Generalized Balanced Partitions of Two Sets of Points in the Plane

We consider the following problem. Let n ? 2, b ? 1 and q ? 2 be integers. Let R and B be two disjoint sets of nred points and bn blue points in the plane, respectively such that no three points of R?B lie on the same line. Let n = n1 + n2 + ... + nq be an integer-partition of n such that 1 ? ni for every 1 ? i ? q. Then we want to partition R?B into qdisjoint subsets P1 ? P2 ? ... ? Pqthat satisfy the following two conditions: (i)conv (Pi)? conv (Pj)= ? for all 1 ? i < j ? q, where conv(Pi) denotes the convex hull of Pi; and (ii) each Pi contains exactly ni red points and bni blue points for every 1 ? i ? q.We shall prove that the above partition exits in the case where (i) 2 ? n ? 8 and 1 ? ni ? n/2 for every 1 ? i ? qand (ii) n1 = n2 = ... = nq-1 = 2 and nq=1.