{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:AYWWRG2Y3UU2YHQGLH35KJ57RP","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"2d9d5dc94ab0a22b5f0d97b7b2bf76d53d41fb027fc21ad75f609ebbfb5f10d6","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2009-01-22T07:33:53Z","title_canon_sha256":"44c974e4e8cbb66a5a50f42e6423c632109d64eb972625a53225265a22abadbd"},"schema_version":"1.0","source":{"id":"0901.3419","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0901.3419","created_at":"2026-05-18T02:40:11Z"},{"alias_kind":"arxiv_version","alias_value":"0901.3419v1","created_at":"2026-05-18T02:40:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0901.3419","created_at":"2026-05-18T02:40:11Z"},{"alias_kind":"pith_short_12","alias_value":"AYWWRG2Y3UU2","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"AYWWRG2Y3UU2YHQG","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"AYWWRG2Y","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:a9f3e43ef31bc28404e3575a3a3543324cd5ab52de008a59438be9ec3f96130f","target":"graph","created_at":"2026-05-18T02:40:11Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let K be a d-dimensional convex body, and let $K^{(n)}$ be the intersection of n halfspaces containing $K$ whose bounding hyperplanes are independent and identically distributed. Under suitable distributional assumptions, we prove an asymptotic formula for the expectation of the difference of the mean widths of $K^{(n)}$ and K, and another asymptotic formula for the expectation of the number of facets of $K^{(n)}$. These results are achieved by establishing an asymptotic result on weighted volume approximation of $K$ and by \"dualizing\" it using polarity.","authors_text":"Daniel Hug, Ferenc Fodor, K\\'aroly J. B\\\"or\\\"oczky","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2009-01-22T07:33:53Z","title":"The mean width of random polytopes circumscribed around a convex body"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0901.3419","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:8dfc148287457ecc72830ad5717cb7533329c7cf35f90d9174e40d6bb19d8f91","target":"record","created_at":"2026-05-18T02:40:11Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"2d9d5dc94ab0a22b5f0d97b7b2bf76d53d41fb027fc21ad75f609ebbfb5f10d6","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2009-01-22T07:33:53Z","title_canon_sha256":"44c974e4e8cbb66a5a50f42e6423c632109d64eb972625a53225265a22abadbd"},"schema_version":"1.0","source":{"id":"0901.3419","kind":"arxiv","version":1}},"canonical_sha256":"062d689b58dd29ac1e0659f7d527bf8bf387195d7d6fca82ad1f72881a78541e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"062d689b58dd29ac1e0659f7d527bf8bf387195d7d6fca82ad1f72881a78541e","first_computed_at":"2026-05-18T02:40:11.238169Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:40:11.238169Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Cun0Litc2w/YWNRBBFhVW6s6PaIO0iuxmQvYiRW3bmt6ixr3aeV1lLGwSBNhAFWbaVKk2u6jgAeedqExyAw1Dg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:40:11.238596Z","signed_message":"canonical_sha256_bytes"},"source_id":"0901.3419","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8dfc148287457ecc72830ad5717cb7533329c7cf35f90d9174e40d6bb19d8f91","sha256:a9f3e43ef31bc28404e3575a3a3543324cd5ab52de008a59438be9ec3f96130f"],"state_sha256":"26839101b7ed3ae073933b53263ba77a3e4fdba8e6d7c1da3ca695e3d54195b6"}