{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:GIGRJCGUC25JMP2VHQJV2QB62K","short_pith_number":"pith:GIGRJCGU","schema_version":"1.0","canonical_sha256":"320d1488d416ba963f553c135d403ed2b380d6f7c3a54349d79729c7772404dc","source":{"kind":"arxiv","id":"1708.01549","version":5},"attestation_state":"computed","paper":{"title":"Fine properties of the curvature of arbitrary closed sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MG"],"primary_cat":"math.DG","authors_text":"Mario Santilli","submitted_at":"2017-08-04T15:17:42Z","abstract_excerpt":"Given an arbitrary closed set A of $\\mathbf{R}^{n}$, we establish the relation between the eigenvalues of the approximate differential of the spherical image map of A and the principal curvatures of A introduced by Hug-Last-Weil, thus extending a well known relation for sets of positive reach by Federer and Zaehle. Then we provide for every $ m = 1, \\ldots , n-1 $ an integral representation for the support measure $ \\mu_{m} $ of A with respect to the m dimensional Hausdoff measure. Moreover a notion of second fundamental form $Q_{A} $ for an arbitrary closed set A is introduced so that the fin"},"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":"1708.01549","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-08-04T15:17:42Z","cross_cats_sorted":["math.AP","math.MG"],"title_canon_sha256":"a469f71234fb2ac1e5605c3e6be394c533cfb81ba17ec0c23cc7738d47737d2c","abstract_canon_sha256":"2703657ed60a201785fc43ae9fd4c67ef7c046542ff95c9766cbff14ba7d2b3b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:19:48.801391Z","signature_b64":"oRw6e0VZ54t7jWYyqZpD6m411Yg6GXQkhdzT5yPSiiUpH+KF5tf2MV9P23FTn6FEB8OEnQXVm3FyTML8BAG7Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"320d1488d416ba963f553c135d403ed2b380d6f7c3a54349d79729c7772404dc","last_reissued_at":"2026-07-05T01:19:48.800833Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:19:48.800833Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fine properties of the curvature of arbitrary closed sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MG"],"primary_cat":"math.DG","authors_text":"Mario Santilli","submitted_at":"2017-08-04T15:17:42Z","abstract_excerpt":"Given an arbitrary closed set A of $\\mathbf{R}^{n}$, we establish the relation between the eigenvalues of the approximate differential of the spherical image map of A and the principal curvatures of A introduced by Hug-Last-Weil, thus extending a well known relation for sets of positive reach by Federer and Zaehle. Then we provide for every $ m = 1, \\ldots , n-1 $ an integral representation for the support measure $ \\mu_{m} $ of A with respect to the m dimensional Hausdoff measure. Moreover a notion of second fundamental form $Q_{A} $ for an arbitrary closed set A is introduced so that the fin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.01549","kind":"arxiv","version":5},"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/1708.01549/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1708.01549","created_at":"2026-07-05T01:19:48.800891+00:00"},{"alias_kind":"arxiv_version","alias_value":"1708.01549v5","created_at":"2026-07-05T01:19:48.800891+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.01549","created_at":"2026-07-05T01:19:48.800891+00:00"},{"alias_kind":"pith_short_12","alias_value":"GIGRJCGUC25J","created_at":"2026-07-05T01:19:48.800891+00:00"},{"alias_kind":"pith_short_16","alias_value":"GIGRJCGUC25JMP2V","created_at":"2026-07-05T01:19:48.800891+00:00"},{"alias_kind":"pith_short_8","alias_value":"GIGRJCGU","created_at":"2026-07-05T01:19:48.800891+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"1908.09795","citing_title":"Uniqueness of critical points of the anisotropic isoperimetric problem for finite perimeter sets","ref_index":31,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GIGRJCGUC25JMP2VHQJV2QB62K","json":"https://pith.science/pith/GIGRJCGUC25JMP2VHQJV2QB62K.json","graph_json":"https://pith.science/api/pith-number/GIGRJCGUC25JMP2VHQJV2QB62K/graph.json","events_json":"https://pith.science/api/pith-number/GIGRJCGUC25JMP2VHQJV2QB62K/events.json","paper":"https://pith.science/paper/GIGRJCGU"},"agent_actions":{"view_html":"https://pith.science/pith/GIGRJCGUC25JMP2VHQJV2QB62K","download_json":"https://pith.science/pith/GIGRJCGUC25JMP2VHQJV2QB62K.json","view_paper":"https://pith.science/paper/GIGRJCGU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1708.01549&json=true","fetch_graph":"https://pith.science/api/pith-number/GIGRJCGUC25JMP2VHQJV2QB62K/graph.json","fetch_events":"https://pith.science/api/pith-number/GIGRJCGUC25JMP2VHQJV2QB62K/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GIGRJCGUC25JMP2VHQJV2QB62K/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GIGRJCGUC25JMP2VHQJV2QB62K/action/storage_attestation","attest_author":"https://pith.science/pith/GIGRJCGUC25JMP2VHQJV2QB62K/action/author_attestation","sign_citation":"https://pith.science/pith/GIGRJCGUC25JMP2VHQJV2QB62K/action/citation_signature","submit_replication":"https://pith.science/pith/GIGRJCGUC25JMP2VHQJV2QB62K/action/replication_record"}},"created_at":"2026-07-05T01:19:48.800891+00:00","updated_at":"2026-07-05T01:19:48.800891+00:00"}