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).
Similar content being viewed by others
References
L. Danzer, B. Grünbaum andV. Klee, Helly’s theorem and its relatives,Proc. Symposia in Pure Math., Vol. VII (Convexity) (1963), 101–180.
M. M. Day, Polygons circumscribed about closed convex curves,Trans. Amer. Math. Soc.,62 (1947), 315–319.
B. Grünbaum, Common secants for families of polyhedra,Arch. Math.,15 (1964), 76–80.
H. Hadwiger andH. Debrunner,Kombinatorische Geometrie in der Ebene, Monographies de l’Enseignement Mathematique, No.2, Geneva, 1960.
H. Hadwiger, H. Debrunner andV. Klee,Combinatorial geometry in the plane, New York, 1964.
O. Ore,The four-color problem, New York 1967.
Author information
Authors and Affiliations
Additional information
Dedicated to Tibor Gallai on his seventieth birthday