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.
Citation
Shintaro Naya. Nobuhiro Innami. "A comparison theorem for Steiner minimum trees in surfaces with curvature bounded below." Tohoku Math. J. (2) 65 (1) 131 - 157, 2013. https://doi.org/10.2748/tmj/1365452629
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