{"paper":{"title":"Dimension-free estimates for covering functionals of simplices and $\\ell_p$ balls","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Chan He, Feifei Chen, Senlin Wu","submitted_at":"2026-05-31T07:58:54Z","abstract_excerpt":"We study \\(\\Gamma_{2^n}(K)\\), the least positive number \\(\\gamma>0\\) such that an \\(n\\)-dimensional convex body \\(K\\) can be covered by \\(2^n\\) translates of \\(\\gamma K\\). For \\(n\\)-simplices \\(\\Delta_n\\), we prove that \\(\\Gamma_{2^n}(\\Delta_n)\\), as a sequence in \\(n\\), tends to \\(1/2\\). For the cross-polytope \\(B_1^n\\), we show that \\(\\Gamma_{2^n}(B_1^n)\\leq5/6\\) holds for all \\(n\\geq2\\), and that \\(\\limsup_{n\\to\\infty}\\Gamma_{2^n}(B_1^n)\\leq0.641\\cdots\\). Finally, we prove the existence of a constant \\(\\kappa_*<1\\) such that \\(\\Gamma_{2^n}(B_p^n)\\leq\\kappa_*\\) for all \\(n\\geq2\\) and all \\(p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.01082","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.01082/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}