{"paper":{"title":"Coverings: variations on a result of Rogers and on the Epsilon-net theorem of Haussler and Welzl","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"J\\'anos Nagy, M\\'arton Nasz\\'odi, N\\'ora Frankl","submitted_at":"2016-07-11T10:18:05Z","abstract_excerpt":"We consider four problems. Rogers proved that for any convex body $K$, we can cover ${\\mathbb R}^d$ by translates of $K$ of density very roughly $d\\ln d$. First, we extend this result by showing that, if we are given a family of positive homothets of $K$ of infinite total volume, then we can find appropriate translation vectors for each given homothet to cover ${\\mathbb R}^d$ with the same (or, in certain cases, smaller) density.\n  Second, we extend Rogers' result to multiple coverings of space by translates of a convex body: we give a non-trivial upper bound on the density of the most economi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.02888","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"}