Skip to main content
SpringerLink
Log in

Intersection patterns of convex sets

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

LetK 1,…Kn be convex sets inR d. For 0≦i<n denote byf ithe number of subsetsS of {1,2,…,n} of cardinalityi+1 that satisfy ∩{K i∶i∈S}≠Ø. We prove:Theorem.If f d+r=0 for somer r>=0, then {fx161-1} This inequality was conjectured by Katchalski and Perles. Equality holds, e.g., ifK 1=…=Kr=Rd andK r+1,…,Kn aren−r hyperplanes in general position inR d. The proof uses multilinear techniques (exterior algebra). Applications to convexity and to extremal set theory are given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. H. Abbot and M. Katchalski,A Turán type problem for interval graphs, Discrete Math.25, (1979), 85–88.

    Article  MathSciNet  Google Scholar 

  2. N. Bourbaki,Algebra I Addison-Wesley, 1974.

  3. J. Eckhoff,Über kombinatorisch-geometrische Eigenschaften von Komplexen und Familien konvexer Mengen, J. Reine Angew. Math.313 (1980), 171–188.

    MATH  MathSciNet  Google Scholar 

  4. O. Gabber and G. Kalai,Geometric analogs for extremal set theoretic results, in preparation.

  5. G. Kalai,Characterization of f-vectors of families of convex sets in R d. Part I: Necessity of Eckhoff's conditions, Isr. J. Math.48 (1984), 175–195.

    MATH  MathSciNet  Google Scholar 

  6. G. Kalai,Hyperconnectivity of graphs, to appear.

  7. M. Katchalski and A. Liu,A problem of geometry in R d, Proc. Am. Math. Soc.75 (1979), 284–288.

    Article  MATH  MathSciNet  Google Scholar 

  8. M. Katchalski,Boxes in R n—a ‘fractional’ theorem, Can. J. Math.32 (1980), 831–838.

    MATH  MathSciNet  Google Scholar 

  9. J. Leray,Sur la forme des espaces topologiques et sur le points fixes des représentations, J. Math. Pures Appl.24 (1945), 95–167.

    MATH  MathSciNet  Google Scholar 

  10. N. Linial and B. R. Rothchild,Incidence matrices of subsets—a rank formula, SIAM J. Alg. Disc. Math.2 (1981), 333–340.

    Article  MATH  Google Scholar 

  11. L. Lovász,Flats in matroids and geometric graphs, inCombinational Surveys (P. J. Cameron, ed.), Academic Press, 1977.

  12. M. Marcus,Finite Dimensional Multilinear Algebra, Part II, Ch. 4, M. Dekker Inc., New York, 1975.

    MATH  Google Scholar 

  13. P. McMullen and G. C. ShephardConvex Polytopes and the Upper Bound Conjecture, London Math. Soc. Lecture Note Series, Vol. 3, Cambridge Univ. Press, London/New York, 1971.

    MATH  Google Scholar 

  14. G. Wegner,d-collapsing and nerves of families of convex sets, Arch. Math.26 (1975), 317–327.

    Article  MATH  MathSciNet  Google Scholar 

  15. P. Frankl,An extremal problem for two families of sets, Eur. J. Comb.3 (1982), 125–127.

    MATH  MathSciNet  Google Scholar 

  16. N. Alon and G. Kalai,A simple proof of the upper bound theorem, Eur. J. Comb., to appear.

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics The Hebrew University of Jerusalem, Jerusalem, Israel

    Gil Kalai

Authors
  1. Gil Kalai
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kalai, G. Intersection patterns of convex sets. Israel J. Math. 48, 161–174 (1984). https://doi.org/10.1007/BF02761162

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02761162

Keywords

Search

Navigation