On hyperbolic virtual polytopes and hyperbolic fans

Gaiane Panina 1
  • 1 Institute for Informatics and Automation


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.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [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).


Journal + Issues