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Absorbing Angles, Steiner Minimal Trees, and Antipodality

Author

Listed:
  • H. Martini

    (Technische Universität Chemnitz)

  • K. J. Swanepoel

    (Technische Universität Chemnitz)

  • P. Oloff Wet

    (University of South Africa)

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 ℓ 1 and ℓ ∞ sum of finitely many copies of ℝ.

Suggested Citation

  • H. Martini & K. J. Swanepoel & P. Oloff Wet, 2009. "Absorbing Angles, Steiner Minimal Trees, and Antipodality," Journal of Optimization Theory and Applications, Springer, vol. 143(1), pages 149-157, October.
    • Handle: RePEc:spr:joptap:v:143:y:2009:i:1:d:10.1007_s10957-009-9552-1
    DOI: 10.1007/s10957-009-9552-1
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    References listed on IDEAS

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    1. H. Martini & K.J. Swanepoel & G. Weiss, 2002. "The Fermat–Torricelli Problem in Normed Planes and Spaces," Journal of Optimization Theory and Applications, Springer, vol. 115(2), pages 283-314, November.
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