Piercing Convex Sets

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 1 

• Type: Conference
• Year: 1992
• Where: COMPGEOM
• Authors: Noga Alon, Daniel J. Kleitman
• Keywords: Bergmann Memorial Grant, BSF Grant, COMPGEOM 1992, Discrete Geometry, Tel Aviv