## Tohoku Mathematical Journal

### 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.

#### Article information

Source
Tohoku Math. J. (2), Volume 65, Number 1 (2013), 131-157.

Dates
First available in Project Euclid: 8 April 2013

https://projecteuclid.org/euclid.tmj/1365452629

Digital Object Identifier
doi:10.2748/tmj/1365452629

Mathematical Reviews number (MathSciNet)
MR3049644

Zentralblatt MATH identifier
1278.53076

Subjects