Информация о цитировании статьи получена из
Web of Science,
Scopus
Статья опубликована в журнале из списка Web of Science и/или Scopus
Дата последнего поиска статьи во внешних источниках: 18 июля 2013 г.
Аннотация:For a metric space $(X,\r)$ and any finite subset $N\ss X$ by
$\r(\SMT_N)$ and $\r(\MST_N)$ we denote respectively the lengths of a
Steiner minimal tree and a minimal spanning tree with the boundary $N$.
The {\em Steiner ratio $m(X,\r)$} of the metric space
is the value $\inf_{\{N:N\ss X\}}\frac{\r(\SMT_N)}{\r(\MST_N)}$.
In this paper we prove the following results describing the Steiner ratio
of some manifolds:
(1) the Steiner ratio of an arbitrary $n$-dimensional connected Riemannian
manifold $M$ does not exceed the Steiner ratio of $\R^n$;
(2) the Steiner ratio of the base of a locally isometric covering is
more or equal than the Steiner ratio of the total space;
(3) the Steiner ratio of a flat two-dimensional torus, a flat Klein
bottle, a projective plain having constant positive curvature is equal to
$\sqrt3/2$;
(4) the Steiner ratio of the curvature $-1$ Lobachevsky space does not
exceed $3/4$;
(5) the Steiner ratio of an arbitrary surface of constant negative
curvature $-1$ is strictly less than $\sqrt3/2$.