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Generalization of Sabitov’s theorem to polyhedra of arbitrary dimensions. (English) Zbl 1314.52008
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, 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:
1.
\(K\) is a pseudomanifold, meaning \(K\) is pure and each codimension-one face is contained in exactly two top-dimensional faces;
2.
\(K\) is strongly connected, meaning its facet-ridge graph is connected; and
3.
\(K\) is oriented.
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 [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.

MSC:
52B11 \(n\)-dimensional polytopes
52A38 Length, area, volume and convex sets (aspects of convex geometry)
05E45 Combinatorial aspects of simplicial complexes
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