Geodesic
Approximation of three-dimensional convex bodies by affine-regular prisms
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 10, Tome 353 (2008), pp. 126-131 
V. V. Makeev. Approximation of three-dimensional convex bodies by affine-regular prisms. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 10, Tome 353 (2008), pp. 126-131. http://geodesic.mathdoc.fr/item/ZNSL_2008_353_a10/
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     title = {Approximation of three-dimensional convex bodies by affine-regular prisms},
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     year = {2008},
     volume = {353},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2008_353_a10/}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Let $K\subset\mathbb R^3$ be a convex body of unit volume. It is proved that $K$ contains an affine-regular pentagonal prism of volume $4(5-2\sqrt5)/9>0.2346$ and an affine-regular pentagonal antiprism of volume $4(3\sqrt5-5)/27>0.253$. Furthermore, $K$ is contained in an affine-regular pentagonal prism of volume $6(3-\sqrt5)<4.5836$, and in an affine-regular heptagonal prism of volume $21(2\cos\pi/7-1)/4<4.2102$. If $K$ is a tetrahedron, then the latter estimate is sharp. Bibl. – 8 titles.

[1] A. Bielecki, K. Radziszewski, “Sur les parallélépipédes inscrits dans les corps convexes”, Ann. Univ. Mariae Curie-Sklodowska Sect. A, 7 (1954), 97–100 | MR

[2] V. V. Makeev, “Vpisannye i opisannye mnogogranniki vypuklogo tela i svyazannye s nimi zadachi”, Mat. zametki, 51:5 (1992), 67–71 | MR | Zbl

[3] V. V. Makeev, “O nekotorykh geometricheskikh svoistvakh vypuklykh trekhmernykh tel”, Algebra i analiz, 14:5 (2002), 96–109 | MR | Zbl

[4] V. V. Makeev, “O nekotorykh geometricheskikh svoistvakh vypuklykh tel. II”, Algebra i analiz, 15:6 (2003), 74–85 | MR | Zbl

[5] V. V. Makeev, “O priblizhenii krugami i ellipsami dvumernykh sechenii vypuklykh tel”, Algebra i analiz, 16:6 (2004), 162–171 | MR

[6] V. V. Makeev, “O priblizhenii trekhmernogo vypuklogo tela tsilindrami”, Algebra i analiz, 17:2 (2005), 133–144 | MR

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

[8] V. V. Makeev, “Skol krugluyu ten imeet vypukloe telo”, Zap. nauchn. semin. POMI, 329, 2005, 67–78 | MR | Zbl