Steiner ratio for hyperbolic surfaces.
type: Item
instance of: scholarly article
title: Steiner ratio for hyperbolic surfaces.
zbMATH Open document ID: 1114.53034
author: Nobuhiro Innami
author: Byung Hak Kim
published in: Proceedings of the Japan Academy. Series A
publication date: 2007-02-07T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.pja/1155820123
review text: Let \(M\) be a complete Riemannian manifold without boundary, \(P\) a finite set of points on \(M\). A shortest network interconnecting \(P\) is called a Steiner minimum tree, shortly \(\text{SMT}(P)\); it may have vertices which are not in \(P\) (``Steiner points''). A shortest tree with vertex set \(P\) is called a minimum spanning tree on \(P\), shortly \(\text{MST}(P)\). The Steiner ratio \(\rho=\rho(M)\) of \(M\) is given by the infimum (with respect to all finite point sets \(P\subset M\)) of the quotient ``total length of the edges in \(\text{SMT}(P)/\) total length of the edges in \(\text{MST}(P)\)''. It is well-known that \(\rho(M)\geq{1\over 2}\) [\textit{D.-Z. Du} and \textit{F. K. Hwang}, Proc. Natl. Acad. Sci. USA 87, No. 23, 9464--9466 (1990; Zbl 0707.05018)]. The authors prove (Theorem 1): the Steiner ratio for a simply connected complete surface of negative constant curvature without boundary (``hyperbolic space'') is \({1\over 2}\) by considering a sequence of regular geodesic polygons in the Poincaré disk.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 05C05
zbMATH DE Number: 5123007
zbMATH Keywords: Steiner ratio
zbMATH Keywords: Steiner tree
zbMATH Keywords: Riemannian geometry
zbMATH Keywords: geodesic
zbMATH Keywords: hyperbolic geometry
zbMATH Keywords: Poincaré disk
MaRDI profile type: Publication
OpenAlex ID: W2065969435
cites work: The Steiner ratio conjecture of Gilbert and Pollak is true.
cites work: Steiner Minimal Trees
cites work: The Steiner ratio for manifolds.
cites work: Compression theorems and Steiner ratios on spheres
DOI: 10.3792/PJAA.82.77
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description: scientific article; zbMATH DE number 5123007