A colorful theorem on transversal lines to plane convex sets

In 1982 (see [1]) Imre Bárány observed that some of the classical theorems in convexity admit interesting and mysterious generalizations which he called “colorful theorems”. For example, the Colorful Helly Theorem says that if a family (repetitions of the same sets are allowed) of compact convex sets in Rk is colored (properly) with k+1 colors and it has the property that any choice of k+1 differently colored sets have non void intersection, then there exists a color such that all the convex sets of that color have non void intersection. In the case that any convex set of the family is repeated k+1 times and they are colored with the k+1 colors, we obtain Helly’s classical theorem. So, the colorful version is indeed a generalization. Bárány attributed this theorem to László Lovász (see [2] for his elegant proof) and he proved a colorful version of Carathéodory’s Theorem. Since then, several papers have been published on this matter. See, for example [3], [7] and [5]. However, in the study of colorful theorems there was a missing piece. Does Hadwiger’s Theorem on transversal lines to plane convex sets admits