Discrete & Computational Geometry

, Volume 49, Issue 3, pp 478–484 | Cite as

Combinatorial Generalizations of Jung’s Theorem



We consider combinatorial generalizations of Jung’s theorem on covering a set by a ball. We prove the “fractional” and “colorful” versions of the theorem.


Jung’s theorem Helly’s theorem Covering by a ball 


  1. 1.
    Bárány, I.: A generalization of Carathéodory’s theorem. Discrete Math. 40(2–3), 141–152 (1982)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Conway, J., Sloane, N., Bannai, E.: Sphere Packings, Lattices, and Groups, vol. 290. Springer, New York (1999)MATHGoogle Scholar
  3. 3.
    Danzer, L., Grünbaum, B., Klee, V.: Helly’s theorem and its relatives. In: Proceedings of Symposia in Pure Mathematics, vol. VII, pp. 101–180. American Mathematical Society, Providence, RI (1963)Google Scholar
  4. 4.
    Ericson, T., Zinoviev, V.: Codes on Euclidean spheres. North-Holland, Amsterdam (2001)MATHGoogle Scholar
  5. 5.
    Fischer, K., Gärtner, B.: The smallest enclosing ball of balls: combinatorial structure and algorithms. Int. J. Comput. Geom. Appl. 14(4–5), 341–378 (2004)MATHCrossRefGoogle Scholar
  6. 6.
    Hadwiger, H., Debrunner, H.: Combinatorial Geometry in the Plane, vol. 46. Holt, Rinehart and Winston, New York (1964)Google Scholar
  7. 7.
    Kalai, G.: Intersection patterns of convex sets. Israel J. Math. 48(2), 161–174 (1984)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Katchalski, M., Liu, A.: A problem of geometry in $\mathbb{R}^n$. Proc. Am. Math. Soc. 75(2), 284–288 (1979)MathSciNetMATHGoogle Scholar
  9. 9.
    Matoušek, J.: Lectures on Discrete Geometry, vol. 212. Springer, New York (2002)MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Institute for Information Transmission ProblemsRASMoscowRussia
  2. 2.B. N. Delone International Laboratory “Discrete and Computational Geometry”P. G. Demidov Yaroslavl State UniversityYaroslavl’Russia

Personalised recommendations