Dimension and structure of higher-order Voronoi cells on discrete sites

TL;DR

This paper investigates the geometric structure of higher-order Voronoi cells on discrete point sets in Euclidean space, revealing new properties and open questions about their dimensions and configurations.

Contribution

It provides new theoretical insights into the dimensions and relations of higher-order Voronoi cells, especially in complex configurations like points on a sphere.

Findings

Higher order cells of dimension n-1 do not exist.

High-order Voronoi cells can have empty interior.

The study highlights complex relations between cells of different orders.

Abstract

We study the structure of higher-order Voronoi cells on a discrete set of sites in $\mathbb{R}^n$, focussing on the relations between cells of different order, and paying special attention to the ill-posed case when a large number of points lie on a sphere. In particular, we prove that higher order cells of dimension $n-1$ do not exist, even though high-order Voronoi cells may have empty interior. We also present a number of open questions.