Some applications of a formula for the volume of a polyhedron
Let \(P\) be a polyhedron in \(\mathbb R^3\) with triangular faces and given edge lengths. The author shows that for every `small' diagonal there is a polynomial equation, with coefficients depending only on the combinatorial structure, the edge lengths, and the volume of \(P\), such that the length of the diagonal is a root of this polynomial (a `small' diagonal is a diagonal between vertices of two faces with a common edge which do not belong to the edge and, of cause, are not joined by an edge). Some consequences of this fact are given. This is an English translation of the author's article published in the book [\textit{Yu. G. Reshetnyak} et al., Proceedings of the conference `Geometry and applications' dedicated to the 70th anniversary of Prof. Victor Toponogov. Novosibirsk: Izdatel'stvo Instituta Matematiki, 183--197 (2001; Zbl 0991.52005)].
scientific article; zbMATH DE number 2097412