An upper bound on the coarseness of point sets

Let S be a two colored (red and blue) set of n points in the plane. A subset I of S is an island if there is a convex set C such that I = C∩S. The discrepancy of an island is the absolute value of the number of red minus the number of blue points it contains. A convex partition of S is a partition of S into islands, with disjoint convex hulls. The discrepancy of a convex partition is the discrepancy of its island of minimum discrepancy. The coarseness of S is the discrepancy of the convex partition of S with maximum discrepancy. This concept of was recently defined by Bereg et al. [CGTA 2013]. In this paper we prove an almost tight upper bound of O(n1/4 √ logn) on this parameter. This is obtained by relating the coarseness of a point set to the discrepancy of a certain class of islands.