The density of iterated crossing points and a gap result for triangulations of finite point sets

Abstract

Consider a plane graph G, drawn with straight lines. For every pair a,b of vertices of G, we compare the shortest-path distance between a and b in G (with Euclidean edge lengths) to their actual distance in the plane. The worst-case ratio of these two values, for all pairs of points, is called the dilation of G. All finite plane graphs of dilation 1 have been classified. They are closely related to the following iterative procedure. For a given point set P ⊆ R2, we connect every pair of points in P by a line segment and then add to P all those points where two such line segments cross. Repeating this process infinitely often, yields a limit point set P∞⊇P. This limit set P∞ is finite if and only if P is contained in the vertex set of a triangulation of dilation 1.The main result of this paper is the following gap theorem: For any finite point set P in the plane for which P∞ is infinite, there exists a threshold λ > 1 such that P is not contained in the vertex set of any finite plane graph of dilation at most λ. As a first ingredient to our proof, we show that such an infinite P∞ must lie dense in a certain region of the plane. In the second, more difficult part, we then construct a concrete point set P0 such that any planar graph that contains this set amongst its vertices must have a dilation larger than 1.0000047.