Around the A.D. Alexandrov's theorem on a characterization of a sphere

Abstract

This is a survey paper on various results relates to the following theorem first proved by A.D. Alexandrov: \textit{Let $S$ be an analytic convex sphere-homeomorphic surface in ${\mathbb{R}}^3$ and let $k_1(\mathbf{x})⩽k_2(\mathbf{x})$ be its principal curvatures at the point $\mathbf{x}$. If the inequalities $k_1(\mathbf{x})⩽k⩽k_2(\mathbf{x})$ hold true with some constant $k$ for all $\mathbf{x}\in S$ then $S$ is a sphere.} The imphases is on a result of Y. Martinez-Maure who first proved that the above statement is not valid for convex $C^2$-surfaces. For convenience of the reader, in addendum we give a Russian translation of that paper by Y. Martinez-Maure originally published in French in \textit{C. R. Acad. Sci., Paris, Sér. I, Math.} {\bf 332} (2001), 41--44.

arXiv commentIn Russian. 12 pages, 2 figures

References in corpus (1)

• On the Invariant Theory of Weingarten Surfaces in Euclidean Space