On hyperbolic virtual polytopes and hyperbolic fans

Gaiane Panina

Open Mathematics (2006)

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

Abstract

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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.

How to cite

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Gaiane 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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  1. [1] A.D. Alexandrov: “On uniqueness theorem for closed surfaces”, Doklady Akad. Nauk SSSR, Vol. 22, (1939), pp. 99–102 (Russian). 
  2. [2] A.D. Alexandrov: Konvexe Polyeder, Berlin, Akademie-Verlag, 1958. 
  3. [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. [4] A. Khovanskii and A. Pukhlikov: “Finitely additive measures of virtual polytopes”, St. Petersburg Math. J., Vol. 4(2), (1993), pp. 337–356. Zbl0791.52010
  5. [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. [6] Y. Martinez-Maure: “Théorie des hérissons et polytopes”, C.R. Acad. Sci. Paris Serie 1, Vol. 336, (2003), pp. 41–44. 
  7. [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. [8] G. Panina: “Virtual polytopes and some classical problems” St. Petersburg Math. J., Vol. 14(5), (2003), pp. 823–834. 
  9. [9] G. Panina: “New counterexamples to A.D. Alexandrov’s hypothesis”, Adv. Geometry, Vol. 5, (2005), pp. 301–317. Zbl1077.52003
  10. [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. [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. [12] H. Radström: “An embedding theorem for spaces of convex sets”, Proc. AMS, Vol. 3(1), (1952), pp. 165–169. Zbl0046.33304
  13. [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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