{"paper":{"title":"Packing Disks into Disks with Optimal Worst-Case Density","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Christian Scheffer, Phillip Keldenich, S\\'andor P. Fekete","submitted_at":"2019-03-19T09:59:06Z","abstract_excerpt":"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 $\\delta\\leq 1/2$ can always be packed into a disk of area 1; on the other hand, for any $\\varepsilon>0$ there are sets of disks of area $1/2+\\varepsilon$ 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 a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.07908","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}