Almost all Delaunay triangulations have stretch factor greater than π/2

Consider the Delaunay triangulation T of a set P of points in the plane as a Euclidean graph, in which the weight of every edge is its length. It has long been conjectured that the stretch factor in T of any pair p,p′∈P, which is the ratio of the length of the shortest path from p to p′ in T over the Euclidean distance ‖pp′‖, can be at most π/2≈1.5708. In this paper, we show how to construct point sets in convex position with stretch factor >1.5810 and in general position with stretch factor >1.5846. Furthermore, we show that a sufficiently large set of points drawn independently from any distribution will in the limit approach the worst-case stretch factor for that distribution.