Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 49, No. 1, pp. 59-70 (2008)
A. D. Alexandrov's uniqueness theorem for convex polytopes and its refinements
Gaiane PaninaInstitute for Informatics and Automation, V.O. 14 line 39, St. Petersburg, 199178, Russia, e-mail: firstname.lastname@example.org
Abstract: In 1937, A. D. Alexandrov proved that if no parallel faces of two 3-dimensional convex polytopes can be placed strictly one into another via a translation, then the polytopes are translates of one another.
The theory of hyperbolic virtual polytopes elucidates this theorem and suggests natural ways of its refinement.
Namely, we present an example of two different 3-dimensional polytopes such that, for each pair of their parallel faces, there exists at most one translation placing one of the faces into another.
Another refinement: given two polytopes, if for any pair of parallel faces, there exists at most one translation placing the face of the first polytope strictly in the face of the second one, and there exists no translation placing the face of the second polytope strictly in the face of the first one, then the polytopes are translates of one another.
Keywords: virtual polytope, saddle surface, hyperbolic polytope
Classification (MSC2000): 52B10, 52B70
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Electronic version published on: 26 Feb 2008. This page was last modified: 28 Jan 2013.