Geodesic
Around the A.\,D.~Alexandrov's theorem on a characterization of a sphere
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 9 (2012), pp. 639-652

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This is a survey paper on various results relates to the following theorem first proved by A.D. Alexandrov: 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 $k_1(\boldsymbol{x})\leqslant k\leqslant k_2(\boldsymbol{x})$ thold 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 C. R. Acad. Sci., Paris, Sér. I, Math. 332 (2001), 41–44.
Mots-clés : normal section, Weingarten surface, convex surface, herisson
Keywords: principal curvature, virtual polytope. 
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     title = {Around the {A.\,D.~Alexandrov's} theorem on a characterization of a sphere},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
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     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2012_9_a16/}
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V. A. Aleksandrov. Around the A.\,D.~Alexandrov's theorem on a characterization of a sphere. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 9 (2012), pp. 639-652. http://geodesic.mathdoc.fr/item/SEMR_2012_9_a16/