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Piercing convex sets
1992
Bulletin of the American Mathematical Society
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 R d which has the (p, q) property there is a set of at most c points in R d that intersects each member of F. This extends Helly's Theorem and settles an old problem of Hadwiger and Debrunner.
doi:10.1090/s0273-0979-1992-00304-x
fatcat:gtfryswzgvbc5gjtuduzarg7xi