  For a convex domain $D$ bounded by the hypersurface $\partial D$ 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 $\partial D$, provided that
normal curvatures of $\partial D$ are pinched by two positive constants.
Furthermore, in the Euclidean case we also present sharp estimates for the
quotient $R/r$. From the obtained estimates we derive stability results for
almost umbilical hypersurfaces in the constant curvature spaces.
