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Around A. D. Alexandrov’s uniqueness theorem for convex polytopes
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Gaiane Panina
Published/Copyright:
December 11, 2012
Abstract
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.
Keywords: Pseudo triangulation; regular triangulation; convex polytope; virtual polytope; hyperbolic virtual polytope.
Published Online: 2012-12-11
Published in Print: 2012-10
© 2012 by Walter de Gruyter GmbH & Co.
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- Barycenters in Alexandrov spaces of curvature bounded below
- Isoperimetric problems in sectors with density
- Around A. D. Alexandrov’s uniqueness theorem for convex polytopes
- A relative isoperimetric inequality for certain warped product spaces
- Markov’s inequality in the o-minimal structure of convergent generalized power series
- Minimal area conics in the elliptic plane
- Geometry of semi-tube domains in ℂ2
- Flag transitive c:F4(2)-geometries
- Majorana representations of L3(2)
- A characterization of multiple (n – k)-blocking sets in projective spaces of square order
Keywords for this article
Pseudo triangulation;
regular triangulation;
convex polytope;
virtual polytope;
hyperbolic virtual polytope.
Articles in the same Issue
- Masthead
- Barycenters in Alexandrov spaces of curvature bounded below
- Isoperimetric problems in sectors with density
- Around A. D. Alexandrov’s uniqueness theorem for convex polytopes
- A relative isoperimetric inequality for certain warped product spaces
- Markov’s inequality in the o-minimal structure of convergent generalized power series
- Minimal area conics in the elliptic plane
- Geometry of semi-tube domains in ℂ2
- Flag transitive c:F4(2)-geometries
- Majorana representations of L3(2)
- A characterization of multiple (n – k)-blocking sets in projective spaces of square order
