Some sharp estimates for convex hypersurfaces of pinched normal curvature

For a convex domain $D$ bounded by the hypersurface $\mathrm{\partial }D$ in a space of constant curvature we give sharp bounds on the width $R-r$ of a spherical shell with radii $R$ and $r$ that can enclose $\mathrm{\partial }D$, provided that normal curvatures of $\mathrm{\partial }D$ are pinched by two positive constants. Furthermore, in the Euclidean case we also present sharp estimates for the quotient $R\mathrm{/}r$. From the obtained estimates we derive stability results for almost umbilical hypersurfaces in the constant curvature spaces.