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We prove that if $E$ is a finite-dimensional normed space with an arbitrary norm, the trace of $\\theta$ is bounded, then $\\theta$ has finite length, i.e. is rectifiable, thus answering positively the question raised in~\\cite{Lemenant16sc-rectif}."},"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":"1707.04922","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2017-07-16T17:46:48Z","cross_cats_sorted":[],"title_canon_sha256":"8f87f8a6cb68239a2867089cb345c513929d9b84f81d0d68fba6ea46a22173ae","abstract_canon_sha256":"55055e0075f46daaebdb9c2f8ed02ae8a4c3885118a91746774adf3ef23cec14"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:40:11.771946Z","signature_b64":"UC4p3itcE13ofo6zsBzNxG3bys8GBBiIEUwv3RkxLY/URmN/NlxVnppG0xg+SQMD6OQLfxImam8zH8o2uSogCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"71b9e0ea7218c6667b487f5299a2edcabf1f112cd328f2355ff6d27a9954b3be","last_reissued_at":"2026-05-18T00:40:11.771235Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:40:11.771235Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Self-contracted curves have finite length","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Eugene Stepanov, Yana Teplitskaya","submitted_at":"2017-07-16T17:46:48Z","abstract_excerpt":"A curve $\\theta$: $I\\to E$ in a metric space $E$ equipped with the distance $d$, where $I\\subset \\R$ is a (possibly unbounded) interval, is called self-contracted, if for any triple of instances of time $\\{t_i\\}_{i=1}^3\\subset I$ with $t_1\\leq t_2\\leq t_3$ one has $d(\\theta(t_3),\\theta(t_2))\\leq d(\\theta(t_3),\\theta(t_1))$. We prove that if $E$ is a finite-dimensional normed space with an arbitrary norm, the trace of $\\theta$ is bounded, then $\\theta$ has finite length, i.e. is rectifiable, thus answering positively the question raised in~\\cite{Lemenant16sc-rectif}."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.04922","kind":"arxiv","version":1},"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":"1707.04922","created_at":"2026-05-18T00:40:11.771344+00:00"},{"alias_kind":"arxiv_version","alias_value":"1707.04922v1","created_at":"2026-05-18T00:40:11.771344+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.04922","created_at":"2026-05-18T00:40:11.771344+00:00"},{"alias_kind":"pith_short_12","alias_value":"OG46B2TSDDDG","created_at":"2026-05-18T12:31:34.259226+00:00"},{"alias_kind":"pith_short_16","alias_value":"OG46B2TSDDDGM62I","created_at":"2026-05-18T12:31:34.259226+00:00"},{"alias_kind":"pith_short_8","alias_value":"OG46B2TS","created_at":"2026-05-18T12:31:34.259226+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/OG46B2TSDDDGM62IP5JJTIXNZK","json":"https://pith.science/pith/OG46B2TSDDDGM62IP5JJTIXNZK.json","graph_json":"https://pith.science/api/pith-number/OG46B2TSDDDGM62IP5JJTIXNZK/graph.json","events_json":"https://pith.science/api/pith-number/OG46B2TSDDDGM62IP5JJTIXNZK/events.json","paper":"https://pith.science/paper/OG46B2TS"},"agent_actions":{"view_html":"https://pith.science/pith/OG46B2TSDDDGM62IP5JJTIXNZK","download_json":"https://pith.science/pith/OG46B2TSDDDGM62IP5JJTIXNZK.json","view_paper":"https://pith.science/paper/OG46B2TS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1707.04922&json=true","fetch_graph":"https://pith.science/api/pith-number/OG46B2TSDDDGM62IP5JJTIXNZK/graph.json","fetch_events":"https://pith.science/api/pith-number/OG46B2TSDDDGM62IP5JJTIXNZK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OG46B2TSDDDGM62IP5JJTIXNZK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OG46B2TSDDDGM62IP5JJTIXNZK/action/storage_attestation","attest_author":"https://pith.science/pith/OG46B2TSDDDGM62IP5JJTIXNZK/action/author_attestation","sign_citation":"https://pith.science/pith/OG46B2TSDDDGM62IP5JJTIXNZK/action/citation_signature","submit_replication":"https://pith.science/pith/OG46B2TSDDDGM62IP5JJTIXNZK/action/replication_record"}},"created_at":"2026-05-18T00:40:11.771344+00:00","updated_at":"2026-05-18T00:40:11.771344+00:00"}