# 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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