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Around the A. D. Alexandrov’s theorem on a characterization of a sphere. (Russian. English summary) Zbl 1330.53007
Summary: 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 $$k_1(\mathbf{x}) \leq k_2 (\mathbf{x})$$ be its principal curvatures at the point $$\mathbf{x}$$. If the inequalities $$k_1 (\mathbf{x})\leq k\leq k_2(\mathbf{x})$$ 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 [C. R. Acad. Sci., Paris, Sér. I, Math. 332, No. 1, 41–44 (2001; Zbl 1008.53002)].
##### MSC:
 53A05 Surfaces in Euclidean and related spaces
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