Optimal densities of packings consisting of highly unequal objects

Let $Δ$ be the optimal packing density of $\mathbb R^n$ by unit balls. We show the optimal packing density using two sizes of balls approaches $Δ+ (1 - Δ) Δ$ 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 $Δ_{\{rB\} \cup D}$ of packings consisting of congruent copies of the bodies from $\{rB\} \cup D$ converges to $Δ_D + (1 - Δ_D) Δ_{\{B\}}$ as $r$ tends to zero.

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