Delaunay simplices in diagonally distorted lattices

TL;DR

This paper investigates the properties of Delaunay triangulations of distorted integer lattices, revealing that optimal quality measures occur at a specific distortion where the lattice aligns with the permutahedral lattice.

Contribution

It demonstrates that the Delaunay protection and other quality measures are maximized at a particular distortion, linking the optimal lattice to the permutahedral lattice.

Findings

Maximum quality measures at a specific distortion parameter.

The optimal lattice is isometric to the permutahedral lattice.

Delaunay protection properties are characterized for distorted lattices.

Abstract

Delaunay protection is a measure of how far a Delaunay triangulation is from being degenerate. In this short paper we study the protection properties and other quality measures of the Delaunay triangulations of a family of lattices that is obtained by distorting the integer grid in $\mathbb{R}^d$. We show that the quality measures of this family are maximized for a certain distortion parameter, and that for this parameter, the lattice is isometric to the permutahedral lattice, which is a well-known object in discrete geometry.