{"paper":{"title":"The set of packing and covering densities of convex disks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"W{\\l}odzimierz Kuperberg","submitted_at":"2013-09-02T01:14:10Z","abstract_excerpt":"For every convex disk $K$ (a convex compact subset of the plane, with non-void interior), the packing density $\\delta(K)$ and covering density $\\vartheta(K)$ form an ordered pair of real numbers, {\\em i.e.}, a point in ${\\mathbb R}^2$. The set $\\Omega$ consisting of points assigned this way to all convex disks is the subject of this article. A few known inequalities on $\\delta(K)$ and $\\vartheta(K)$ jointly outline a relatively small convex polygon $P$ that contains $\\Omega$, while the exact shape of $\\Omega$ remains a mystery. Here we describe explicitly a leaf-shaped convex region $\\Lambda$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.0281","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"}