{"paper":{"title":"Geometric Bounds for the Mean Gauge and the Mean Width in Isotropic Position","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Christos Pandis, Silouanos Brazitikos","submitted_at":"2026-08-07T13:34:34Z","abstract_excerpt":"Let $K \\subset \\mathbb{R}^n$ be an origin-symmetric convex body and assume that its uniform probability measure is isotropic in the probabilistic normalization, namely \\[\n  \\int_K x \\otimes x \\, d\\mu_K(x) = \\mathrm{Id}_n. \\] We give deterministic geometric proofs of \\[\n  M(K) \\leq C \\frac{\\log(n)}{\\sqrt{n}}\n  \\qquad \\text{and} \\qquad\n  M^*(K) \\leq C \\sqrt{n} \\, \\log(n), \\] where \\[\n  M(K) = \\int_{\\mathbb{S}^{n-1}} \\|\\theta\\|_K \\, d\\sigma(\\theta),\n  \\qquad\n  M^*(K) = \\int_{\\mathbb{S}^{n-1}} h_K(\\theta) \\, d\\sigma(\\theta). \\] Combining both estimates yields \\[\n  M(K) M^*(K) \\leq C \\log^2(n). \\]\n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.07216","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/2608.07216/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"}