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.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2016
- DOI:
- 10.48550/arXiv.1603.01094
- arXiv:
- arXiv:1603.01094
- Bibcode:
- 2016arXiv160301094D
- Keywords:
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- Mathematics - Metric Geometry;
- 52C17