{"paper":{"title":"Kirszbraun extensions preserving uniform distance in Hilbert spaces","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.FA","authors_text":"Krzysztof J. Ciosmak","submitted_at":"2026-07-20T08:24:35Z","abstract_excerpt":"Let $X$ be a subset of a real Hilbert space and let $v\\colon X\\to Y$, where $Y$ is a real Hilbert space. We prove that the following conditions are equivalent: whenever $A\\subset X$, $\\rho\\geq0$, and $u\\colon A\\to Y$ is $1$-Lipschitz with $\\left\\lVert u(x)-v(x)\\right\\lVert\\leq\\rho$ for $x\\in A$, there is a $1$-Lipschitz extension $\\widetilde u\\colon X\\to Y$ with $\\left\\lVert \\widetilde u(x)-v(x)\\right \\lVert\\leq\\rho$ for $x\\in X$; and for every $1\\leq k\\leq\\dim Y$, $$\n  \\left\\lVert v(x_0)-\\sum_{i=1}^k t_i v(x_i)\\right\\lVert\n  \\leq\n  \\left\\lVert x_0-\\sum_{i=1}^k t_i x_i \\right \\lVert$$ whenever"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.17672","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/2607.17672/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"}