Improved stretch factor of Delaunay triangulations of points in convex position

Let S be a set of n points in the plane, and let DT(S) be the planar graph of the Delaunay triangulation of S. For a pair of points $$a, b \in S$$ a , b ∈ S , denote by |ab| the Euclidean distance between a and b. Denote by DT(a, b) the shortest path in DT(S) between a and b, and let |DT(a, b)| be the total length of DT(a, b). Dobkin et al. were the first to show that DT(S) can be used to approximate the complete graph of S in the sense that the stretch factor $$\frac{|DT(a, b)|}{|a b|}$$ | D T ( a , b ) | | a b | is upper bounded by $$((1 + \sqrt{5})/2) \pi \approx 5.08$$ ( ( 1 + 5 ) / 2 ) π ≈ 5.08 . Recently, Xia improved this factor to 1.998. Amani et al. have also shown that if the points of S are in convex position (i.e., they form the vertices of a convex polygon), then a planar graph with these vertices can be constructed such that its stretch factor is 1.88. In this paper, we prove that if the points of S are in convex position, then the stretch factor of DT(S) is less than 1.84, improving upon the previously known factors of Delaunay triangulations or planar graphs in the convex case.