{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:LX45RCUZBXSCIQ4IJHDFULQKNG","short_pith_number":"pith:LX45RCUZ","schema_version":"1.0","canonical_sha256":"5df9d88a990de424438849c65a2e0a69871f41910e9c0ba0c0b8f3a3b2f233c5","source":{"kind":"arxiv","id":"1604.08841","version":2},"attestation_state":"computed","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1604.08841","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-04-29T14:13:11Z","cross_cats_sorted":[],"title_canon_sha256":"c5e0982515e7fbb2f432f40571e05906f61008a1a7add745010d1511adae15ae","abstract_canon_sha256":"241eb58348c81504723c33d6eb9976f286beddd0e9e876488ccf2916a7d57d72"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:37:48.557720Z","signature_b64":"Gl5HGOFaLLV8ZT1GKeDXTvmOuoA+1t0TrIxN11cUxk+e/ls33RtimYLeGFcRU6ypRKIF3+If+86vUhNcdhHUDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5df9d88a990de424438849c65a2e0a69871f41910e9c0ba0c0b8f3a3b2f233c5","last_reissued_at":"2026-05-18T00:37:48.557203Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:37:48.557203Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1604.08841","created_at":"2026-05-18T00:37:48.557290+00:00"},{"alias_kind":"arxiv_version","alias_value":"1604.08841v2","created_at":"2026-05-18T00:37:48.557290+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.08841","created_at":"2026-05-18T00:37:48.557290+00:00"},{"alias_kind":"pith_short_12","alias_value":"LX45RCUZBXSC","created_at":"2026-05-18T12:30:29.479603+00:00"},{"alias_kind":"pith_short_16","alias_value":"LX45RCUZBXSCIQ4I","created_at":"2026-05-18T12:30:29.479603+00:00"},{"alias_kind":"pith_short_8","alias_value":"LX45RCUZ","created_at":"2026-05-18T12:30:29.479603+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/LX45RCUZBXSCIQ4IJHDFULQKNG","json":"https://pith.science/pith/LX45RCUZBXSCIQ4IJHDFULQKNG.json","graph_json":"https://pith.science/api/pith-number/LX45RCUZBXSCIQ4IJHDFULQKNG/graph.json","events_json":"https://pith.science/api/pith-number/LX45RCUZBXSCIQ4IJHDFULQKNG/events.json","paper":"https://pith.science/paper/LX45RCUZ"},"agent_actions":{"view_html":"https://pith.science/pith/LX45RCUZBXSCIQ4IJHDFULQKNG","download_json":"https://pith.science/pith/LX45RCUZBXSCIQ4IJHDFULQKNG.json","view_paper":"https://pith.science/paper/LX45RCUZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1604.08841&json=true","fetch_graph":"https://pith.science/api/pith-number/LX45RCUZBXSCIQ4IJHDFULQKNG/graph.json","fetch_events":"https://pith.science/api/pith-number/LX45RCUZBXSCIQ4IJHDFULQKNG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LX45RCUZBXSCIQ4IJHDFULQKNG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LX45RCUZBXSCIQ4IJHDFULQKNG/action/storage_attestation","attest_author":"https://pith.science/pith/LX45RCUZBXSCIQ4IJHDFULQKNG/action/author_attestation","sign_citation":"https://pith.science/pith/LX45RCUZBXSCIQ4IJHDFULQKNG/action/citation_signature","submit_replication":"https://pith.science/pith/LX45RCUZBXSCIQ4IJHDFULQKNG/action/replication_record"}},"created_at":"2026-05-18T00:37:48.557290+00:00","updated_at":"2026-05-18T00:37:48.557290+00:00"}