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Remarks on soft ball packings in dimensions 2 and 3

Károly Bezdek, Zsolt Lángi

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.

Remarks on soft ball packings in dimensions 2 and 3

Abstract

We study translative arrangements of centrally symmetric convex domains in the plane (resp., of congruent balls in the Euclidean -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 -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.
Paper Structure (4 sections, 9 theorems, 11 equations, 3 figures)

This paper contains 4 sections, 9 theorems, 11 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:lattice']}
  3. Proof of Theorem \ref{['thm:density']}
  4. Proof of Theorem \ref{['thm:localmaximum']}

Key Result

Theorem 1

For any $\mathbf{o}$-symmetric planar convex body $\mathbf{M}$ with $3$-fold rotational symmetry and $\lambda\geq 0$, there is a lattice packing $\mathcal{P}$ of translates of $\mathbf{M}$ such that In other words, we have $\bar{\delta}_{\mathbb{M}}(\lambda) = \bar{\delta}_{\mathbb{M}}^{\mathrm{lattice}}(\lambda)$ for any $\mathbf{o}$-symmetric planar convex body $\mathbf{M}$ with $3$-fold rotati

Figures (3)

  • Figure 1: An illustration for the Molnár decomposition of a point system in the case that $\mathbf{M}$ is a Euclidean disk. In the figure, the edges of the cells, the circumcircles of the cells and the separating sides are denoted by solid, dotted and dashed lines, respectively.
  • Figure 2: An illustration for Lemma \ref{['lem:legs']}. The dotted line indicates the bisector of $[\mathbf{a},\mathbf{b}]$. Dashed lines show the boundaries of $\mathbf{a} + \mu \mathbf{M}$ and $\mathbf{b} + \mu \mathbf{M}$.
  • Figure 3: An illlustration for Lemma \ref{['rem:geomobs']} for the case $\mathbf{x}=\mathbf{x}'$.

Theorems & Definitions (21)

  • Definition 1
  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Conjecture 1
  • Theorem 2
  • Theorem 3
  • Conjecture 2
  • Remark 4
  • ...and 11 more