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For instance, in the Euclidean case, if $f \\in \\mathcal{C}^3(\\mathbb{R}^n, \\mathbb{R})$ and 0 is a regular value of $f$, then the intrinsic volume of degree $n-k$ of the sublevel set $M^0 = f^{-1}(]-\\infty, 0])$, if the latter is compact, is given by \\begin{equation*} \\mathcal{L}_{n-k}(M^0) = \\frac{\\Gamma(k/2)}{2 \\pi^{k/2} (k-1)!} \\int_{M^0} \\operatorname{div} \\left( \\frac{P_{n, k}(\\operatorn"},"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":"1903.01592","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-03-04T23:54:27Z","cross_cats_sorted":[],"title_canon_sha256":"9481b80458c00683a2116038097df5cfa5ee71eb2599f876e63fff9575ea4b58","abstract_canon_sha256":"311c6c207e7e9af193af800bbd64935ee70691d05d070f5299ce2987e8abde81"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:20:19.779964Z","signature_b64":"J0PVkB+cHhTzIPEt3xCM306FkVGKKTPoC+M1pGx182+vUOnWAFvRojSfAwfuMO1z+f1fojekL8FDTMy9q9h+Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"983d03a68792c36184001a2e4f976b3d6e292fa7698716408ee90784f1215cb1","last_reissued_at":"2026-07-05T08:20:19.779561Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:20:19.779561Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Intrinsic volumes of sublevel sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Beno\\^it Jubin","submitted_at":"2019-03-04T23:54:27Z","abstract_excerpt":"We establish formulas that give the intrinsic volumes, or curvature measures, of sublevel sets of functions defined on Riemannian manifolds as integrals of functionals of the function and its derivatives. For instance, in the Euclidean case, if $f \\in \\mathcal{C}^3(\\mathbb{R}^n, \\mathbb{R})$ and 0 is a regular value of $f$, then the intrinsic volume of degree $n-k$ of the sublevel set $M^0 = f^{-1}(]-\\infty, 0])$, if the latter is compact, is given by \\begin{equation*} \\mathcal{L}_{n-k}(M^0) = \\frac{\\Gamma(k/2)}{2 \\pi^{k/2} (k-1)!} \\int_{M^0} \\operatorname{div} \\left( \\frac{P_{n, k}(\\operatorn"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.01592","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1903.01592/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":"1903.01592","created_at":"2026-07-05T08:20:19.779619+00:00"},{"alias_kind":"arxiv_version","alias_value":"1903.01592v2","created_at":"2026-07-05T08:20:19.779619+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.01592","created_at":"2026-07-05T08:20:19.779619+00:00"},{"alias_kind":"pith_short_12","alias_value":"TA6QHJUHSLBW","created_at":"2026-07-05T08:20:19.779619+00:00"},{"alias_kind":"pith_short_16","alias_value":"TA6QHJUHSLBWDBAA","created_at":"2026-07-05T08:20:19.779619+00:00"},{"alias_kind":"pith_short_8","alias_value":"TA6QHJUH","created_at":"2026-07-05T08:20:19.779619+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/TA6QHJUHSLBWDBAADIXE7F3LHV","json":"https://pith.science/pith/TA6QHJUHSLBWDBAADIXE7F3LHV.json","graph_json":"https://pith.science/api/pith-number/TA6QHJUHSLBWDBAADIXE7F3LHV/graph.json","events_json":"https://pith.science/api/pith-number/TA6QHJUHSLBWDBAADIXE7F3LHV/events.json","paper":"https://pith.science/paper/TA6QHJUH"},"agent_actions":{"view_html":"https://pith.science/pith/TA6QHJUHSLBWDBAADIXE7F3LHV","download_json":"https://pith.science/pith/TA6QHJUHSLBWDBAADIXE7F3LHV.json","view_paper":"https://pith.science/paper/TA6QHJUH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1903.01592&json=true","fetch_graph":"https://pith.science/api/pith-number/TA6QHJUHSLBWDBAADIXE7F3LHV/graph.json","fetch_events":"https://pith.science/api/pith-number/TA6QHJUHSLBWDBAADIXE7F3LHV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TA6QHJUHSLBWDBAADIXE7F3LHV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TA6QHJUHSLBWDBAADIXE7F3LHV/action/storage_attestation","attest_author":"https://pith.science/pith/TA6QHJUHSLBWDBAADIXE7F3LHV/action/author_attestation","sign_citation":"https://pith.science/pith/TA6QHJUHSLBWDBAADIXE7F3LHV/action/citation_signature","submit_replication":"https://pith.science/pith/TA6QHJUHSLBWDBAADIXE7F3LHV/action/replication_record"}},"created_at":"2026-07-05T08:20:19.779619+00:00","updated_at":"2026-07-05T08:20:19.779619+00:00"}