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arXiv · 1603.01094

Optimal densities of packings consisting of highly unequal objects

David de Laat
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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.

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David de Laat. 2016-03-03. Optimal densities of packings consisting of highly unequal objects. https://arxiv.org/abs/1603.01094

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