The Voronoi Functional is Maximized by the Delaunay Triangulation in the Plane

TL;DR

This paper introduces the Voronoi functional for triangulations in the Euclidean plane and proves that the Delaunay triangulation maximizes this functional among all geometric triangulations, highlighting a unique optimality property.

Contribution

It establishes a new optimality property of Delaunay triangulations via the Voronoi functional in the Euclidean plane.

Findings

Delaunay triangulation maximizes the Voronoi functional among geometric triangulations.

The result does not extend to topological triangulations or higher dimensions.

The Voronoi functional provides a new perspective on triangulation optimality.

Abstract

We introduce the Voronoi functional of a triangulation of a finite set of points in the Euclidean plane and prove that among all geometric triangulations of the point set, the Delaunay triangulation maximizes the functional. This result neither extends to topological triangulations in the plane nor to geometric triangulations in three and higher dimensions.