On the union complexity of diametral disks

Let S be a set of n points in the plane, and let D be an arbitrary set of disks, each having a pair of points of S as a diameter. We show that the combinatorial complexity of the union of the disks in D is O(n^3/2), and provide a lower bound construction with (n^4/3) complexity. If the points of S are in convex position, the upper bound drops to O(n log n). The problem. Let S be a set of n points in the plane, and let E be a collection of unordered pairs of points of S. For each pair {a, b} ∈ E let Dab denote the disk with diameter ab; we refer to D ^ab as the diametral disk determined by {a, b}. Let D ^ab denote the resulting collection of disks, and let U denote their union. Let the complexity of U be measured by the number of its vertices, namely intersection points of two disk boundaries that lie on the boundary ∂U of U. In this paper we establish the following upper and lower bounds on the complexity of U.