Transversals to Colorful Intersecting Convex Sets

Abstract Let K be a compact convex set in R^2 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 F₈ A ∈ F i and B F₉ 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 F₉ 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, , F₃ F 1, ⋯, F d are finite families of convex sets in R^d R d such that for every choice of sets C₁ F₁, , C₃ F₃ C 1 ∈ F 1, ⋯, C d ∈ F d the intersection ₈=₁^d C₈ ⋂ 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 ₈=₁^n F₈ ⋃ 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.