{"paper":{"title":"Proximinality and uniformly approximable sets in $L^p$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Guillaume Grelier, Jaime San Mart\\'in","submitted_at":"2022-08-09T17:36:34Z","abstract_excerpt":"For any $p\\in[1,\\infty]$, we prove that the set of simple functions taking at most $k$ different values is proximinal in $L^p$ for all $k\\geq 1$. We introduce the class of uniformly approximable subsets of $L^p$, which is larger than the class of uniformly integrable sets. This new class is characterized in terms of the $p$-variation if $p\\in[1,\\infty)$ and in terms of covering numbers if $p=\\infty$. We study properties of uniformly approximable sets. In particular, we prove that the convex hull of a uniformly approximable bounded set is also uniformly approximable and that this class is stabl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2209.02598","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/2209.02598/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"}