Some sharp estimates for convex hypersurfaces of pinched normal curvatures

Abstract

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 \(R/r\).