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Journal of Optimization Theory and Applications

, Volume 143, Issue 1, pp 149–157 | Cite as

Absorbing Angles, Steiner Minimal Trees, and Antipodality

  • H. Martini
  • K. J. Swanepoel
  • P. Oloff de Wet
Article

Abstract

We give a new proof that a star {op i :i=1,…,k} in a normed plane is a Steiner minimal tree of vertices {o,p 1,…,p k } if and only if all angles formed by the edges at o are absorbing (Swanepoel in Networks 36: 104–113, 2000). The proof is simpler and yet more conceptual than the original one.

We also find a new sufficient condition for higher-dimensional normed spaces to share this characterization. In particular, a star {op i :i=1,…,k} in any CL-space is a Steiner minimal tree of vertices {o,p 1,…,p k } if and only if all angles are absorbing, which in turn holds if and only if all distances between the normalizations \(\frac{1}{\Vert p_{i}\Vert}p_{i}\) equal 2. CL-spaces include the mixed 1 and sum of finitely many copies of ℝ.

Keywords

Steiner minimal trees Absorbing angles Antipodality Face antipodality Minkowski geometry 

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • H. Martini
    • 1
  • K. J. Swanepoel
    • 1
  • P. Oloff de Wet
    • 2
  1. 1.Fakultät für MathematikTechnische Universität ChemnitzChemnitzGermany
  2. 2.Department of Decision SciencesUniversity of South AfricaUnisaSouth Africa

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