{"paper":{"title":"Minimal $N$-Point Diameters and $f$-Best-Packing Constants in $R^d$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"A.V. Bondarenko, D.P. Hardin, E.B. Saff","submitted_at":"2012-04-19T16:23:39Z","abstract_excerpt":"In terms of the minimal $N$-point diameter $D_d(N)$ for $R^d,$ we determine, for a class of continuous real-valued functions $f$ on $[0,+\\infty],$ the $N$-point $f$-best-packing constant $\\min\\{f(\\|x-y\\|)\\, :\\, x,y\\in \\R^d\\}$, where the minimum is taken over point sets of cardinality $N.$ We also show that $$ N^{1/d}\\Delta_d^{-1/d}-2\\le D_d(N)\\le N^{1/d}\\Delta_d^{-1/d}, \\quad N\\ge 2,$$ where $\\Delta_d$ is the maximal sphere packing density in $\\R^d$. Further, we provide asymptotic estimates for the $f$-best-packing constants as $N\\to\\infty$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.4403","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"}