Optimal densities of packings consisting of highly unequal objects

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.