On the stretch factor of Delaunay triangulations of points in convex position

Abstract

Let S be a finite set of points in the Euclidean plane. Let G be a geometric graph in the plane whose point set is S. The stretch factor of G is the maximum ratio, among all points p and q in S, of the length of the shortest path from p to q in G over the Euclidean distance | pq | . Keil and Gutwin in 1989 [11] proved that the stretch factor of the Delaunay triangulation of a set of points S in the plane is at most 2 π / ( 3 cos ( π / 6 ) ) ≈ 2.42 . Improving on this upper bound remains an intriguing open problem in computational geometry. In this paper we consider the special case when the points in S are in convex position. We prove that in this case the stretch factor of the Delaunay triangulation of S is at most ρ = 2.33 .