{"paper":{"title":"On strengthened versions of Klee's convex body problem in Banach spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Chulei Liu, Lixin Cheng, Wuyi He, Zhizheng Yu","submitted_at":"2026-06-05T15:12:54Z","abstract_excerpt":"In a recent article, Cheng, Jiang and Yuan gave an affirmative answer to Klee's convex bodies problem of Banach spaces in the sense of strict convexity and G\\^{a}teaux smoothness. In this paper, we continue to study this problem in strong senses, such as local uniform convexity, uniform convexity, Fr\\'{e}chet smoothness and uniform smoothness. As a result, we show\n  (1) Every convex body in a Banach space $X$ is approximated by locally uniformly convex bodies with respect to the Hausdorff metric if and only if $X$ admits an equivalent locally uniformly convex norm; (2) Every convex body in $X$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.07370","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/2606.07370/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"}