Steiner ratio for hyperbolic surfaces.
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.
scientific article; zbMATH DE number 5123007