Comparison theorem for support functions of hypersurfaces

For a convex domain $D$ that is enclosed by the hypersurface $\mathrm{\partial }D$ of bounded normal curvature, we prove an angle comparison theorem for angles between $\mathrm{\partial }D$ and geodesic rays starting from some fixed point in $D$, and the corresponding angles for hypersurfaces of constant normal curvature. Also, we obtain a comparison theorem for support functions of such surfaces. As a corollary, we present a proof of Blaschke’s Rolling Theorem.