Geodesic
Locally Minimal Trees in $n$-Normed Spaces
Matematičeskie zametki, Tome 74 (2003) no. 5, pp. 656-668

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The locally minimal trees in normed spaces $({\mathbb R}^2,\rho)$, where the unit circle $\Sigma=\{x\in{\mathbb R}^2\mid{\rho}(x)=1\}$ in the norm $\rho$ coincides with the regular $m$-gon ($m = 2n$) inscribed in the Euclidean unit circle $S^1$, are completely classified. 
@article{MZM_2003_74_5_a1,
     author = {D. P. Il'yutko},
     title = {Locally {Minimal} {Trees} in $n${-Normed} {Spaces}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {656--668},
     publisher = {mathdoc},
     volume = {74},
     number = {5},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2003_74_5_a1/}
}
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D. P. Il'yutko. Locally Minimal Trees in $n$-Normed Spaces. Matematičeskie zametki, Tome 74 (2003) no. 5, pp. 656-668. http://geodesic.mathdoc.fr/item/MZM_2003_74_5_a1/