Displaying similar documents to “A. D. Alexandrov's uniqueness theorem for convex polytopes and its refinements.”

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.

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,...

Construction of a Φ-function for two convex polytopes

Y. Stoyan, J. Terno, M. Gil, T. Romanova, G. Scheithauer (2002)

Applicationes Mathematicae

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The analytical description of Φ-functions for two convex polytopes is investigated. These Φ-functions can be used for mathematical modelling of packing problems in the three-dimensional space. Only translations of the polytopes are considered. The approach consists of two stages. First the 0-level surface of a Φ-function is constructed, and secondly, the surface is extended to get the Φ-function. The method for constructing the 0-level surface is described in detail.

Coating by cubes.

Bezdek, K., Hausel, T. (1994)

Beiträge zur Algebra und Geometrie

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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.