Absorbing angles, Steiner minimal trees, and antipodality

TL;DR

This paper provides a simpler proof for characterizing Steiner minimal trees in normed planes using absorbing angles and extends the characterization to higher-dimensional CL-spaces, including mixed  and inf spaces.

Contribution

It offers a more conceptual proof for the planar case and introduces a new sufficient condition for higher-dimensional spaces to share this characterization.

Findings

A star is a Steiner minimal tree if and only if all angles are absorbing.

In CL-spaces, the same characterization holds with all distances between normalized points equal to 2.

The proof simplifies previous results and extends them to higher dimensions.

Abstract

We give a new proof that a star $\{op_i:i=1,...,k\}$ 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 $\{op_i: i=1,...,k\}$ 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 $\frac{1}{\|p_i\|}p_i$ equal 2. CL-spaces include the mixed $\ell_1$ and $\ell_\infty$ sum of finitely many copies of $R^1$.