Cutting families of convex sets

Abstract

A family of convex sets in the plane admits a common transversal if there is a straight line which intersects (cuts) each member of the family. It is shown that there is a positive integer k such that for any compact convex set C in the plane and for any finite family A \mathcal {A} of pairwise disjoint translates of C: If each 3-membered subfamily of A \mathcal {A} admits a common transversal then there is a subfamily B \mathcal {B} of A \mathcal {A} such that B \mathcal {B} admits a common transversal and | A ∖ B | ⩽ k |\mathcal {A}\backslash \mathcal {B}| \leqslant k .