{"paper":{"title":"On the structure of sets with positive reach","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Jan Rataj, Ludek Zajicek","submitted_at":"2016-04-29T14:13:11Z","abstract_excerpt":"We give a complete characterization of compact sets with positive reach (=proximally $C^1$ sets) in the plane and of one-dimensional sets with positive reach in ${\\mathbb R}^d$. Further, we prove that if $\\emptyset \\neq A\\subset{\\mathbb R}^d$ is a set of positive reach of topological dimension $0< k \\leq d$, then $A$ has its \"$k$-dimensional regular part\" $\\emptyset \\neq R \\subset A$ which is a $k$-dimensional \"uniform\" $C^{1,1}$ manifold open in $A$ and $A\\setminus R$ can be locally covered by finitely many $(k-1)$-dimensional DC surfaces. We also show that if $A \\subset {\\mathbb R}^d$ has po"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.08841","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"}