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Two-sided bounds for the volume of right-angled hyperbolic polyhedra. (English. Russian original) Zbl 1252.51014

Math. Notes 89, No. 1, 31-36 (2011); translation from Mat. Zametki 89, No. 1, 12-18 (2011).
Summary: For a compact right-angled polyhedron \(R\) in the Lobachevskii space \(\mathbb H^{3}\), let \((R)\) denote its volume and vert\((R)\) the number of its vertices. Upper and lower bounds for \(vol(R)\) were recently obtained by Atkinson in terms of vert\((R)\). In constructing a two-parameter family of polyhedra, we show that the asymptotic upper bound \(5v _{3}/8\), where \(v _{3}\) is the volume of the ideal regular tetrahedron in \(\mathbb H^{3}\), is a double limit point for the ratios \(\text{vol}(R)/\text{vert}(R)\). Moreover, we improve the lower bound in the case vert\((R) \leq 56\).

MSC:

51M25 Length, area and volume in real or complex geometry
51M20 Polyhedra and polytopes; regular figures, division of spaces
51M10 Hyperbolic and elliptic geometries (general) and generalizations
52B10 Three-dimensional polytopes
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References:

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