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The proof of the “bellows” conjecture for polyhedra of low topological genus. (English. Russian original) Zbl 0961.52007

Dokl. Math. 57, No. 1, 132-135 (1998); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 358, No. 6, 743-746 (1998).
From the text: The “bellows” conjecture states that, when a polyhedron bends, its generalized volume (i.e., the sum of volumes of consistently oriented tetrahedra with a common vertex and bases on faces of the polyhedron) remains constant. This conjecture was stated in 1978 in R. Konnelli’s report at the International Mathematical Congress in Helsinki. The author suggested examining this conjecture in a more general statement, namely, that the generalized volume of any polyhedron is a root of some polynomial whose coefficients depend only on the metric of the polyhedron. The first partial result concerning this conjecture was announced by the author in [Usp. Mat. Nauk 50, No. 2, 223-224 (1995; Zbl 0857.52004)]. A more general result, which is valid, in particular, for all polyhedra homeomorphic to a sphere, was announced in [Fundam. Prikl. Mat. 2, No. 1, 305-307 (1996; Zbl 0903.52005)].
In this paper, we give a detailed scheme of the proof of the following theorem.
Theorem. If an orientable polyhedron with triangular faces is homeomorphic to the handlebody of genus \(g\), where \(g<4\), then its generalized volume \(V\) is a root of a polynomial equation of the form \[ V^N+ \sum^N_{i=1} a_iV^{N-i}=0, \] whose coefficients depend only on the lengths of the polyhedron edges (to be more precise, \(a_i\) are polynomials in \(l\), where \(l\) is the set of lengths of the polyhedron edges).

MSC:

52C25 Rigidity and flexibility of structures (aspects of discrete geometry)
52A38 Length, area, volume and convex sets (aspects of convex geometry)
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