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An illustrated theory of hyperbolic virtual polytopes. (English) Zbl 1145.52004

The group of virtual polytopes in \({\mathbb R}^3\) consists of all formal expressions \(K\oplus L^{-1}\), where \(K\) and \(L\) are convex polytopes in \({\mathbb R}^3\). They may be visualized as closed polytopal surfaces together with a fan which describes the structure of the set of normal vectors. The support function of \(K\oplus L^{-1}\) is defined to be the difference of the support functions of \(K\) and \(L\).
The authors study in particular hyperbolic virtual polytopes, which are characterized by a certain saddle property of their support function. They present non-trivial examples of hyperbolic virtual polytopes with 6 and 8 “horns”. Such polytopes may be used to construct counterexamples to a conjecture of A. D. Alexandrov (1939), as the second author has already previously shown [Adv. Geom. 5, No. 2, 301–317 (2005; Zbl 1077.52003) and Cent. Eur. J. Math. 4, No. 2, 270–293 (2006; Zbl 1107.52002)].

MSC:

52B10 Three-dimensional polytopes
52B70 Polyhedral manifolds
52A15 Convex sets in \(3\) dimensions (including convex surfaces)
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References:

[1] Alexandrov A.D., Sur les théoremes d’unicité pour les surfaces fermées, C. R. (Dokl.) Acad. Sci. URSS, 1939, 22, 99-102; · JFM 65.0828.03
[2] Martinez-Maure Y., A counterexample to a conjectured characterization of the sphere, C. R. Acad. Sci. Paris Sér. I Math., 2001, 332, 41-44 (in French); · Zbl 1008.53002
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[6] Panina G., On hyperbolic virtual polytopes and hyperbolic fans, Cent. Eur. J. Math., 2006, 4, 270-293 http://dx.doi.org/10.2478/s11533-006-0006-9; · Zbl 1107.52002
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