{"paper":{"title":"Universal convex coverings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.MG"],"primary_cat":"math.NT","authors_text":"Roland Bacher (IF)","submitted_at":"2008-12-18T13:09:56Z","abstract_excerpt":"In every dimension $d\\ge1$, we establish the existence of a constant $v_d>0$ and of a subset $\\mathcal U_d$ of $\\mathbb R^d$ such that the following holds: $\\mathcal C+\\mathcal U_d=\\mathbb R^d$ for every convex set $\\mathcal C\\subset \\mathbb R^d$ of volume at least $v_d$ and $\\mathcal U_d$ contains at most $\\log(r)^{d-1}r^d$ points at distance at most $r$ from the origin, for every large $r$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0812.3525","kind":"arxiv","version":3},"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"}