{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:Z5TMHMJLSXQSKXCFRSKHWTR34T","short_pith_number":"pith:Z5TMHMJL","schema_version":"1.0","canonical_sha256":"cf66c3b12b95e1255c458c947b4e3be4fcb6e4aa79f45c24f7cd1fcb1bdebc6d","source":{"kind":"arxiv","id":"1707.02253","version":2},"attestation_state":"computed","paper":{"title":"Expected intrinsic volumes and facet numbers of random beta-polytopes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.MG","authors_text":"Christoph Thaele, Daniel Temesvari, Zakhar Kabluchko","submitted_at":"2017-07-07T16:16:39Z","abstract_excerpt":"Let $X_1,\\ldots,X_n$ be i.i.d.\\ random points in the $d$-dimensional Euclidean space sampled according to one of the following probability densities: $$ f_{d,\\beta} (x) = \\text{const} \\cdot (1-\\|x\\|^2)^{\\beta}, \\quad \\|x\\|\\leq 1, \\quad \\text{(the beta case)} $$ and $$ \\tilde f_{d,\\beta} (x) = \\text{const} \\cdot (1+\\|x\\|^2)^{-\\beta}, \\quad x\\in\\mathbb{R}^d, \\quad \\text{(the beta' case).} $$ We compute exactly the expected intrinsic volumes and the expected number of facets of the convex hull of $X_1,\\ldots,X_n$. Asymptotic formulae where obtained previously by Affentranger [The convex hull of r"},"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":"1707.02253","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2017-07-07T16:16:39Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"b4d9e75c6e0b06ed9559757c590bbc3a827f70696b0dcfa32d382fc45fc86f0c","abstract_canon_sha256":"d357efb8c7882c0601c98afa3644681aee29dd86cd97f7a9fbd7fcb0581ffe1e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:27:30.329729Z","signature_b64":"iW7Xrjjy8qOjAnX8vtqdxFgG3geI8Jm0isBGqLdm4Hdk74ZIemHY0psRrkG0EaxFunOuzDbDEZL/Iws5qBNbBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cf66c3b12b95e1255c458c947b4e3be4fcb6e4aa79f45c24f7cd1fcb1bdebc6d","last_reissued_at":"2026-05-18T00:27:30.328842Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:27:30.328842Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Expected intrinsic volumes and facet numbers of random beta-polytopes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.MG","authors_text":"Christoph Thaele, Daniel Temesvari, Zakhar Kabluchko","submitted_at":"2017-07-07T16:16:39Z","abstract_excerpt":"Let $X_1,\\ldots,X_n$ be i.i.d.\\ random points in the $d$-dimensional Euclidean space sampled according to one of the following probability densities: $$ f_{d,\\beta} (x) = \\text{const} \\cdot (1-\\|x\\|^2)^{\\beta}, \\quad \\|x\\|\\leq 1, \\quad \\text{(the beta case)} $$ and $$ \\tilde f_{d,\\beta} (x) = \\text{const} \\cdot (1+\\|x\\|^2)^{-\\beta}, \\quad x\\in\\mathbb{R}^d, \\quad \\text{(the beta' case).} $$ We compute exactly the expected intrinsic volumes and the expected number of facets of the convex hull of $X_1,\\ldots,X_n$. Asymptotic formulae where obtained previously by Affentranger [The convex hull of r"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.02253","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":"1707.02253","created_at":"2026-05-18T00:27:30.328967+00:00"},{"alias_kind":"arxiv_version","alias_value":"1707.02253v2","created_at":"2026-05-18T00:27:30.328967+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.02253","created_at":"2026-05-18T00:27:30.328967+00:00"},{"alias_kind":"pith_short_12","alias_value":"Z5TMHMJLSXQS","created_at":"2026-05-18T12:31:59.375834+00:00"},{"alias_kind":"pith_short_16","alias_value":"Z5TMHMJLSXQSKXCF","created_at":"2026-05-18T12:31:59.375834+00:00"},{"alias_kind":"pith_short_8","alias_value":"Z5TMHMJL","created_at":"2026-05-18T12:31:59.375834+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/Z5TMHMJLSXQSKXCFRSKHWTR34T","json":"https://pith.science/pith/Z5TMHMJLSXQSKXCFRSKHWTR34T.json","graph_json":"https://pith.science/api/pith-number/Z5TMHMJLSXQSKXCFRSKHWTR34T/graph.json","events_json":"https://pith.science/api/pith-number/Z5TMHMJLSXQSKXCFRSKHWTR34T/events.json","paper":"https://pith.science/paper/Z5TMHMJL"},"agent_actions":{"view_html":"https://pith.science/pith/Z5TMHMJLSXQSKXCFRSKHWTR34T","download_json":"https://pith.science/pith/Z5TMHMJLSXQSKXCFRSKHWTR34T.json","view_paper":"https://pith.science/paper/Z5TMHMJL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1707.02253&json=true","fetch_graph":"https://pith.science/api/pith-number/Z5TMHMJLSXQSKXCFRSKHWTR34T/graph.json","fetch_events":"https://pith.science/api/pith-number/Z5TMHMJLSXQSKXCFRSKHWTR34T/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Z5TMHMJLSXQSKXCFRSKHWTR34T/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Z5TMHMJLSXQSKXCFRSKHWTR34T/action/storage_attestation","attest_author":"https://pith.science/pith/Z5TMHMJLSXQSKXCFRSKHWTR34T/action/author_attestation","sign_citation":"https://pith.science/pith/Z5TMHMJLSXQSKXCFRSKHWTR34T/action/citation_signature","submit_replication":"https://pith.science/pith/Z5TMHMJLSXQSKXCFRSKHWTR34T/action/replication_record"}},"created_at":"2026-05-18T00:27:30.328967+00:00","updated_at":"2026-05-18T00:27:30.328967+00:00"}