{"paper":{"title":"Definable versions of theorems by Kirszbraun and Helly","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.LO","authors_text":"Andreas Fischer, Matthias Aschenbrenner","submitted_at":"2009-06-05T17:02:47Z","abstract_excerpt":"Kirszbraun's Theorem states that every Lipschitz map $S\\to\\mathbb R^n$, where $S\\subseteq \\mathbb R^m$, has an extension to a Lipschitz map $\\mathbb R^m \\to \\mathbb R^n$ with the same Lipschitz constant. Its proof relies on Helly's Theorem: every family of compact subsets of $\\mathbb R^n$, having the property that each of its subfamilies consisting of at most $n+1$ sets share a common point, has a non-empty intersection. We prove versions of these theorems valid for definable maps and sets in arbitrary definably complete expansions of ordered fields."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0906.1168","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}