Packing Disks into Disks with Optimal Worst-Case Density

03/19/2019

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by   Sandor P. Fekete, et al.

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We provide a tight result for a fundamental problem arising from packing disks into a circular container: The critical density of packing disks in a disk is 0.5. This implies that any set of (not necessarily equal) disks of total area δ≤ 1/2 can always be packed into a disk of area 1; on the other hand, for any ε>0 there are sets of disks of area 1/2+ε that cannot be packed. The proof uses a careful manual analysis, complemented by a minor automatic part that is based on interval arithmetic. Beyond the basic mathematical importance, our result is also useful as a blackbox lemma for the analysis of recursive packing algorithms.