Displaying similar documents to “Compact hyperbolic Coxeter n -polytopes with n + 3 facets.”

On hyperbolic virtual polytopes and hyperbolic fans

Gaiane Panina (2006)

Open Mathematics

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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,...

An illustrated theory of hyperbolic virtual polytopes

Marina Knyazeva, Gaiane Panina (2008)

Open Mathematics

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The paper gives an illustrated introduction to the theory of hyperbolic virtual polytopes and related counterexamples to A.D. Alexandrov’s conjecture.

Rigidity and flexibility of virtual polytopes

G. Panina (2003)

Open Mathematics

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All 3-dimensional convex polytopes are known to be rigid. Still their Minkowski differences (virtual polytopes) can be flexible with any finite freedom degree. We derive some sufficient rigidity conditions for virtual polytopes and present some examples of flexible ones. For example, Bricard's first and second flexible octahedra can be supplied by the structure of a virtual polytope.