Article

# Absorbing Angles, Steiner Minimal Trees, and Antipodality

(Impact Factor: 1.51). 08/2011; 143(1). DOI: 10.1007/s10957-009-9552-1
Source: arXiv

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
$R^1$.