## Tohoku Mathematical Journal 2013 March SECOND SERIES VOL. 65, NO. 1

 Tohoku Math. J. 65 (2013), 131-157

Title A COMPARISON THEOREM FOR STEINER MINIMUM TREES IN SURFACES WITH CURVATURE BOUNDED BELOW

Author Shintaro Naya and Nobuhiro Innami

(Received October 19, 2010, revised May 7, 2012)
Abstract. Let $D$ be a compact polygonal Alexandrov surface with curvature bounded below by $\kappa$. We study the minimum network problem of interconnecting the vertices of the boundary polygon $\partial D$ in $D$. We construct a smooth polygonal surface $\widetilde D$ with constant curvature $\kappa$ such that the length of its minimum spanning trees is equal to that of $D$ and the length of its Steiner minimum trees is less than or equal to $D$'s. As an application we show a comparison theorem of Steiner ratios for polygonal surfaces.

2000 Mathematics Subject Classification. Primary 53C20; Secondary 05C05.

Key words and phrases. Steiner tree, geodesic, Alexandrov space.