Points surrounding the origin

Abstract

Suppose d > 2, n > d+1, and we have a set P of n points in d-dimensional Euclidean space. Then P contains a subset Q of d points such that for any pP, the convex hull of Q∪{p} does not contain the origin in its interior.

We also show that for non-empty, finite point sets A 1, ..., A d+1 in ℝd, if the origin is contained in the convex hull of A i A j for all 1≤i<jd+1, then there is a simplex S containing the origin such that |SA i |=1 for every 1≤id+1. This is a generalization of Bárány’s colored Carathéodory theorem, and in a dual version, it gives a spherical version of Lovász’ colored Helly theorem.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    J. L. Arocha, J. Bracho and V. Neumann-Lara: On the minimum size of tight hypergraphs, J. Graph Theory 16 (1992), 319–326.

    MATH  Article  MathSciNet  Google Scholar 

  2. [2]

    I. Bárány: A generalization of Carathéodory’s theorem, Discrete Math. 40 (1982), 141–152.

    MATH  Article  MathSciNet  Google Scholar 

  3. [3]

    D. W. Barnette: The minimal number of vertices of a simple polytope, Israel J. Math. 10 (1971), 121–125.

    MATH  Article  MathSciNet  Google Scholar 

  4. [4]

    T. K. Dey: Improved bounds on planar k-sets and related problems, Discrete Comput. Geom. 19 (1998), 373–382.

    MATH  Article  MathSciNet  Google Scholar 

  5. [5]

    O. Devillers, F. Hurtado, Gy. Károlyi and C. Seara: Chromatic variants of the Erdős-Szekeres theorem on points in convex position, Comput. Geom. Th. & Appls. 26 (2003), 193–208.

    MATH  Article  Google Scholar 

  6. [6]

    J. Eckhoff: Helly, Radon, and Carathéodory type theorems; in Handbook of Convex Geometry (P. M. Gruber, J. M. Wills, eds.), Vol. A (1993), 389–448.

  7. [7]

    J. Matoušek: Lectures on discrete geometry, Graduate Texts in Mathematics 212, Springer-Verlag, New York, 2002.

    Google Scholar 

  8. [8]

    L. Lovász: On the number of halving lines, Ann. Univ. Sci. Budapest Eötvös Sect. Math. 14 (1971), 107–108.

    Google Scholar 

  9. [9]

    M. Sharir, S. Smorodinsky and G. Tardos: An improved bound for k-sets in three dimensions, Discrete Comput. Geom. 26(2) (2001), 195–204.

    MATH  MathSciNet  Google Scholar 

  10. [10]

    R. Strausz, personal communication, February, 2007.

  11. [11]

    G. Ziegler: Lectures on Polytopes, Graduate Texts in Mathematics 152, Springer-Verlag, New York, 1995.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to János Pach.

Additional information

Dedicated to Imre Bárány, Gábor Fejes Tóth, László Lovász, and Endre Makai on the occasion of their sixtieth birthdays.

Supported by the Norwegian research council project number: 166618, and BK 21 Project, KAIST. Part of the research was conducted while visiting the Courant Institute of Mathematical Sciences.

Supported by NSF Grant CCF-05-14079, and by grants from NSA, PSC-CUNY, the Hungarian Research Foundation OTKA, and BSF.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Holmsen, A.F., Pach, J. & Tverberg, H. Points surrounding the origin. Combinatorica 28, 633–644 (2008). https://doi.org/10.1007/s00493-008-2427-5

Download citation

Mathematics Subject Classification (2000)

  • 52A35
  • 52C35
  • 52A25