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St. Petersburg Mathematical Journal

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Some geometric properties of convex bodies. II


Author: V. V. Makeev
Translated by: N. Yu. Netsvetaev
Original publication: Algebra i Analiz, tom 15 (2003), nomer 6.
Journal: St. Petersburg Math. J. 15 (2004), 867-874
MSC (2000): Primary 52A10, 52A15
DOI: https://doi.org/10.1090/S1061-0022-04-00836-2
Published electronically: November 16, 2004
MathSciNet review: 2044632
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Abstract | References | Similar Articles | Additional Information

Abstract: Topological means are used for the study of approximation of $2$-dimensional sections of a $3$-dimensional convex body by affine-regular pentagons and approximation of a centrally symmetric convex body by a prism. Also, the problem of estimating the relative surface area of the sphere in a normed $3$-space, the problem on universal covers for sets of unit diameter in Euclidean space, and some related questions are considered.


References [Enhancements On Off] (What's this?)

  • 1. V. V. Makeev, The Knaster problem and almost spherical sections, Mat. Sb. 180 (1989), no. 3, 424-431; English transl., Math. USSR-Sb. 66 (1990), no. 2, 431-438. MR 0993234 (90d:55005)
  • 2. -, Affine-inscribed and affine-circumscribed polygons and polyhedra, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 231 (1995), 286-298; English transl., J. Math. Sci. 91 (1998), no. 6, 3518-3525. MR 1434300 (98b:52004)
  • 3. J. Pal, Über ein elementares Variationsproblem, Danske Vid. Selsk. Mat.-Fys. Medd. 3 (1920), no. 2, 35 pp.
  • 4. V. V. Makeev, On affine images of a rhombo-dodecahedron circumscribed about a three-dimensional convex body, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 246 (1997), 191-195; English transl., J. Math. Sci. 100 (2000), no. 3, 2307-2309. MR 1631812 (99e:52005)
  • 5. -, Some special configurations of planes that are associated with convex compacta, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 252 (1998), 165-174; English transl., J. Math. Sci. 104 (2001), no. 4, 1358-1363. MR 1756722 (2001f:52013)
  • 6. T. Hausel, E. Makai, and A. Szücs, Polyhedra inscribed and circumscribed to convex bodies, Proceedings of the 3rd International Workshop on Differential Geometry and its Applications and the 1st German-Romanian Seminar on Geometry (Sibiu, 1997), Gen. Math. 5 (1997), 183-190. MR 1723608
  • 7. G. Kuperberg, Circumscribing constant-width bodies with polytopes, New York J. Math. 5 (1999), 91-100; Preprint arXiv: math.MG/9809165. MR 1703205 (2000h:52002)
  • 8. V. V. Makeev, Three-dimensional polytopes inscribed in and circumscribed about compact convex sets, Algebra i Analiz 12 (2000), no. 4, 1-15; English transl., St. Petersburg Math. J. 12 (2001), no. 4, 507-518. MR 1793615 (2001k:52017)
  • 9. -, Universal coverings. I, Ukrain. Geom. Sb. No. 24 (1981), 70-79. (Russian) MR 0629813 (83e:52009)
  • 10. -, On geometric properties of three-dimensional convex bodies, Algebra i Analiz 14 (2002), no. 5, 96-109; English transl., St. Petersburg Math. J. 14 (2003), no. 5, 781-790. MR 1970335 (2004c:52001)
  • 11. -, Inscribed and circumscribed polygons of a convex body, Mat. Zametki 55 (1994) no. 4, 128-130; English transl., Math. Notes 55 (1994), no. 3-4, 423-425. MR 1296224 (95h:52002)

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Additional Information

V. V. Makeev
Affiliation: Universitetskiĭ Pr. 27, St. Petersburg 190000, Russia

DOI: https://doi.org/10.1090/S1061-0022-04-00836-2
Keywords: Convex body, figure, field of convex bodies, relative surface area
Received by editor(s): December 25, 2002
Published electronically: November 16, 2004
Additional Notes: The paper was revised by the author for the English edition.
Article copyright: © Copyright 2004 American Mathematical Society

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