Generalization of Sabitov's theorem to polyhedra of arbitrary dimensions
type: Item
instance of: scholarly article
title: Generalization of Sabitov's theorem to polyhedra of arbitrary dimensions
zbMATH Open document ID: 1314.52008
author: A. A. Gaifullin
published in: Discrete \& Computational Geometry
publication date: 2014-10-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/1210.5408
review text: Heron's formula from classical geometry expresses the square of the area of a triangle as a polynomial in the squares of the lengths of its sides. More generally, \textit{I. Kh. Sabitov} [Vestn. Mosk. Univ., Ser. I 1996, No. 6, 89--91 (1996; Zbl 0897.51009)]; Discrete Comput. Geom. 20, No. 4, 405--425 (1998; Zbl 0922.52006)] proved that if \(P\) is a simplicial polyhedron in \(\mathbb{R}^3\), then there exists a polynomial relation of the form \[ V^{2N} + a_1(\ell)V^{2N-2} + \cdots + a_n(\ell) = 0, \] where \(V\) is the volume of the polyhedron, \(\ell\) is the set of the squares its edge lengths, and the coefficients \(a_i(\ell)\) are polynomials with rational coefficients. This paper generalizes Sabitov's result to higher dimensions. Specifically, let \(n \geq 3\) and let \(K\) be a simplicial complex of dimension \(n-1\). The objects of study are those complexes \(K\) such that: {\parindent=6mm \begin{itemize}\item[1.] \(K\) is a pseudomanifold, meaning \(K\) is pure and each codimension-one face is contained in exactly two top-dimensional faces; \item[2.] \(K\) is strongly connected, meaning its facet-ridge graph is connected; and \item[3.] \(K\) is oriented. \end{itemize}} A simplicial polyhedron of combinatorial type \(K\) is a map \(P: |K| \rightarrow \mathbb{R}^n\) that is linear on the faces of \(|K|\) (but \(P\) need not be injective). The main result of this paper shows that for such a simplicial complex \(K\) and a polyhedron \(P\) of combinatorial type \(K\), there exists an analogous monic polynomial relation to the one shown above that involves the volume of \(P(K)\) and the set of squares of the edge lengths of \(P(K)\). The Bellows Conjecture [\textit{R. Connelly} et al., Beitr. Algebra Geom. 38, No. 1, 1--10 (1997; Zbl 0939.52009)] claims that the generalized oriented volume of a flexible polyhedron in \(\mathbb{R}^n\) remains constant under its flexes. This follows from the main result of the paper.
reviewed by: Steven Klee
Mathematics Subject Classification ID: 52B11
Mathematics Subject Classification ID: 52A38
Mathematics Subject Classification ID: 05E45
zbMATH DE Number: 6362290
zbMATH Keywords: flexible polyhedron
zbMATH Keywords: Sabitov polynomial
zbMATH Keywords: Bellows conjecture
zbMATH Keywords: a place of field
zbMATH Keywords: simplicial collapse
MaRDI profile type: Publication
OpenAlex ID: W2065249655
arXiv ID: 1210.5408
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DOI: 10.1007/S00454-014-9609-2
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description: scientific article; zbMATH DE number 6362290