{"paper":{"title":"Mean Curvature and Closed Geodesics in Convex Hypersurfaces","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.DG","authors_text":"James Dibble, Joseph Ansel Hoisington","submitted_at":"2026-06-02T18:19:20Z","abstract_excerpt":"We give a sharp lower bound for the total mean curvature of a convex hypersurface in Euclidean space in terms of the length of a shortest nontrivial closed geodesic, generalizing a result of Paiva \\cite{Paiva97} for convex surfaces. This result is based on a sharp lower bound for the mean width of a convex hypersurface in terms of its Birkhoff invariant, which also gives sharp lower bounds for a broader array of total curvature functionals. We also characterize spheres as the unique convex hypersurfaces whose planar sections containing chords of maximal length are all as long as possible."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.04112","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/2606.04112/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"}