{"paper":{"title":"On the $\\ell_0$ Isoperimetric Coefficient of Measurable Sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"math.MG","authors_text":"Manuel Fernandez V","submitted_at":"2023-10-29T01:27:44Z","abstract_excerpt":"In this paper we prove that the $\\ell_0$ isoperimetric coefficient for any axis-aligned cubes, $\\psi_{\\mathcal{C}}$, is $\\Theta(n^{-1/2})$ and that the isoperimetric coefficient for any measurable body $K$, $\\psi_K$, is of order $O(n^{-1/2})$. As a corollary we deduce that axis-aligned cubes essentially \"maximize\" the $\\ell_0$ isoperimetric coefficient: There exists a positive constant $q > 0$ such that $\\psi_K \\leq q \\cdot \\psi_{\\mathcal{C}}$, whenever $\\mathcal{C}$ is an axis-aligned cube and $K$ is any measurable set. Lastly, we give immediate applications of our results to the mixing time "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2312.00015","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2312.00015/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"}