Absorbing angles, Steiner minimal trees, and antipodality

Abstract

We give a new proof that a star $\{o{p}_i:i=1,...,k\}$ in a normed plane is a Steiner minimal tree of its vertices $\{o,p_1,...,p_k\}$ if and only if all angles formed by the edges at o are absorbing [Swanepoel, Networks \textbf{36} (2000), 104--113]. The proof is more conceptual and simpler than the original one. We also find a new sufficient condition for higher-dimensional normed spaces to share this characterization. In particular, a star $\{o{p}_i:i=1,...,k\}$ in any CL-space is a Steiner minimal tree of its 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 ${\ell }_1$ and ${\ell }_{\infty }$ sum of finitely many copies of $R^1$.

7 pages, 4 figures. An old paper already published in 2009. This is a preprint version

References in corpus (3)

• The Fermat-Torricelli problem in normed planes and spaces
• The local Steiner problem in finite-dimensional normed spaces
• Low-degree minimal spanning trees in normed spaces

Cited by in corpus (1)

• Combinatorial distance geometry in normed spaces