Balanced partitions of two sets of points in the plane
Computational Geometry: Theory and Applications
2-Dimension Ham Sandwich Theorem for Partitioning into Three Convex Pieces
JCDCG '98 Revised Papers from the Japanese Conference on Discrete and Computational Geometry
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Let R and B be two disjoint sets of red points and blue points, respectively, in the plane such that no three points of R ∪ B are co-linear. Suppose ag ≤ |R| ≤ (a+1)g, bg ≤ |B| ≤ (b+1)g. Then without loss of generality, we can express |R| = a(g. + g.) + (a + 1)g., |B| = bg. + (b + 1)(g. + g.), where g = g. + g. + g., g. ≥ 0, g. ≥ 0, g. ≥ 0 and g. + g. + g. ≥ 1. We show that the plane can be subdivided into g disjoint convex polygons X.∪...∪Xg1 ∪Y.∪...∪Yg2 ∪Z.∪...∪Zg3 such that every Xi contains a red points and b blue points, every Yi contains a red points and b+1 blue points and every Zi contains a+1 red points and b + 1 blue points.