We construct new polyhedra such that some of their similar or affine images can be inscribed in (or circumscribed about) every centrally symmetric convex body.
One of the main theorems is as follows. If K is a centrally symmetric, three-dimensional, convex body, then either an affine-regular dodecahedron is inscribed in K or there are two affine-regular dodecahedra D1 and D2 such that nine pairs of opposite vertices of Di, i = 1, 2, lie on the boundary of K. Furthermore, the remaining two vertices of D1 lie outside K, while the remaining two vertices of D2 lie inside K.
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References
L. Fejes Tóth, Lagerungen in der Ebene auf der Kugel und im Raum, Springer-Verlag, Berlin, Göttingen, Heidelberg (1953).
B. Grünbaum, “Affine-regular polygons inscribed in plane convex sets,” Riveon Lematimatika, 13, 20–24 (1959).
V. V. Makeev, “Knaster problem and almost spherical sections ,” Mat. Sb., 180, 424–431 (1989).
T. Hausel, E. Makai, and A. Szücz, “Inscribing cubes and covering by rhombic dodecahedra via equivariant topology,” Mathematika, 47, 371–397 (2000).
H. Griffiths, “The topology of square pegs in round holes,” Proc. London Math. Soc., 62, 647–672 (1991).
V. V. Makeev, “Three-dimensional polyhedra inscribed in and circumscribed about convex compacta,” Algebra Analiz, 12, 1–15 (2000).
V. V. Makeev, “On certain geometric properties of convex bodies,” Algebra Analiz, 14, 96–109 (2002).
V. V. Makeev, “Three-dimensional polyhedra inscribed in and circumscribed about convex compacta. II,” Algebra Analiz 13, 110–133 (2001).
V. V. Makeev, “On certain geometric properties of convex bodies. II,” Algebra Analiz, 15, 74–85 (2003).
V. V. Makeev, “On geometry of finite-dimensional normed spaces and continuous functions on the sphere in the Euclidean space,” Zap. Nauchn. Semin. POMI, 329, 107–117 (2005).
V. V. Makeev, “Inscribed and circumscribed polyhedra for a convex body and the problem on continuous functions on the sphere in the Euclidean space,” Algebra Analiz, 18, 187–204 (2006).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 415, 2013, pp. 54–61.
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Makeev, V.V., Netsvetaev, N.Y. Inscribed and Circumscribed Polyhedra for a Centrally Symmetric Convex Body. J Math Sci 212, 552–557 (2016). https://doi.org/10.1007/s10958-016-2687-3
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DOI: https://doi.org/10.1007/s10958-016-2687-3