Cutting Families of Convex Sets
Meir Katchalski, Ted Lewis · Proceedings of the American Mathematical Society · 1980
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 $\mathcal {A}$ of pairwise disjoint translates of C: If each 3-membered subfamily of $\mathcal {A}$ admits a common transversal then there is a subfamily $\mathcal {B}$ of $\mathcal {A}$ such that $\mathcal {B}$ admits a common transversal and $|\mathcal {A}\backslash \mathcal {B}| \leqslant k$.