k -Bisectors of Finite Planar Sets

Let V be a finite set of points in the plane, not all on one line, and let l be a line that contains at least 2 points of V. We say that l is a k-bisector of V if there are at least k points of V on each one of the two open half-planes bounded by l. For k≥ 6 we construct planar sets of 2 k+ 4 points having no k-bisector (this might be best possible). Furthermore, we show that if | V| > 3 k, then in every triangulation of convV with vertex set V there is an edge whose loading line is a k-bisector of V. This is best possible for all k.