Optimal densities of packings consisting of highly unequal objects

TL;DR

This paper analyzes how the optimal packing density of objects with highly unequal sizes approaches a specific limit, revealing a formula that combines densities of individual objects as size ratios tend to extreme values.

Contribution

It provides a mathematical formula describing the limiting behavior of packing densities for objects of vastly different sizes, extending previous understanding of packings.

Findings

Optimal packing density approaches a specific limit as size ratio tends to infinity.

Generalizes the result to arbitrary bodies and finite sets of bodies.

Provides a formula for the limiting density involving individual densities.

Abstract

Let $\Delta$ be the optimal packing density of $\mathbb R^n$ by unit balls. We show the optimal packing density using two sizes of balls approaches $\Delta + (1 - \Delta) \Delta$ as the ratio of the radii tends to infinity. More generally, if $B$ is a body and $D$ is a finite set of bodies, then the optimal density $\Delta_{\{rB\} \cup D}$ of packings consisting of congruent copies of the bodies from $\{rB\} \cup D$ converges to $\Delta_D + (1 - \Delta_D) \Delta_{\{B\}}$ as $r$ tends to zero.