2026/07/01 by Andreĭ Vesnin, A. Yu. Vesnin, A. A. Egorov
Mathematics · #Analytic and geometric function theory #Mathematical Dynamics and Fractals #Mathematics and Applications
paper · doi:10.1134/s0037446626040026
crossref issued 2026/07/01 · crossref published 2026/07/01 · crossref published-print 2026/07/01 · openalex publication_date 2026/07/01 · crossref published-online 2026/07/30 · crossref created 2026/07/30 · crossref deposited 2026/07/30 · crossref indexed 2026/07/30 · openalex created_date 2026/07/31 · openalex updated_date 2026/08/01
We consider three-dimensional hyperbolic polyhedra of finite volume with finitely many vertices. The normalized volume of a polyhedron is the ratio of its volume to the number of vertices. Given some set of hyperbolic polyhedra, we can associate with it the set of normalized volumes of the polyhedra belonging to it. We call this set the spectrum of normalized volumes of the set under consideration. We focus on right-angled hyperbolic polyhedra. For the subset of ideal polyhedra, we find bounds for the spectrum of normalized volumes and prove that they are sharp. Moreover, we show that the spectrum splits into discrete and dense parts. For the subset of compact polyhedra, we obtain estimates for the spectrum of normalized volumes, prove that the upper bound is sharp, and also establish numerical intervals on which the spectrum is discrete and dense. The endpoints of the indicated numerical intervals are expressed in terms of the volume of the regular ideal hyperbolic tetrahedron and the volume of the regular ideal hyperbolic octahedron.