Absorbing angles, Steiner minimal trees, and antipodality

We give a new proof that a star $\{o{p}_i:i=1,\mathrm{.}\mathrm{.}\mathrm{.},k\}$ in a normed plane is a Steiner minimal tree of its vertices $\{o,p_1,\mathrm{.}\mathrm{.}\mathrm{.},p_k\}$ if and only if all angles formed by the edges at o are absorbing [Swanepoel, Networks $\mathbf{\text{36}}$ (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 $\{o{p}_i:i=1,\mathrm{.}\mathrm{.}\mathrm{.},k\}$ in any CL-space is a Steiner minimal tree of its vertices $\{o,p_1,\mathrm{.}\mathrm{.}\mathrm{.},p_k\}$ if and only if all angles are absorbing, which in turn holds if and only if all distances between the normalizations $\frac{1}{\mathrm{\parallel }{p}_i\mathrm{\parallel }}{p}_i$ equal 2. CL-spaces include the mixed ${\mathrm{\ell }}_1$ and ${\mathrm{\ell }}_{\mathrm{\infty }}$ sum of finitely many copies of $R^1$.