On the optimality of functionals over triangulations of Delaunay sets

In this short paper we consider the functional density on sets of uniformly bounded triangulations with fixed sets of vertices. We prove that if a functional attains its minimum on the Delaunay triangulation for every finite set in the plane, then for infinite sets the density of this functional attains its minimum also on the Delaunay triangulations. A Delaunay set in E is a set of points X for which there are positive numbers r and R such that every open d-ball of radius r contains at most one point and every closed d-ball of radius R contains at least one point of X. In this paper we consider Delaunay sets in general position, that is, no d + 2 points in X lie on a common (d− 1)-sphere. By a triangulation of X we mean a simplicial complex whose vertex set is X. For finite sets the simplices decompose the convex hull of the set, while for Delaunay sets X the simplices decompose E. We say that a triangulation T is uniformly bounded if there exists a positive number q = q(T ) that is greater than or equal to the circumradii of all d-simplices in the triangulation: R(S) 6 q for all d-simplices S of T . We denote the family of all uniformly bounded triangulations of X by Θ(X). Delaunay sets were introduced byBorisDelaunay (1924), who called them (r, R)-systems. He proved that for any Delaunay set X there exists a unique Delaunay tesselation DT (X) (see, for instance, [1]). If X is in general position, then DT (X) is a triangulation of X in the sense defined above. Since the circumradius of any simplex is at most R, the Delaunay triangulation is uniformly bounded with q = R, that is, DT (X) ∈ Θ(X). We note that every Delaunay set also has triangulations that are not uniformly bounded, and it is not difficult to construct them. We want to remind the reader of a related open problem about Delaunay sets: is it true that for every planar Delaunay set X and every positive number C there exists a triangle ∆ that contains none of the points in X and has area greater than C? While we heard of this question from Michael Boshernitzan, it is sometimes referred to as Danzer’s problem. Let F be a functional defined on d-simplices S. (For instance, F (S) may be the sum of squares of edge lengths multiplied by the volume of S.) We only consider functionals that are continuous with respect to the parameters describing the simplices, for example, the lengths of their edges. Let X be a finite set in E and T any triangulation of X. Then F can be defined on T as F (T ) = ∑ S∈T F (S). It is clear that this definition cannot be used for infinite sets. We therefore define the (lower) density of F for a uniformly bounded triangulation T of a Delaunay set X as