{"paper":{"title":"On a biased edge isoperimetric inequality for the discrete cube","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"David Ellis, Nathan Keller, Noam Lifshitz","submitted_at":"2017-02-06T16:05:32Z","abstract_excerpt":"The `full' edge isoperimetric inequality for the discrete cube (due to Harper, Bernstein, Lindsay and Hart) specifies the minimum size of the edge boundary $\\partial A$ of a set $A \\subset \\{0,1\\}^n$, as a function of $|A|$. A weaker (but more widely-used) lower bound is $|\\partial A| \\geq |A| \\log_2(2^n/|A|)$, where equality holds iff $A$ is a subcube. In 2011, the first author obtained a sharp `stability' version of the latter result, proving that if $|\\partial A| \\leq |A| (\\log(2^n/|A|)+\\epsilon)$, then there exists a subcube $C$ such that $|A \\Delta C|/|A| = O(\\epsilon /\\log(1/\\epsilon))$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.01675","kind":"arxiv","version":2},"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"}