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

TL;DR

This survey reviews various results related to Alexandrov's theorem on characterizing spheres through principal curvatures, highlighting counterexamples for convex C^2-surfaces and discussing contributions by Martinez-Maure.

Contribution

The paper compiles and discusses key results and counterexamples related to Alexandrov's sphere characterization theorem, emphasizing Martinez-Maure's findings.

Findings

Martinez-Maure proved the theorem does not hold for convex C^2-surfaces.

The survey summarizes multiple approaches and results in the area.

Includes a Russian translation of Martinez-Maure's original paper.

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 $k_1(\boldsymbol{x})\leqslant k_2(\boldsymbol{x})$ be its principal curvatures at the point $\boldsymbol{x}$. If the inequalities $k_1(\boldsymbol{x})\leqslant k\leqslant k_2(\boldsymbol{x})$ hold true with some constant $k$ for all $\boldsymbol{x}\in S$ 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, S\'{e}r. I, Math.} {\bf 332} (2001), 41--44.