Geodesic
On parallelepipeds and centrally symmetric hexagonal prisms circumscribed about a three-dimensional centrally symmetric convex body
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 11, Tome 372 (2009), pp. 103-107 
V. V. Makeev. On parallelepipeds and centrally symmetric hexagonal prisms circumscribed about a three-dimensional centrally symmetric convex body. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 11, Tome 372 (2009), pp. 103-107. http://geodesic.mathdoc.fr/item/ZNSL_2009_372_a9/
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     title = {On parallelepipeds and centrally symmetric hexagonal prisms circumscribed about a~three-dimensional centrally symmetric convex body},
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Let $K$ be a three-dimensional centrally symmetric compact convex set of unit volume. It is proved that $K$ is contained in a centrally symmetric hexagonal prism or a parallelepiped with volume $4/\root3\of3<2.7735$. This fact implies that $K$ admits a lattice packing in space with density at least $\root3\of3/4>0.3605$. Furthermore, $K$ is contained in a parallelepiped with volume $4(3+6/(\sqrt3(1+\operatorname{ctg}(\pi/12))))^{2/3}/3<3.2082$. Bibl. – 6 titles.

[1] V. V. Makeev, “Trëkhmernye mnogogranniki, vpisannye i opisannye vokrug vypuklykh kompaktov. 2”, Algebra i analiz, 13:5 (2001), 110–133 | MR | Zbl

[2] V. V. Makeev, “O parallelepipedakh, opisannykh vokrug trëkhmernogo vypuklogo tela”, Zap. nauchn. semin. POMI, 329, 2005, 79–87 | MR | Zbl

[3] V. V. Makeev, “Ploskie secheniya vypuklykh tel i universalnye rassloeniya”, Zap. nauchn. semin. POMI, 280, 2001, 219–233 | MR | Zbl

[4] V. V. Makeev, “Nekotorye ekstremalnye zadachi dlya vektornykh rassloenii”, Algebra i analiz, 19:2 (2007), 131–155 | MR | Zbl

[5] I. Fáry, “Sur la densité des réseaux de domains convexes”, Bull. Soc. Math. France, 78 (1950), 152–161 | MR | Zbl

[6] I. K. Babenko, “Asimptoticheskii ob'ëm torov i geometriya vypuklykh tel”, Mat. zametki, 44:2 (1988), 177–190 | MR | Zbl