Geodesic
On inscribing a regular octahedron in a three-dimensional convex body with smooth boundary
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 6, Tome 279 (2001), pp. 183-186

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A norm $\|\cdot\|$ and a convex body $K$ with smooth boundary in the standard Euclidean space $\mathbb R^3$ are considered. It is proved that the boundary $\partial K$ of $K$ contains the vertices $AA'BB'CC'$ of a regular octahedron with $\|AA'\|=\|BB'\|\ge\|CC'\|$ (respectively, $\|AA'\|=\|BB'\|\le\|CC'\|$). 
@article{ZNSL_2001_279_a10,
     author = {V. V. Makeev},
     title = {On inscribing a regular octahedron in a three-dimensional convex body with smooth boundary},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {183--186},
     publisher = {mathdoc},
     volume = {279},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2001_279_a10/}
}
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V. V. Makeev. On inscribing a regular octahedron in a three-dimensional convex body with smooth boundary. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 6, Tome 279 (2001), pp. 183-186. http://geodesic.mathdoc.fr/item/ZNSL_2001_279_a10/