Advertisement

Discrete & Computational Geometry

, Volume 45, Issue 2, pp 340–347 | Cite as

Tverberg-Type Theorems for Intersecting by Rays

  • R. N. Karasev
Article

Abstract

In this paper we consider some results on intersection between rays and a given family of convex, compact sets. These results are similar to the central point theorem, and Tverberg’s theorem on partitions of a point set.

Keywords

Central point theorem Tverberg’s theorem Helly’s theorem 

References

  1. 1.
    Aharoni, R., Duchet, P., Wajnryb, B.: Successive projections on hyperplanes. J. Math. Anal. Appl. 103, 134–138 (1984) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bárány, I., Shlosman, S.B., Szücz, S.: On a topological generalization of a theorem of Tverberg. J. Lond. Math. Soc., II. Ser. 23, 158–164 (1981) zbMATHCrossRefGoogle Scholar
  3. 3.
    Bartsch, T.: Topological Methods for Variational Problems with Symmetries. Springer, Berlin (1993) zbMATHGoogle Scholar
  4. 4.
    Fulek, R., Holmsen, A.F., Pach, J.: Intersecting convex sets by rays. Discrete Comput. Geom. 42(3), 343–358 (2009) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Grünbaum, B.: Partitions of mass-distributions and of convex bodies by hyperplanes. Pac. J. Math. 10, 1257–1261 (1960) zbMATHGoogle Scholar
  6. 6.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002) zbMATHGoogle Scholar
  7. 7.
    Hsiang, W.Y.: Cohomology Theory of Topological Transformation Groups. Springer, Berlin (1975) zbMATHGoogle Scholar
  8. 8.
    Karasev, R.N.: Dual theorems on central points and their generalizations. Sb. Math. 199(10), 1459–1479 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Karasev, R.N.: Analogues of the central point theorem for families with d-intersection property in ℝd. arXiv:0906.2262v1 (2009)
  10. 10.
    Karasev, R.N.: Theorems of Borsuk-Ulam type for flats and common transversals. Sb. Math. 200(10), 39–58 (2009) (In Russian); translated in arXiv:0905.2747v1 CrossRefMathSciNetGoogle Scholar
  11. 11.
    Krasnosel’skii, M.A.: On the estimation of the number of critical points of functionals. Usp. Mat. Nauk 7(2), 157–164 (1952) (In Russian) MathSciNetGoogle Scholar
  12. 12.
    Luke, G., Mishchenko, A.S.: Vector Bundles and Their Applications. Springer, Berlin (1998) zbMATHGoogle Scholar
  13. 13.
    Matoušek, J.: Using the Borsuk-Ulam Theorem. Springer, Berlin (2003) zbMATHGoogle Scholar
  14. 14.
    Milnor, J., Stasheff, J.: Characteristic Classes. Princeton University Press, Princeton (1974) zbMATHGoogle Scholar
  15. 15.
    Neumann, B.H.: On an invariant of plane regions and mass distributions. J. Lond. Math. Soc. 20, 226–237 (1945) zbMATHCrossRefGoogle Scholar
  16. 16.
    Rado, R.: A theorem on general measure. J. Lond. Math. Soc. 21, 291–300 (1946) CrossRefMathSciNetGoogle Scholar
  17. 17.
    Rousseeuw, P.J., Hubert, M.: Depth in an arrangement of hyperplanes. Discrete Comput. Geom. 22, 167–176 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Schwartz, A.S.: Some estimates of the genus of a topological space in the sense of Krasnosel’skii. Usp. Mat. Nauk 12:4(76), 209–214 (1957) (In Russian) Google Scholar
  19. 19.
    Schwartz, A.S.: The genus of a fibre space. Tr. Mosk. Mat. Obsc. 11, 99–126 (1962); translation in Am. Math. Soc. Trans. 55, 49–140 (1966) Google Scholar
  20. 20.
    Tverberg, H.: A generalization of Radon’s theorem. J. Lond. Math. Soc. 41, 123–128 (1966) zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Volovikov, A.Yu.: On a topological generalization of the Tverberg theorem. Math. Not. 59(3), 324–326 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Volovikov, A.Yu.: On the Cohen-Lusk theorem. Fundam. Appl. Math. 13(8), 61–67 (2007) (In Russian) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Dept. of MathematicsMoscow Institute of Physics and TechnologyDolgoprudnyRussia

Personalised recommendations