Discrete & Computational Geometry

, Volume 44, Issue 4, pp 805–811

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 determined by them is balanced if in each open half-plane bounded by 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.


Balanced partitions Halving triangles Generalized Lower Bound Theorem Circular sequences Allowable sequences Sliding rotations 


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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad de AlcaláAlcalá de HenaresSpain
  2. 2.Instituto de FísicaUniversidad Autónoma de San Luis PotosíSan Luis PotosíMexico

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