Absorbing angles, Steiner minimal trees and antipodality

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 equal 2. CL-spaces include the mixed ℓ 1 and ℓ ∞ sum of finitely many copies of ℝ.

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http://eprints.lse.ac.uk/25403/