This is an announcement for the paper "Absorbing angles, Steiner minimal trees, and antipodality" by Horst Martini, Konrad J. Swanepoel, and P. Oloff de Wet.

Abstract: We give a new proof that a star ${op_i:i=1,\dots,k}$ in a normed plane is a Steiner minimal tree of its vertices ${o,p_1,\dots,p_k}$ 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 ${op_i: i=1,\dots,k}$ in any CL-space is a Steiner minimal tree of its vertices ${o,p_1,\dots,p_k}$ 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 $R^1$.

Archive classification: math.MG math.FA

Mathematics Subject Classification: 46B20 (Primary). 05C05, 49Q10, 52A21 (Secondary)

Citation: Journal of Optimization Theory and Applications, 143 (2009),

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1108.5046

or

http://arXiv.org/abs/1108.5046