Packing Disks into Disks with Optimal Worst-Case Density

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