{"paper":{"title":"Optimal densities of packings consisting of highly unequal objects","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"David de Laat","submitted_at":"2016-03-03T13:31:03Z","abstract_excerpt":"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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.01094","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"}