Piercing intersecting convex sets
Imre Bárány, Travis Dillon, Dömötör Pálvölgyi, Dániel Varga · Linear Algebra and its Applications · 2025
Assume two finite families A and B of convex sets in R 3 have the property that A ∩ B ≠ ∅ for every A ∈ A and B ∈ B . Is there a constant γ > 0 (independent of A and B ) such that there is a line intersecting γ | A | sets in A or γ | B | sets in B ? This is an intriguing Helly-type question from a paper by Martínez, Roldan and Rubin. We confirm this in the special case when all sets in A lie in parallel planes and all sets in B lie in parallel planes; in fact, one of the two families has a transversal by a single line.