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 ∈ 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 | D T ( a , b ) | | a b | is upper bounded by ( ( 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.