic properties of convex sets in Riemannian spaces

We investigate analytic properties of the boundary $bC$ of a locally convex set $C$ in a Riemannian space $M^n$, $n\geqslant2$, in particular, its mean curvature $H$ as a function of the set. For an $M^3$ of nonnegative curvature we prove the inequality
$$ 4\pi\chi(bC)t_0\leq H(bC)+\Omega(C), $$
where $\chi$ is the Euler characteristic, $t_0$ the radius of the largest ball inscribed in $C$, and $\Omega(C)$ the scalar curvature of $C$.
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