Geodesic
Metric characteristics of hyperbolic polygons and polyhedra
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference "Geometric Methods in Control Theory and Mathematical Physics" dedicated to the 70th anniversary of S.L. Atanasyan, the 70th anniversary of I.S. Krasilshchik, the 70th anniversary of A.V. Samokhin, and the 80th anniversary of V.T. Fomenko. S.A. Esenin Ryazan State University, Ryazan, September 25–28, 2018. Part 2, Tome 169 (2019), pp. 31-38.

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In this paper, we consider some properties of hyperbolic polyhedra, both common with Euclidean and specific. Asymptotic behavior of metric characteristics of polyhedra in the $n$-dimensional hyperbolic space is examined in the cases where parameters of the polyhedra change and the dimension of the space unboundedly increases; in particular, the radius of the inscribed sphere of a polyhedron is estimated and its asymptotic behavior is obtained. In connection with this, the problem of estimating the minimal number of faces of the described polyhedron in the $n$-dimensional hyperbolic space depending on the radius of the inscribed sphere is posed. We also consider some properties of hyperbolic polygons, both belonging to absolute geometry and only hyperbolic.
Keywords: Lobachevsky space, hyperbolic trigonometry, polyhedron, sphere
Mots-clés : polygon, simplex. 
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E. A. Kostina; N. N. Kostina. Metric characteristics of hyperbolic polygons and polyhedra. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference "Geometric Methods in Control Theory and Mathematical Physics" dedicated to the 70th anniversary of S.L. Atanasyan, the 70th anniversary of I.S. Krasilshchik, the 70th anniversary of A.V. Samokhin, and the 80th anniversary of V.T. Fomenko. S.A. Esenin Ryazan State University, Ryazan, September 25–28, 2018. Part 2, Tome 169 (2019), pp. 31-38. http://geodesic.mathdoc.fr/item/INTO_2019_169_a4/

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