Remarks on soft ball packings in dimensions 2 and 3

TL;DR

This paper investigates soft packing densities of symmetric convex shapes in 2D and 3D, showing optimal configurations are achieved by specific lattice arrangements, with implications for understanding packing limits.

Contribution

It introduces the concept of soft density for arrangements of convex domains and proves maximal soft densities occur in particular lattice packings, including the FCC lattice.

Findings

Maximum soft density occurs in soft lattice packings.

FCC lattice locally maximizes soft density for congruent soft balls.

Optimal arrangements depend on symmetry and the soft parameter.

Abstract

We study translative arrangements of centrally symmetric convex domains in the plane (resp., of congruent balls in the Euclidean $3$-space) that neither pack nor cover. We define their soft density depending on a soft parameter and prove that the largest soft density for soft translative packings of a centrally symmetric convex domain with $3$-fold rotational symmetry and given soft parameter is obtained for a proper soft lattice packing. Furthermore, we show that among the soft lattice packings of congruent soft balls with given soft parameter the soft density is locally maximal for the corresponding face centered cubic (FCC) lattice.