General Balanced Subdivision of Two Sets of Points in the Plane
نویسندگان
چکیده
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(g1 + g2) + (a + 1)g3, |B| = bg1 + (b + 1)(g2 + g3), where g = g1 + g2 + g3, g1 ≥ 0, g2 ≥ 0, g3 ≥ 0 and g1+g2+g3 ≥ 1. We show that the plane can be subdivided into g disjoint convex polygons X1 ∪ · · · ∪Xg1 ∪ Y1 ∪ · · · ∪ Yg2 ∪ Z1 ∪ · · · ∪ 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.
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