On the Stretch Factor of Convex Delaunay Graphs

Let C be a compact and convex set in the plane that contains the origin in its interior, and let S be a finite set of points in the plane. The Delaunay graph DG_C(S) of S is defined to be the dual of the Voronoi diagram of S with respect to the convex distance function defined by C. We prove that DG_C(S) is a t-spanner for S, for some constant t that depends only on the shape of the set C. Thus, for any two points p and q in S, the graph DG_C(S) contains a path between p and q whose Euclidean length is at most t times the Euclidean distance between p and q.