Strongly Proximal Edelsbrunner-Harer Nerves in Voronoi Tessellations

TL;DR

This paper explores the properties of Edelsbrunner-Harer nerves within Voronoi tessellations, revealing that maximal nucleus clusters form strongly proximal nerves sharing the same homotopy type as their unions.

Contribution

It introduces the concept of strongly proximal Edelsbrunner-Harer nerves in Voronoi tessellations and establishes their topological equivalence with unions of nucleus clusters.

Findings

Maximal nucleus clusters are strongly proximal Edelsbrunner-Harer nerves.

Nerves and unions of sets in MNCs share the same homotopy type.

Abstract

This paper introduces Edelsbrunner-Harer nerve in collections of Voronoi regions (called nucleus clusters) endowed with one or more proximity relations. The main results in this paper are that a maximal nucleus cluster (MNC) in a Voronoi Tessellation is a strongly proximal Edelsbrunner-Harer nerve, each MNC nerve and the union of the sets in the MNC have the same homotopy type.