{"paper":{"title":"Interpolating Thin-Shell and Sharp Large-Deviation Estimates For Isotropic Log-Concave Measures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Emanuel Milman, Olivier Gu\\'edon","submitted_at":"2010-11-03T17:12:07Z","abstract_excerpt":"Given an isotropic random vector $X$ with log-concave density in Euclidean space $\\Real^n$, we study the concentration properties of $|X|$ on all scales, both above and below its expectation. We show in particular that: \\[ \\P(\\abs{|X| -\\sqrt{n}} \\geq t \\sqrt{n}) \\leq C \\exp(-c n^{1/2} \\min(t^3,t)) \\;\\;\\; \\forall t \\geq 0 ~, \\] for some universal constants $c,C>0$. This improves the best known deviation results on the thin-shell and mesoscopic scales due to Fleury and Klartag, respectively, and recovers the sharp large-deviation estimate of Paouris. Another new feature of our estimate is that i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.0943","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"}