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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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References:
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