Transversals to Colorful Intersecting Convex Sets

Let K be a compact convex set in R² R 2 and let F₁, F₂, F₃ F 1 , F 2 , F 3 be finite families of translates of K such that A ∩ B ≠ ∅ A ∩ B ≠ ∅ for every A ∈ Fᵢ A ∈ F i and B ∈ Fⱼ B ∈ F j with i ≠ j i ≠ j . A conjecture by Dol’nikov is that, under these conditions, there is always some j ∈ \ 1,2,3 \ j ∈ { 1 , 2 , 3 } such that Fⱼ 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 F₁, , Fd F 1 , ⋯ , F d are finite families of convex sets in Rᵈ R d such that for every choice of sets C₁ ∈ F₁, , Cd ∈ Fd C 1 ∈ F 1 , ⋯ , C d ∈ F d the intersection ᵢ₌₁ᵈ Cᵢ ⋂ i = 1 d C i is non-empty, then either there exists j ∈ \ 1,2, , n \ j ∈ { 1 , 2 , ⋯ , n } such that Fⱼ F j can be pierced by few points or ᵢ₌₁ⁿ Fᵢ ⋃ i = 1 n F i can be crossed by few lines. We give optimal values for the number of piercing points and crossing lines needed when $$d=2$$ d = 2 and also consider the problem restricted to special families of convex sets.