Type: Article

Transversals to colorful intersecting convex sets

Gomez-Navarro, Cuauhtemoc; Roldán-Pensado, Edgardo;

Abstract:

Let K be a compact convex set in and let be finite families of translates of K such that for every and with . A conjecture by Dol’nikov is that, under these conditions, there is always some such that can be pierced by 3 points. In this paper we prove a stronger version of this conjecture when K is a body of constant width or when it is close in Banach-Mazur distance to a disk. We also show that the conjecture is true with 8 piercing points instead of 3. Along the way we prove more general statements both in the plane and in higher dimensions. A related result was given by Martínez-Sandoval, Roldán-Pensado and Rubin. They showed that if are finite families of convex sets in such that for every choice of sets the intersection is non-empty, then either there exists such that can be pierced by few points or can be crossed by few lines. We give optimal values for the number of piercing points and crossing lines needed when and also consider the problem restricted to special families of convex sets.