{"paper":{"title":"Digesting the proof of the sharp thin-shell inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.PR"],"primary_cat":"math.MG","authors_text":"Boaz Klartag, Yuansi Chen","submitted_at":"2026-07-25T17:48:30Z","abstract_excerpt":"We present a proof that determines the optimal value of the universal constant in the thin-shell theorem for log-concave distributions in high dimensions. We prove that for any log-concave random vector $X = (X_1,\\ldots,X_n)$ in $\\mathbb{R}^n$ with mean zero and identity covariance, $$ {\\rm Var}( |X|^2 ) \\leq 8 n. $$ The constant $8$ is optimal: equality is attained when $X_1,\\ldots,X_n$ are independent, identically distributed, standard, centered exponential random variables. Moreover, among isotropic random vectors distributed uniformly on convex bodies in $\\mathbb{R}^n$, the quantity ${\\rm "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.23307","kind":"arxiv","version":1},"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/2607.23307/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"}