{"paper":{"title":"On the Total Perimeter of Homothetic Convex Bodies in a Convex Container","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Adrian Dumitrescu, Csaba D. T\\'oth","submitted_at":"2014-05-15T19:10:58Z","abstract_excerpt":"For two planar convex bodies, $C$ and $D$, consider a packing $S$ of $n$ positive homothets of $C$ contained in $D$. We estimate the total perimeter of the bodies in $S$, denoted ${\\rm per}(S)$, in terms of ${\\rm per}(D)$ and $n$. When all homothets of $C$ touch the boundary of the container $D$, we show that either ${\\rm per}(S)=O(\\log n)$ or ${\\rm per}(S)=O(1)$, depending on how $C$ and $D$ \"fit together,\" and these bounds are the best possible apart from the constant factors. Specifically, we establish an optimal bound ${\\rm per}(S)=O(\\log n)$ unless $D$ is a convex polygon and every side o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.3950","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"}