Discrete & Computational Geometry

, Volume 44, Issue 4, pp 805–811

The Number of Generalized Balanced Lines

Article

Abstract

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.

Keywords

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

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References

  1. 1.
    Erdős, P., Lovász, L., Simmons, A., Strauss, E.G.: Dissection graphs on planar point sets. In: A Survey of Combinatorial Theory, pp. 139–149. North Holland, Amsterdam (1973) Google Scholar
  2. 2.
    Orden, D., Ramos, P., Salazar, G.: Balanced lines in two-coloured point sets. arXiv:0905.3380v1 [math.CO]
  3. 3.
    Pach, J., Pinchasi, R.: On the number of balanced lines. Discrete Comput. Geom. 25, 611–628 (2001) MATHMathSciNetGoogle Scholar
  4. 4.
    Sharir, M., Welzl, E.: Balanced lines, halving triangles, and the generalized lower bound theorem. In: Aronov, B., Basu, S., Pach, J., Sharir, M. (eds.) Discrete and Computational Geometry—The Goodman–Pollack Festschrift, pp. 789–798. Springer, Heidelberg (2003) Google Scholar

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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