Touching perfect matchings and halving lines

Let V be a set of 2m (1 ≤ m < ∞) points in the plane. Two segments I, J with endpoints in V cross if relint I ∩ relint J is a singleton. A (perfect) cross-matching M on V is a set of m segments with endpoints in V such that every two segments in M cross. A halving line of V is a line l spanned by two points of V such that each one of the two open half planes bounded by l contains fewer than m points of V . Pach and Solymosi proved that if V is in general position, then V admits a perfect cross-matching iff V has exactly m halving lines. The aim of this note is to extend this result to the general case (where V is unrestricted).