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Piercing convex sets and the Hadwiger Debrunner (p, q)-problem Department of Mathematics

Summary: Piercing convex sets and the Hadwiger Debrunner (p, q)-problem
Noga Alon
Department of Mathematics
Raymond and Beverly Sackler Faculty of Exact Sciences
Tel Aviv University, Tel Aviv, Israel
Daniel J. Kleitman
Department of Mathematics
MIT, Cambridge, Ma, 02139
A family of sets has the (p, q) property if among any p members of the family some q have
a nonempty intersection. It is shown that for every p q d + 1 there is a c = c(p, q, d) <
such that for every family F of compact, convex sets in Rd
which has the (p, q) property there
is a set of at most c points in Rd
that intersects each member of F. This settles an old problem
of Hadwiger and Debrunner.

Research supported in part by a United States Israel BSF Grant and by a Bergmann Memorial Grant


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University


Collections: Mathematics