Abstract
Three theorems on approximation of plane sections of convex bodies by affine-regular polygons, ellipses, or circles are proved by topological means. In particular, it is proved that if K is a convex body in ℝ3 (resp., ℝ4), then for every interior point O of K there is a plane cross section of K through O which is circumscribed about an affine-regular hexagon (resp., octagon) with center O. Bibliography: 8 titles.
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References
I. Yaglom and V. Boltyanskii, Convex Figures [in Russian], Gostekhizdat, Moscow (1951).
V. Makeev, “Affine-inscribed and affine-circumscribed polygons and polyhedra,” Zap. Nauchn. Semin. POMI, 231, 286–298 (1996).
B. Grünbaum, Essays in Combinatorial Geometry and the Theory of Convex Bodies [in Russian], Nauka, Moscow (1971).
V. Dol'nikov, “Transversals of families of sets in ℝn and the relation between the theorems of Helly and Borsuk,” Mat. Sb., 184, No. 5, 111–131 (1993).
D. Bourgin, “Deformation and mapping theorems,” Fund. Math., 46, 285–303 (1959).
S. Vrecia and R. Zivaljevic, “The ham sandwich theorem revisited,” Israel J. Math., 78, 21–32 (1992).
V. Makeev, “Estimates for asphericity of the cross sections of convex bodies,” Ukr. Geom. Sb., 28, 76–79 (1985).
F. John, “An inequality for convex bodies,” Univ. Kentucky Res. Club Bull., 6, 26 (1940).
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 246, 1997, pp. 174–183.
Translated by N. Yu. Netsvetaev.
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Makeev, V.V. Approximation of plane cross sections of a convex body. J Math Sci 100, 2297–2302 (2000). https://doi.org/10.1007/s10958-000-0013-5
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DOI: https://doi.org/10.1007/s10958-000-0013-5