On the coarseness of bicolored point sets

Let R be a set of red points and B a set of blue points on the plane. In this paper we introduce a new concept, which we call coarseness, for measuring how blended   the elements of S=R∪BS=R∪B are. For X⊆SX⊆S, let ∇(X)=||X∩R|−|X∩B||∇(X)=||X∩R|−|X∩B|| be the bichromatic discrepancy of X  . We say that a partition Π={S1,S2,…,Sk}Π={S1,S2,…,Sk} of S is convex if the convex hulls of its members are pairwise disjoint. The discrepancy of a convex partition Π of S   is the minimum ∇(Si)∇(Si) over the elements of Π. The coarseness of S is the discrepancy of the convex partition of S with maximum discrepancy. We study the coarseness of bicolored point sets, and relate it to well blended point sets. In particular, we show combinatorial results on the coarseness of general configurations and give efficient algorithms for computing the coarseness of two specific cases, namely when the set of points is in convex position and when the measure is restricted to convex partitions with two elements.