On the stabbing number of a random Delaunay triangulation

We consider a Delaunay triangulation defined on n points distributed independently and uniformly on a planar compact convex set of positive volume. Let the stabbing number be the maximal number of intersections between a line and edges of the triangulation. We show that the stabbing number Sn is Θ( √ n) in the mean, and provide tail bounds for P{Sn ≥ t √ n}. Applications to planar point location, nearest neighbor searching, range queries, planar separator determination, approximate shortest paths, and the diameter of the Delaunay triangulation are discussed.