Steiner ratio for Riemannian manifolds (Q2759261)

Given a finite set of points in a Riemanian manifold, the shortest network interconnecting them is called the Steiner minimum tree. The Steiner ratio is the largest lower bound for the ratio between lengths of the Steiner minimum tree and the minimum spanning tree for the same set of points. It is proved in this paper that (1) the Steiner ratio of an arbitrary connected \(n\)-dimensional manifold does not exceed that of the \(n\)-dimensional Euclidean space; (2) the Steiner ration of the Lobachevski space of curvature \(-1\) does not exceed \(3/4\); (3) the Steiner ratio of an arbitrary two-dimensional surface of constant curvature \(-1\) is strictly less than \(\sqrt{3}/2\).