Rigidity and flexibility of virtual polytopes
type: Item
instance of: scholarly article
title: Rigidity and flexibility of virtual polytopes
zbMATH Open document ID: 1032.52011
DOI: 10.2478/BF02476005
author: D. Massart
published in: Central European Journal of Mathematics
publication date: 2004-03-11T00:00:00Z
review text: A virtual polytope is the Minkowski difference of two convex compact polytopes in \(\mathbb R^n\). The notion of a virtual polytope was independently introduced by several authors [see, e.g., \textit{P. McMullen}, Adv. Math. 78, 76-130 (1989; Zbl 0686.52005)] or \textit{A. V. Pukhlikov} and \textit{A. G. Khovanskij}, St. Petersbg. Math. J. 4, 337-356 (1993; Zbl 0791.52010)]. Virtual polytopes form an abelian group with respect to the Minkowski addition \(\otimes\). It is possible to introduce mutually consistent notions of face, edge, volume, mixed volume, support function, and fan for virtual polytopes. The volume \(V(\lambda K\otimes \mu L)\) of a linear combination of virtual polytopes \(K\) and \(L\) is a polynomial in \(\lambda\) and \(\mu\). A virtual polytope \(K\) is said to be flexible if there exists a nontrivial (i.e., not generated by a motion) continuous family of virtual polytopes \(K_t\), \(t\in [0,1]\), \(K_0=K\), such that every two \(K_t\) are combinatorially equivalent and their correspondent faces can be obtained by a motion one from another. The author proves that a virtual polytope in \(\mathbb R^3\) with a convex fan is not flexible, which is a refinement of the rigidity theorem for herissons obtained in [\textit{L. Rodriguez} and \textit{H. Rosenberg}, Comment. Math. Helv. 75, 478-503 (2000; Zbl 0968.52018)]. She also gives examples of flexible virtual polytopes which are similar to Bricard's flexible octahedra.
reviewed by: Victor Alexandrov
Mathematics Subject Classification ID: 52C25
Mathematics Subject Classification ID: 52B10
Mathematics Subject Classification ID: 52B20
zbMATH DE Number: 1978729
zbMATH Keywords: polytopal function
zbMATH Keywords: Bricard's flexible octahedron
zbMATH Keywords: Minkowski addition
zbMATH Keywords: Grothendieck group
zbMATH Keywords: support function
zbMATH Keywords: normal vector
zbMATH Keywords: connected fan
zbMATH Keywords: disconnected net
zbMATH Keywords: Cauchy lemma
zbMATH Keywords: Euler characteristic
zbMATH Keywords: virtual polytope
zbMATH Keywords: rigidity theorem
zbMATH Keywords: flexible virtual polytopes
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.2478/bf02476005
OpenAlex ID: W1970364595
description: scientific article; zbMATH DE number 1978729