{"paper":{"title":"Eldan's Stochastic Localization and the KLS Conjecture: Isoperimetry, Concentration and Mixing","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG","cs.DS","math.MG","math.PR"],"primary_cat":"math.FA","authors_text":"Santosh S. Vempala, Yin Tat Lee","submitted_at":"2016-12-05T20:36:28Z","abstract_excerpt":"We show that the Cheeger constant for $n$-dimensional isotropic logconcave measures is $O(n^{1/4})$, improving on the previous best bound of $O(n^{1/3}\\sqrt{\\log n}).$ As corollaries we obtain the same improved bound on the thin-shell estimate, Poincar\\'{e} constant and Lipschitz concentration constant and an alternative proof of this bound for the isotropic (slicing) constant; it also follows that the ball walk for sampling from an isotropic logconcave density in ${\\bf R}^{n}$ converges in $O^{*}(n^{2.5})$ steps from a warm start. The proof is based on gradually transforming any logconcave de"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.01507","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"}