On a conjecture of Karasev

Karasev conjectured that for any set of 3k lines in general position in the plane, which is partitioned into 3 color classes of equal size k, the set can be partitioned into k colorful 3-subsets such that all the triangles formed by the subsets have a point in common. Although the general conjecture is false, we show that Karasev's conjecture is true for lines in convex position. We also discuss possible generalizations of this result.