Higher signature Delaunay decompositions

TL;DR

This paper generalizes Delaunay decompositions by replacing Euclidean balls with other regions, enabling applications in semi-Riemannian geometries, and explores their properties and optimality in higher dimensions.

Contribution

It introduces a new class of Delaunay decompositions adaptable to semi-Riemannian geometries, proving their existence, uniqueness, and analyzing their properties and optimality.

Findings

Existence and uniqueness of higher signature Delaunay decompositions.

Extension of Euclidean Delaunay properties to semi-Riemannian settings.

Description of intersection angle properties in the generalized setting.

Abstract

A Delaunay decomposition is a cell decomposition in R^d for which each cell is inscribed in a Euclidean ball which is empty of all other vertices. This article introduces a generalization of the Delaunay decomposition in which the Euclidean balls in the empty ball condition are replaced by other families of regions bounded by certain quadratic hypersurfaces. This generalized notion is adaptable to geometric contexts in which the natural space from which the point set is sampled is not Euclidean, but rather some other flat semi-Riemannian geometry, possibly with degenerate directions. We prove the existence and uniqueness of the decomposition and discuss some of its basic properties. In the case of dimension d = 2, we study the extent to which some of the well-known optimality properties of the Euclidean Delaunay triangulation generalize to the higher signature setting. In particular, we describe a higher signature generalization of a well-known description of Delaunay decompositions in terms of the intersection angles between the circumscribed circles.