On the 1-density of Unit Ball Covering

TL;DR

This paper investigates the maximum density of space covered by exactly one unit ball in infinite coverings, providing exact results in 2D and conjectures for 3D with numerical estimates.

Contribution

It establishes the exact optimal 1-density in 2D and proposes a conjecture and numerical estimate for the 3D case, extending the understanding of space coverage by unit balls.

Findings

Exact optimal 1-density in 2D is approximately 0.6539.

Introduces the Dodecahedral Cover Conjecture for 3D.

Numerical estimate of 3D 1-density is approximately 0.315.

Abstract

Motivated by modern applications like image processing and wireless sensor networks, we consider a variation of the famous Kepler Conjecture. Given any infinite set of unit balls covering the whole space, we want to know the optimal (lim sup) density of the volume which is covered by exactly one ball (i.e., the maximum such density over all unit ball covers, called the {\em optimal 1-density} and denoted as $\delta_d$, where $d$ is the dimension of the Euclidean space). We prove that in 2D the optimal 1-density $\delta_2=(3\sqrt(3)-\pi)/\pi \approx 0.6539$, which is achieved through a regular hexagonal covering. In 3D, the problem is widely open and we present a Dodecehadral Cover Conjecture which states that the optimal 1-density in 3D, $\delta_3$, is bounded from above by the 1-density of a unit ball whose Voronoi polyhedron is a regular dodecahedron of circum-radius one (determined by 12 extra unit balls). We show numerically that this 1-density $\delta_3(dc)\approx 0.315$.