This article is cited in 3 papers

Some Sharp Estimates for Convex Hypersurfaces of Pinched Normal Curvature

K. Drach<sup>ab

a</sup> V. N. Karazin Kharkiv National University, 4 Svobody Sq., Kharkiv 61022, Ukraine
^b Sumy State University, 2, Rimskogo-Korsakova Str., Sumy 40007, Ukraine

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 $R/r$. From the obtained estimates we derive stability results for almost umbilical hypersurfaces in the constant curvature spaces.

Key words and phrases: convex hypersurface, space of constant curvature, pinched normal curvature, $\lambda$-convexity, spherical shell, stability, almost umbilical hypersurface.

MSC: 53C40

Received: 05.04.2014
Revised: 19.03.2015

Language: English

DOI: 10.15407/mag11.02.111

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