On the Optimality of Functionals over Triangulations of Delaunay Sets

TL;DR

This paper proves that for certain functionals, the Delaunay triangulation minimizes the functional density not only for finite point sets but also extends this optimality to infinite sets in the plane.

Contribution

It establishes the extension of Delaunay triangulation optimality from finite to infinite point sets for specific functionals.

Findings

Delaunay triangulation minimizes the functional for finite sets.

The minimality property extends to infinite sets.

Functional density is minimized on Delaunay triangulations for infinite sets.

Abstract

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.