Steiner ratio for hyperbolic surfaces. (English) Zbl 1114.53034

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 \(\rho(M)\geq{1\over 2}\) [D.-Z. Du and 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.


53C20 Global Riemannian geometry, including pinching
05C05 Trees


Zbl 0707.05018
Full Text: DOI Euclid


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