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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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