Geodesic
Diameters of octahedra
Matematičeskie zametki, Tome 5 (1969) no. 4, pp. 429-436.

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The Kolmogorov $n$-diameters of an $m$-dimensional octahedron $O^m_\alpha$, i.e., of a convex hull of vectors $\{\pm\alpha_1e_1,\dots,\alpha_me_m\}$, are calculated. 
@article{MZM_1969_5_4_a4,
     author = {L. B. Sofman},
     title = {Diameters of octahedra},
     journal = {Matemati\v{c}eskie zametki},
     pages = {429--436},
     publisher = {mathdoc},
     volume = {5},
     number = {4},
     year = {1969},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1969_5_4_a4/}
}
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L. B. Sofman. Diameters of octahedra. Matematičeskie zametki, Tome 5 (1969) no. 4, pp. 429-436. http://geodesic.mathdoc.fr/item/MZM_1969_5_4_a4/