Two-Fold Circle-Covering of the Plane under Congruent Voronoi Polygon Conditions

TL;DR

This paper investigates the minimum density needed for two-fold coverage of the plane using congruent Voronoi polygons, providing a simpler and more rigorous proof than previous work under these specific geometric conditions.

Contribution

It introduces a new, simplified proof for the minimum density of two-fold coverage with congruent Voronoi polygons, advancing understanding of multi-coverage problems.

Findings

Established the minimum density for two-coverage under congruent Voronoi polygons

Provided a simpler, more rigorous proof compared to prior work

Enhanced theoretical understanding of multi-coverage geometric configurations

Abstract

The $k$-coverage problem is to find the minimum number of disks such that each point in a given plane is covered by at least $k$ disks. Under unit disk condition, when $k$=1, this problem has been solved by Kershner in 1939. However, when $k > 1$, it becomes extremely difficult. One tried to tackle this problem with different restrictions. In this paper, we restrict ourself to congruent Voronoi polygon, and prove the minimum density of the two-coverage with such a restriction. Our proof is simpler and more rigorous than that given recently by Yun et al.