Intersection properties of boxes in Rd

Abstract

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).

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References

  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.

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Dedicated to Tibor Gallai on his seventieth birthday

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Danzer, L., Grünbaum, B. Intersection properties of boxes in Rd . Combinatorica 2, 237–246 (1982). https://doi.org/10.1007/BF02579232

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AMS subject classification (1980)

  • 52 A 35
  • 05 B 99