Generalized Balanced Partitions of Two Sets of Points in the Plane

Abstract

We consider the following problem. Let n ≥ 2, b ≥ 1 and q ≥ 2 be integers. Let R and B be two disjoint sets of n red points and bn blue points in the plane, respectively, such that no three points of R∪B lie on the same line. Let n = n ₁ + n ₂ + ... + n _q be an integer-partition of n such that 1 ≤ n _i for every 1 ≤ i ≤ q. Then we want to partition R ∪ B into q disjoint subsets P ₁ ∪ P ₂ ∪ ... ∪ P _q that satisfy the following two conditions: (i) conv (P _i ) ∩ conv (P _j ) = ⊘ for all 1 ≤ i < j ≤ q, where conv(P _i ) denotes the convex hull of P _i ; and (ii) each P _i contains exactly n _i red points and bn _i blue points for every 1 ≤ i ≤ q.

We shall prove that the above partition exists in the case where (i) 2 ≤ n ≤ 8 and 1 ≤ n _i ≤ n/2 for every 1 ≤ i ≤ q, and (ii) n ₁ = n ₂ = ... = n _q-1 = 2 and n _q =1.