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 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.
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Orden, D., Ramos, P., Salazar, G.: Balanced lines in two-coloured point sets. arXiv:0905.3380v1 [math.CO]
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This work was started at the 6th Iberian Workshop on Computational Geometry, in Aveiro, and was concluded while Gelasio Salazar was visiting Department of Mathematics of Alcalá University under the program Giner de los Ríos.
D. Orden and P. Ramos are partially supported by grants MTM2008-04699-C03-02 and HP2008-0060.
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Orden, D., Ramos, P. & Salazar, G. The Number of Generalized Balanced Lines. Discrete Comput Geom 44, 805–811 (2010). https://doi.org/10.1007/s00454-010-9253-4
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DOI: https://doi.org/10.1007/s00454-010-9253-4