 
Summary: Piercing Convex Sets
Noga Alon
and Daniel J. Kleitman
Abstract
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 extends Helly's Theorem
and settles an old problem of Hadwiger and Debrunner.
Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel
Aviv, Israel and Bellcore, Morristown, NJ 07960, USA. Research supported in part by a United States Israel BSF
Grant and by a Bergmann Memorial Grant
Department of Mathematics, MIT, Cambridge, Ma, 02139. Research supported in part by a United States Israel
BSF Grant and by a Bergmann Memorial Grant. 1980 Mathematics Subject Classification (1985 Revision). Primary
52A35
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