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Tohoku Mathematical Journal
SECOND SERIES VOL. 65, NO. 1
|Tohoku Math. J.|
65 (2013), 131-157
A COMPARISON THEOREM FOR STEINER MINIMUM TREES IN SURFACES WITH CURVATURE BOUNDED BELOW
Shintaro Naya and Nobuhiro Innami
(Received October 19, 2010, revised May 7, 2012)
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.
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