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Gaiane Panina

New counterexamples to A. D. Alexandrov’s hypothesis

Keywords: Virtual polytope, fan, hérisson, hyperbolic hérisson, saddle surface

The paper presents a series of principally different C^? -smooth counterexamples to the following hypothesis on a characterization of the sphere: Let K ? ?³ be a smooth convex body. If at every point of ?K, we have R₁? C? R₂ for a constant C, then K is a ball. (R₁ and R₂ stand for the principal curvature radii of ?K.)

The hypothesis was proved by A. D. Alexandrov and H. F. Münzner for analytic bodies. For the case of general smoothness it has been an open problem for years. Recently, Y. Martinez-Maure has presented a C²-smooth counterexample to the hypothesis.

Advances in Geometry, Walter de Gruyter

Print ISSN: 1615-715X
Volume: 5, 04/2005
Pages: 301 - 317

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