The Number of Generalized Balanced Lines

Let S be a set of r red points and b=r+2δ blue points in general position in the plane, with δ≥0. A line ℓ determined by them is balanced if in each open half-plane bounded by ℓ the difference between the number of blue points and red points is δ. We show that every set S as above has at least r balanced lines. The proof is a refinement of the ideas and techniques of Pach and Pinchasi (Discrete Comput. Geom. 25:611–628, 2001), where the result for δ=0 was proven, and introduces a new technique: sliding rotations.