{"paper":{"title":"Optimal mean width and metric entropy estimates for convex bodies","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.FA","math.PR"],"primary_cat":"math.MG","authors_text":"Grigoris Paouris, Reese Pathak","submitted_at":"2026-07-31T15:22:06Z","abstract_excerpt":"We show that there exists a constant $C > 0$ such that for any $n \\geq 1$ and any convex body $K \\subset \\mathbf{R}^n$, \\[ 1 \\leq \\inf_{T \\in \\mathrm{GL}(n)} \\, \\frac{M^\\ast(TK)}{\\mathrm{vr}(TK)}\n  \\leq C\\sqrt{\\log(\\mathrm{e} n)}, \\] where $M^\\ast$ denotes the spherical mean width and $\\mathrm{vr}(\\cdot)$ denotes the volume radius. The righthand side is attained, up to universal constants, by the crosspolytope and the regular $n$-simplex. Analogously, we show that, up to universal constants, the logarithm of the Euclidean covering number is maximized over convex bodies $K \\subset \\mathbf{R}^n$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.29522","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/2607.29522/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"}