Affiliation:
1. 1. Institute for Informatics and Automation, V.O. 14 line 39, St. Petersburg, 199178, Russia.
Abstract
Abstract
The paper presents a series of principally different C
∞ -smooth counterexamples to the following hypothesis on a characterization of the sphere: Let K ⊂ ℝ3 be a smooth convex body. If at every point of ∂K, we have R
1≤ C≤ R
2 for a constant C, then K is a ball. (R
1 and R
2 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 C
2-smooth counterexample to the hypothesis.
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