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

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.