## Tohoku Mathematical Journal

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

### A comparison theorem for Steiner minimum trees in surfaces with curvature bounded below

Shintaro Naya and Nobuhiro Innami

#### 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

**Permanent link to this document**

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**

Primary: 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20]

Secondary: 05C05: Trees

**Keywords**

Steiner tree geodesic Alexandrov space

#### Citation

Naya, Shintaro; Innami, Nobuhiro. A comparison theorem for Steiner minimum trees in surfaces with curvature bounded below. Tohoku Math. J. (2) 65 (2013), no. 1, 131--157. doi:10.2748/tmj/1365452629. https://projecteuclid.org/euclid.tmj/1365452629