# On hyperbolic virtual polytopes and hyperbolic fans

Open Mathematics (2006)

- Volume: 4, Issue: 2, page 270-293
- ISSN: 2391-5455

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topGaiane Panina. "On hyperbolic virtual polytopes and hyperbolic fans." Open Mathematics 4.2 (2006): 270-293. <http://eudml.org/doc/269097>.

@article{GaianePanina2006,

abstract = {Hyperbolic virtual polytopes arose originally as polytopal versions of counterexamples to the following A.D.Alexandrov’s uniqueness conjecture: Let K ⊂ ℝ3 be a smooth convex body. If for a constant C, at every point of ∂K, we have R 1 ≤ C ≤ R 2 then K is a ball. (R 1 and R 2 stand for the principal curvature radii of ∂K.) This paper gives a new (in comparison with the previous construction by Y.Martinez-Maure and by G.Panina) series of counterexamples to the conjecture. In particular, a hyperbolic virtual polytope (and therefore, a hyperbolic hérisson) with odd an number of horns is constructed. Moreover, various properties of hyperbolic virtual polytopes and their fans are discussed.},

author = {Gaiane Panina},

journal = {Open Mathematics},

keywords = {52A15; 52B70; 52B10},

language = {eng},

number = {2},

pages = {270-293},

title = {On hyperbolic virtual polytopes and hyperbolic fans},

url = {http://eudml.org/doc/269097},

volume = {4},

year = {2006},

}

TY - JOUR

AU - Gaiane Panina

TI - On hyperbolic virtual polytopes and hyperbolic fans

JO - Open Mathematics

PY - 2006

VL - 4

IS - 2

SP - 270

EP - 293

AB - Hyperbolic virtual polytopes arose originally as polytopal versions of counterexamples to the following A.D.Alexandrov’s uniqueness conjecture: Let K ⊂ ℝ3 be a smooth convex body. If for a constant C, at every point of ∂K, we have R 1 ≤ C ≤ R 2 then K is a ball. (R 1 and R 2 stand for the principal curvature radii of ∂K.) This paper gives a new (in comparison with the previous construction by Y.Martinez-Maure and by G.Panina) series of counterexamples to the conjecture. In particular, a hyperbolic virtual polytope (and therefore, a hyperbolic hérisson) with odd an number of horns is constructed. Moreover, various properties of hyperbolic virtual polytopes and their fans are discussed.

LA - eng

KW - 52A15; 52B70; 52B10

UR - http://eudml.org/doc/269097

ER -

## References

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- [2] A.D. Alexandrov: Konvexe Polyeder, Berlin, Akademie-Verlag, 1958.
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- [5] Y. Martinez-Maure: “Contre-exemple à une caractérisation conjecturée de la sphère”, C.R. Acad. Sci. Paris, Vol. 332(1), (2001), pp. 41–44.
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- [8] G. Panina: “Virtual polytopes and some classical problems” St. Petersburg Math. J., Vol. 14(5), (2003), pp. 823–834.
- [9] G. Panina: “New counterexamples to A.D. Alexandrov’s hypothesis”, Adv. Geometry, Vol. 5, (2005), pp. 301–317. Zbl1077.52003
- [10] A.V. Pogorelov: “On uniqueness theorem for closed convex surfaces”, Doklady Akad. Nauk SSSR, Vol. 366(5), (1999), pp. 602–604 (Russian). Zbl0976.53072
- [11] R. Langevin, G. Levitt and H. Rosenberg: “Hérissons et multihérissons (enveloppes paramétrées par leur application de Gauss)”, Singularities, Warsaw, Banach Center Publ., Vol. 20, (1985), pp. 245–253. Zbl0658.53004
- [12] H. Radström: “An embedding theorem for spaces of convex sets”, Proc. AMS, Vol. 3(1), (1952), pp. 165–169. Zbl0046.33304
- [13] È. Rozendorn: “Surfaces of negative curvature”, Current Problems Math., Fund. Dir., Vol. 48, (1989), pp. 98–195 (Russian). Zbl0711.53004

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