  Let $S$ be a finite set of points in the Euclidean plane. Let $D$ be a
Delaunay triangulation of $S$. The {\em stretch factor} (also known as {\em
dilation} or {\em spanning ratio}) of $D$ is the maximum ratio, among all
points $p$ and $q$ in $S$, of the shortest path distance from $p$ to $q$ in $D$
over the Euclidean distance $||pq||$. Proving a tight bound on the stretch
factor of the Delaunay triangulation has been a long standing open problem in
computational geometry.
  In this paper we prove that the stretch factor of the Delaunay triangulation
of a set of points in the plane is less than $\rho = 1.998$, improving the
previous best upper bound of 2.42 by Keil and Gutwin (1989). Our bound 1.998 is
better than the current upper bound of 2.33 for the special case when the point
set is in convex position by Cui, Kanj and Xia (2009). This upper bound breaks
the barrier 2, which is significant because previously no family of plane
graphs was known to have a stretch factor guaranteed to be less than 2 on any
set of points.
