Inequalities between the radii of spheres that are connected with a convex surface in a space of constant curvature

With a convex surface $\Phi$ in space of constant curvature, we associate four numbers ($\lambda$, $\Lambda$, $M$, $\mu$), where $\lambda$ is the radius of a largerst sphere freely rolling over the interior side of $\Phi$, $\Lambda$ is the inradius of $\Phi$, $M$ is the outradius of $\Phi$, and $\mu$ is the radius of a sphere over whose interior $\Phi$ may roll freely. Exact inequalities connecting these four numbers are found.