Some Sharp Estimates for Convex Hypersurfaces of Pinched Normal Curvature

For a convex domain D bounded by the hypersurface partial derivative 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 derivative D, provided that normal curvatures of partial derivative 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.