  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 dilation in T of any pair p, p \in 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 {\pi}/2 \approx 1.5708. In this paper, we show
how to construct point sets in convex position with dilation > 1.5810 and in
general position with dilation > 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 dilation for that distribution.
