Discrete & Computational Geometry

, Volume 48, Issue 3, pp 766–776 | Cite as

Kadets-Type Theorems for Partitions of a Convex Body

Article

Abstract

For convex partitions of a convex body B, we try to put a homothetic copy of B into each set of the partition so that the sum of the homothety coefficients is at least 1. In the plane this can be done for an arbitrary partition, while in higher dimensions we need certain restrictions on the partition.

Keywords

The Tarski plank problem The Kadets theorem 

References

  1. 1.
    Alexander, R.: A problem about lines and ovals. Am. Math. Mon. 75, 482–487 (1968) MATHCrossRefGoogle Scholar
  2. 2.
    Ball, K.: The plank problem for symmetric bodies. Invent. Math. 104(3), 535–543 (1991). doi:10.1007/BF01245089 MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bang, T.: A solution of the “plank problem”. Proc. Am. Math. Soc. 2, 990–993 (1951) MathSciNetMATHGoogle Scholar
  4. 4.
    Bezdek, A.: On a generalization of Tarski’s plank problem. Discrete Comput. Geom. 38(2), 189–200 (2007). doi:10.1007/s00454-007-1333-8 MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bezdek, A., Bezdek, K.: A solution of Conway’s fried potato problem. Bull. Lond. Math. Soc. 27(5), 492 (1995) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Bezdek, A., Bezdek, K.: Conway’s fried potato problem revisited. Arch. Math. 66(6), 522–528 (1996) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Bezdek, K.: Classical Topics in Discrete Geometry. Springer, Berlin (2010). doi:10.1007/978-1-4419-0600-7 MATHCrossRefGoogle Scholar
  8. 8.
    Bezdek, K., Schneider, R.: Covering large balls with convex sets in spherical space. Contrib. Algebra Geom. 51(1), 229–235 (2010) MathSciNetMATHGoogle Scholar
  9. 9.
    Brass, P., Moser, W., Pach, J.: Research Problems in Discrete Geometry. Springer, New York (2005) MATHGoogle Scholar
  10. 10.
    Kadets, V.: Coverings by convex bodies and inscribed balls. Proc. Am. Math. Soc. 133(5), 1491–1496 (2005) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Nazarov, F., Sodin, M., Vol’berg, A.: The geometric Kannan–Lovász–Simonovits lemma, dimension-free estimates for the distribution of the values of polynomials, and the distribution of the zeros of random analytic functions. St. Petersburg Math. J. 14(2), 351–366 (2002) MathSciNetGoogle Scholar
  12. 12.
    Tarski, A.: Further remarks about the degree of equivalence of polygons. Odbitka Z. Parametr. 2, 310–314 (1932) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Institute for Information Transmission Problems RASMoscowRussia
  2. 2.Dept. of MathematicsMoscow Institute of Physics and TechnologyDolgoprudnyRussia
  3. 3.Laboratory of Discrete and Computational GeometryYaroslavl State UniversityYaroslavlRussia

Personalised recommendations