On the Dilation of Delaunay Triangulations of Points in Convex Position

Let S be a finite set of points in the Euclidean plane, and let E be the complete graph whose point-set is S. Chew, in 1986, proved a lower bound of π/2 on the stretch factor of the Delaunay triangulation of S (with respect to E), and conjectured that this bound is tight. Dobkin, Friedman, and Supowit, in 1987, showed that the stretch factor of the Delaunay triangulation of S is at most π( √ 5 + 1)/2 ≈ 5.084. This upper bound was later improved by Keil and Gutwin in 1989 to 2π/(3 cos (π/6)) ≈ 2.42. Since then (1989), Keil and Gutwin’s bound has stood as the best upper bound on the stretch factor of Delaunay triangulations, even though Chew’s conjecture is now widely believed to be true. Whether the stretch factor of Delaunay triangulations is π/2 or not remains a challenging and intriguing problem in computational geometry. Bose, in an open-problem session at CCCG 2007, suggested looking at the special case when the points in S are in convex position. In this paper we show that the stretch factor of the Delaunay triangulation of a point-set in convex position is at most ρ = 2.33.