On densest packings of equal balls of $\rb^{n}$ and Marcinkiewicz spaces

TL;DR

This paper explores the densest arrangements of equal spheres in Euclidean spaces using Marcinkiewicz space techniques, providing new proofs for existence theorems of maximal and saturated packings.

Contribution

It introduces a novel approach applying Marcinkiewicz space methods to analyze sphere packings and offers new direct proofs for key existence theorems.

Findings

Established existence of densest sphere packings.

Proved existence of completely saturated packings with maximal density.

Applied Marcinkiewicz techniques to asymptotic density analysis.

Abstract

We investigate, by "a la Marcinkiewicz" techniques applied to the (asymptotic) density function, how dense systems of equal spheres of $\rb^{n}, n \geq 1,$ can be partitioned at infinity in order to allow the computation of their density as a true limit and not a limsup. The density of a packing of equal balls is the norm 1 of the characteristic function of the systems of balls in the sense of Marcinkiewicz. Existence Theorems for densest sphere packings and completely saturated sphere packings of maximal density are given new direct proofs.