Description
Assume two finite families AA and BB of convex sets in R^3 have the property that A intersection B is never empty for A in AA and B in BB. Is there a constant such that there is a line intersecting a positive fraction of the sets from one of the two families? 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 AA lie in parallel planes and all sets in BB lie in parallel planes; in fact, all sets from one of the two families have a line transversal. For the
proof we will use LP duality in an interesting way.
Joint work with Imre Bárány, Travis Dillon, Dániel Varga.