{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:5SNXBJBARZDCTP7A4Q2OSZLER3","short_pith_number":"pith:5SNXBJBA","schema_version":"1.0","canonical_sha256":"ec9b70a4208e4629bfe0e434e965648ee49ae3ba85c80e66bf83bb28b5eb2dcd","source":{"kind":"arxiv","id":"1509.04575","version":3},"attestation_state":"computed","paper":{"title":"Caratheodory's Theorem in Depth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.CG","authors_text":"Clemens Huemer, Ruy Fabila-Monroy","submitted_at":"2015-09-15T14:19:51Z","abstract_excerpt":"Let $X$ be a finite set of points in $\\mathbb{R}^d$. The Tukey depth of a point $q$ with respect to $X$ is the minimum number $\\tau_X(q)$ of points of $X$ in a halfspace containing $q$. In this paper we prove a depth version of Carath\\'eodory's theorem. In particular, we prove that there exists a constant $c$ (that depends only on $d$ and $\\tau_X(q)$) and pairwise disjoint sets $X_1,\\dots, X_{d+1} \\subset X$ such that the following holds. Each $X_i$ has at least $c|X|$ points, and for every choice of points $x_i$ in $X_i$, $q$ is a convex combination of $x_1,\\dots, x_{d+1}$. We also prove dept"},"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":"1509.04575","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2015-09-15T14:19:51Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"eef167f3395ed483f8ee17efc740f48ac4cad4ffb9e477c40a1fae198ecee601","abstract_canon_sha256":"4509788f5c8ca76fd3b27e15516f3cf0ace4571508a29b76f6700918c0f3ac4b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:46:59.834360Z","signature_b64":"2c1vnf3F/2OULk2BfuOr6lw7XPB0fda/SRkb4nhZI7fsCTOpuizVO7P8N18HTX94JPNXjYNlVz4R2Xh3egvfBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ec9b70a4208e4629bfe0e434e965648ee49ae3ba85c80e66bf83bb28b5eb2dcd","last_reissued_at":"2026-05-18T00:46:59.833808Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:46:59.833808Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Caratheodory's Theorem in Depth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.CG","authors_text":"Clemens Huemer, Ruy Fabila-Monroy","submitted_at":"2015-09-15T14:19:51Z","abstract_excerpt":"Let $X$ be a finite set of points in $\\mathbb{R}^d$. The Tukey depth of a point $q$ with respect to $X$ is the minimum number $\\tau_X(q)$ of points of $X$ in a halfspace containing $q$. In this paper we prove a depth version of Carath\\'eodory's theorem. In particular, we prove that there exists a constant $c$ (that depends only on $d$ and $\\tau_X(q)$) and pairwise disjoint sets $X_1,\\dots, X_{d+1} \\subset X$ such that the following holds. Each $X_i$ has at least $c|X|$ points, and for every choice of points $x_i$ in $X_i$, $q$ is a convex combination of $x_1,\\dots, x_{d+1}$. We also prove dept"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.04575","kind":"arxiv","version":3},"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":"1509.04575","created_at":"2026-05-18T00:46:59.833876+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.04575v3","created_at":"2026-05-18T00:46:59.833876+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.04575","created_at":"2026-05-18T00:46:59.833876+00:00"},{"alias_kind":"pith_short_12","alias_value":"5SNXBJBARZDC","created_at":"2026-05-18T12:29:07.941421+00:00"},{"alias_kind":"pith_short_16","alias_value":"5SNXBJBARZDCTP7A","created_at":"2026-05-18T12:29:07.941421+00:00"},{"alias_kind":"pith_short_8","alias_value":"5SNXBJBA","created_at":"2026-05-18T12:29:07.941421+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/5SNXBJBARZDCTP7A4Q2OSZLER3","json":"https://pith.science/pith/5SNXBJBARZDCTP7A4Q2OSZLER3.json","graph_json":"https://pith.science/api/pith-number/5SNXBJBARZDCTP7A4Q2OSZLER3/graph.json","events_json":"https://pith.science/api/pith-number/5SNXBJBARZDCTP7A4Q2OSZLER3/events.json","paper":"https://pith.science/paper/5SNXBJBA"},"agent_actions":{"view_html":"https://pith.science/pith/5SNXBJBARZDCTP7A4Q2OSZLER3","download_json":"https://pith.science/pith/5SNXBJBARZDCTP7A4Q2OSZLER3.json","view_paper":"https://pith.science/paper/5SNXBJBA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.04575&json=true","fetch_graph":"https://pith.science/api/pith-number/5SNXBJBARZDCTP7A4Q2OSZLER3/graph.json","fetch_events":"https://pith.science/api/pith-number/5SNXBJBARZDCTP7A4Q2OSZLER3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5SNXBJBARZDCTP7A4Q2OSZLER3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5SNXBJBARZDCTP7A4Q2OSZLER3/action/storage_attestation","attest_author":"https://pith.science/pith/5SNXBJBARZDCTP7A4Q2OSZLER3/action/author_attestation","sign_citation":"https://pith.science/pith/5SNXBJBARZDCTP7A4Q2OSZLER3/action/citation_signature","submit_replication":"https://pith.science/pith/5SNXBJBARZDCTP7A4Q2OSZLER3/action/replication_record"}},"created_at":"2026-05-18T00:46:59.833876+00:00","updated_at":"2026-05-18T00:46:59.833876+00:00"}