Around A. D. Alexandrov’s uniqueness theorem for convex polytopes

Two dependent examples are presented: 1. Two convex polytopes in ℝ3 such that for each pair of their parallel facets, one of the facets fits strictly into the other. (The example gives a refinement of A. D. Alexandrov’s uniqueness theorem for convex polytopes.) 2. A pointed tiling of the two-sphere...

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Bibliographic Details
Published in:Advances in geometry 2012-10, Vol.14 (4), p.621-637
Main Author: Panina, Gaiane
Format: Article
Language:English
Subjects:
convex polytope
hyperbolic virtual polytope
Pseudo triangulation
regular triangulation
virtual polytope
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Summary:Two dependent examples are presented: 1. Two convex polytopes in ℝ3 such that for each pair of their parallel facets, one of the facets fits strictly into the other. (The example gives a refinement of A. D. Alexandrov’s uniqueness theorem for convex polytopes.) 2. A pointed tiling of the two-sphere S2 generated by a Laman-plus-one graph which can be regularly triangulated without adding extra vertices. The construction uses the combinatorial rigidity theory of spherically embedded graphs and the relationship between the theory of pseudo triangulations and the theory of hyperbolic virtual polytopes.
ISSN:1615-715X
1615-7168
DOI:10.1515/advgeom-2012-0006