Transversals to Colorful Intersecting Convex Sets

Let K be a compact convex set in \(\mathbb {R}^{2}\) and let \(\mathcal {F}_{1}, \mathcal {F}_{2}, \mathcal {F}_{3}\) be finite families of translates of K such that \(A \cap B \ne \emptyset \) for every \(A \in \mathcal {F}_{i}\) and \(B \in \mathcal {F}_{j}\) with \(i \ne j\). A conjecture by Dol’nikov is that, under these conditions, there is always some \(j \in \{ 1,2,3 \}\) such that \(\mathcal {F}_{j}\) 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 \(\mathcal {F}_{1}, \dots , \mathcal {F}_{d}\) are finite families of convex sets in \(\mathbb {R}^{d}\) such that for every choice of sets \(C_{1} \in \mathcal {F}_{1}, \dots , C_{d} \in \mathcal {F}_{d}\) the intersection \(\bigcap _{i=1}^{d} {C_{i}}\) is non-empty, then either there exists \(j \in \{ 1,2, \dots , n \}\) such that \(\mathcal {F}_j\) can be pierced by few points or \(\bigcup _{i=1}^{n} \mathcal {F}_{i}\) can be crossed by few lines. We give optimal values for the number of piercing points and crossing lines needed when \(d=2\) and also consider the problem restricted to special families of convex sets.