Intersection properties of boxes in Rd


A family of sets is calledn-pierceable if there exists a set ofn points such that each member of the family contains at least one of the points. Helly’s theorem on intersections of convex sets concerns 1-pierceable families. Here the following Helly-type problem is investigated: Ifd andn are positive integers, what is the leasth =h(d, n) such that a family of boxes (with parallel edges) ind-space isn-pierceable if each of itsh-membered subfamilies isn-pierceable? The somewhat unexpected solution is: (i)h(d, 2) equals3d for oddd and 3d−1 for evend; (ii)h(2, 3)=16; and (iii)h(d, n) is infinite for all (d, n) withd≧2 andn≧3 except for (d, n)=(2, 3).

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


  1. [1]

    L. Danzer, B. Grünbaum andV. Klee, Helly’s theorem and its relatives,Proc. Symposia in Pure Math., Vol. VII (Convexity) (1963), 101–180.

  2. [2]

    M. M. Day, Polygons circumscribed about closed convex curves,Trans. Amer. Math. Soc.,62 (1947), 315–319.

    MATH  Article  MathSciNet  Google Scholar 

  3. [3]

    B. Grünbaum, Common secants for families of polyhedra,Arch. Math.,15 (1964), 76–80.

    MATH  Article  MathSciNet  Google Scholar 

  4. [4]

    H. Hadwiger andH. Debrunner,Kombinatorische Geometrie in der Ebene, Monographies de l’Enseignement Mathematique, No.2, Geneva, 1960.

  5. [5]

    H. Hadwiger, H. Debrunner andV. Klee,Combinatorial geometry in the plane, New York, 1964.

  6. [6]

    O. Ore,The four-color problem, New York 1967.

Download references

Author information



Additional information

Dedicated to Tibor Gallai on his seventieth birthday

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Danzer, L., Grünbaum, B. Intersection properties of boxes in Rd . Combinatorica 2, 237–246 (1982).

Download citation

AMS subject classification (1980)

  • 52 A 35
  • 05 B 99