Absorbing Angles, Steiner Minimal Trees, and Antipodality

H. Martini (), K. J. Swanepoel () and P. Oloff Wet
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H. Martini: Technische Universität Chemnitz
K. J. Swanepoel: Technische Universität Chemnitz
P. Oloff Wet: University of South Africa

Journal of Optimization Theory and Applications, 2009, vol. 143, issue 1, No 9, 149-157

Abstract: 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.

Keywords: Steiner minimal trees; Absorbing angles; Antipodality; Face antipodality; Minkowski geometry (search for similar items in EconPapers)
Date: 2009
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DOI: 10.1007/s10957-009-9552-1

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