Helly-type theorems for bodies in the plane with common supports (Q1208683)

The author is concerned with the existence of common supporting lines for disjoint families of bodies (compact convex sets) in the plane. He shows that if every 5-element subfamily of a disjoint family has a common support, then the whole family has also a common support. Furthermore, he proves that if every 4-element (3-element) subfamily of a disjoint family of seven or more (237 or more) bodies has a common support, then the whole family has also a common support.