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2006 | 4 | 2 | 270-293
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On hyperbolic virtual polytopes and hyperbolic fans

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EN
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EN
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.
Słowa kluczowe
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Czasopismo
Rocznik
Tom
4
Numer
2
Strony
270-293
Opis fizyczny
Daty
wydano
2006-06-01
online
2006-06-01
Twórcy
Bibliografia
  • [1] A.D. Alexandrov: “On uniqueness theorem for closed surfaces”, Doklady Akad. Nauk SSSR, Vol. 22, (1939), pp. 99–102 (Russian).
  • [2] A.D. Alexandrov: Konvexe Polyeder, Berlin, Akademie-Verlag, 1958.
  • [3] Yu. Burago and S.Z. Shefel: “The geometry of surfaces in Euclidean spaces”, In: Geometry III. Theory of surfaces. Encycl. Math. Sci., Vol. 48, 1992, pp. 1–85 (Russian, English).
  • [4] A. Khovanskii and A. Pukhlikov: “Finitely additive measures of virtual polytopes”, St. Petersburg Math. J., Vol. 4(2), (1993), pp. 337–356.
  • [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.
  • [6] Y. Martinez-Maure: “Théorie des hérissons et polytopes”, C.R. Acad. Sci. Paris Serie 1, Vol. 336, (2003), pp. 41–44.
  • [7] P. McMullen: “The polytope algebra”, Adv. Math., Vol. 78(1), (1989), pp. 76–130. http://dx.doi.org/10.1016/0001-8708(89)90029-7
  • [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.
  • [10] A.V. Pogorelov: “On uniqueness theorem for closed convex surfaces”, Doklady Akad. Nauk SSSR, Vol. 366(5), (1999), pp. 602–604 (Russian).
  • [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.
  • [12] H. Radström: “An embedding theorem for spaces of convex sets”, Proc. AMS, Vol. 3(1), (1952), pp. 165–169.
  • [13] È. Rozendorn: “Surfaces of negative curvature”, Current Problems Math., Fund. Dir., Vol. 48, (1989), pp. 98–195 (Russian).
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-006-0006-9
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