, Volume 32, Issue 6, pp 689–702 | Cite as

Analogues of the central point theorem for families with d-intersection property in ℝ d

  • Roman N. Karasev
Original paper


In this paper we consider families of compact convex sets in ℝ d such that any subfamily of size at most d has a nonempty intersection. We prove some analogues of the central point theorem and Tverberg’s theorem for such families.

Mathematics Subject Classification (2000)

52A20 52A35 52C35 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    I. Bárány, S.B. Shlosman, S. Szücz: On a topological generalization of a theorem of Tverberg, J. London Math. Soc. II. Ser. 23 (1981), 158–164.zbMATHCrossRefGoogle Scholar
  2. [2]
    L. Brouwer: Über abbildung von mannigfaltigkeiten, Mathematische Annalen 71 (1910), 97–115.MathSciNetCrossRefGoogle Scholar
  3. [3]
    V. L. Dol’nikov: Common transversals for families of sets in ℝn and connections between theorems of Helly and Borsuk, (In Russian) Doklady Akademii Nauk USSR 297(4) (1987), 777–780.MathSciNetGoogle Scholar
  4. [4]
    J. Eckhoff: Helly, Radon, and Carathéodory type theorems, Handbook of Convex Geometry, ed. by P.M. Gruber and J.M. Willis, North-Holland, Amsterdam, 1993, 389–448.Google Scholar
  5. [5]
    B. Grünbaum: Partitions of mass-distributions and of convex bodies by hyperplanes, Pacific J. Math. 10 (1960), 1257–1261.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    A. Hatcher: Algebraic Topology, Cambridge University Press, 2002.Google Scholar
  7. [7]
    E. Helly: Über Mengen konvexer Körper mit gemeinschaftlichen Punkten, Jber Deutsch. Math. Verein. 32 (1923), 175–176.zbMATHGoogle Scholar
  8. [8]
    Wu Yi Hsiang: Cohomology theory of topological transformation groups, Springer Verlag, 1975.Google Scholar
  9. [9]
    R. N. Karasev: Tverberg’s transversal conjecture and analogues of nonembeddability theorems for transversals, Discrete and Computational Geometry 38(3) (2007), 513–525.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    R. N. Karasev: Piercing families of convex sets with d-intersection property in ℝd, Discrete and Computational Geometry 39(4) (2008), 766–777.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    R. N. Karasev: Dual theorems on central points and their generalizations, Sbornik: Mathematics 199(10) (2008), 1459–1479; translated and corrected version available at arXiv:0909.4915.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    R. N. Karasev: Tverberg-type theorems for intersecting by rays, Discrete and Computational Geometry 45(2) (2011), 340–347.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    B. M. Mann and R. J. Milgram: On the Chern classes of the regular representations of some finite groups, Proc. Edinburgh Math. Soc., II Ser. 25 (1982), 259–268.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    G. Luke, A.S. Mishchenko: Vector bundles and their applications, Springer Verlag, 1998.Google Scholar
  15. [15]
    J. Matoušek: Using the Borsuk-Ulam theorem, Berlin-Heidelberg, Springer Verlag, 2003.zbMATHGoogle Scholar
  16. [16]
    J. McCleary: A user’s guide to spectral sequences, Cambridge University Press, 2001.Google Scholar
  17. [17]
    J. Milnor, J. Stasheff: Characteristic classes, Princeton University Press, 1974.Google Scholar
  18. [18]
    B. H. Neumann: On an invariant of plane regions and mass distributions, J. London Math. Soc. 20 (1945), 226–237.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    R. Rado: A theorem on general measure, J. London Math. Soc. 21 (1946), 291–300.MathSciNetCrossRefGoogle Scholar
  20. [20]
    H. Tverberg: A generalization of Radon’s theorem, J. London Math. Soc. 41 (1966), 123–128.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    H. Tverberg, S. Vrécica: On generalizations of Radon’s theorem and the ham sandwich theorem, Europ. J. Combinatorics 14 (1993), 259–264.zbMATHCrossRefGoogle Scholar
  22. [22]
    S. T. Vrécica: Tverberg’s conjecture, Discrete and Computational Geometry 29 (2003), 505–510.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    A. Yu. Volovikov: A theorem of Bourgin-Yang type for ℤpn -action, Sbornik Mathematics 76(2) (1993), 361–387.MathSciNetCrossRefGoogle Scholar
  24. [24]
    A. Yu. Volovikov: On a topological generalization of the Tverberg theorem, Mathematical Notes 59:3 (1996), 324–326.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    R. T. Živaljević: The Tverberg-Vrécica problem and the combinatorial geometry on vector bundles, Israel J. Math. 111 (1999), 53–76.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsMoscow Institute of Physics and TechnologyDolgoprudnyRussia
  2. 2.Laboratory of Discrete and Computational GeometryYaroslavl’ State UniversityYaroslavl’Russia

Personalised recommendations