{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:Z7LADLNTWB6N5EC364QE2JUOQT","short_pith_number":"pith:Z7LADLNT","canonical_record":{"source":{"id":"1311.3690","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2013-11-14T22:47:25Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"bd9b2099e3ecf8de4a6aab39a9ffa5a079becbbfae43e4f4a80f9cec86b77d3f","abstract_canon_sha256":"e3726f679dd4d957d768747f357a4b1f2e7b91537edfd1e1c7b9752648f545ab"},"schema_version":"1.0"},"canonical_sha256":"cfd601adb3b07cde905bf7204d268e84dce809e27a0608b38b11d2ad46535598","source":{"kind":"arxiv","id":"1311.3690","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.3690","created_at":"2026-05-18T03:07:03Z"},{"alias_kind":"arxiv_version","alias_value":"1311.3690v1","created_at":"2026-05-18T03:07:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.3690","created_at":"2026-05-18T03:07:03Z"},{"alias_kind":"pith_short_12","alias_value":"Z7LADLNTWB6N","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_16","alias_value":"Z7LADLNTWB6N5EC3","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_8","alias_value":"Z7LADLNT","created_at":"2026-05-18T12:28:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:Z7LADLNTWB6N5EC364QE2JUOQT","target":"record","payload":{"canonical_record":{"source":{"id":"1311.3690","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2013-11-14T22:47:25Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"bd9b2099e3ecf8de4a6aab39a9ffa5a079becbbfae43e4f4a80f9cec86b77d3f","abstract_canon_sha256":"e3726f679dd4d957d768747f357a4b1f2e7b91537edfd1e1c7b9752648f545ab"},"schema_version":"1.0"},"canonical_sha256":"cfd601adb3b07cde905bf7204d268e84dce809e27a0608b38b11d2ad46535598","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:07:03.258590Z","signature_b64":"m4j9d8+zL5giLHaf6Juc83AqiN6wQT1wKRuvuKRsB9/8cDckGcLsLXMdmshW49dS1XUVZn/s3ZbztBUWolA2Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cfd601adb3b07cde905bf7204d268e84dce809e27a0608b38b11d2ad46535598","last_reissued_at":"2026-05-18T03:07:03.258072Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:07:03.258072Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1311.3690","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:07:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"J2frme/GPPtvPSMWVJKhY3IngJyPW7H/l+NxDmpbw721RXPFVNpI2c8oBYEz8dtndFp0FD4OL2qg+2ZQapH7CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T00:44:34.784031Z"},"content_sha256":"30709172ec5ffe0b0b62c110724b70243a0310753546621fb187ff821f9f6265","schema_version":"1.0","event_id":"sha256:30709172ec5ffe0b0b62c110724b70243a0310753546621fb187ff821f9f6265"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:Z7LADLNTWB6N5EC364QE2JUOQT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Volume of the polar of random sets and shadow systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.MG","authors_text":"Dario Cordero-Erausquin, Grigoris Paouris, Matthieu Fradelizi, Peter Pivovarov","submitted_at":"2013-11-14T22:47:25Z","abstract_excerpt":"We obtain optimal inequalities for the volume of the polar of random sets, generated for instance by the convex hull of independent random vectors in Euclidean space. Extremizers are given by random vectors uniformly distributed in Euclidean balls. This provides a random extension of the Blaschke-Santalo inequality which, in turn, can be derived by the law of large numbers. The method involves generalized shadow systems, their connection to Busemann type inequalities, and how they interact with functional rearrangement inequalities."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.3690","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:07:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"s/J+LmNXFpYi2vb5FtDT+IfMZHvFEDg12bssSgxZVsCWEaqDTWyD6VwIaPLL56+Zj/fxejK5RKdfHFaCbJ7vBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T00:44:34.784738Z"},"content_sha256":"972eaaa8304a4214ca001cda3c0c88d0fa9f3632a472628686c247a455db80c7","schema_version":"1.0","event_id":"sha256:972eaaa8304a4214ca001cda3c0c88d0fa9f3632a472628686c247a455db80c7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Z7LADLNTWB6N5EC364QE2JUOQT/bundle.json","state_url":"https://pith.science/pith/Z7LADLNTWB6N5EC364QE2JUOQT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Z7LADLNTWB6N5EC364QE2JUOQT/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-27T00:44:34Z","links":{"resolver":"https://pith.science/pith/Z7LADLNTWB6N5EC364QE2JUOQT","bundle":"https://pith.science/pith/Z7LADLNTWB6N5EC364QE2JUOQT/bundle.json","state":"https://pith.science/pith/Z7LADLNTWB6N5EC364QE2JUOQT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Z7LADLNTWB6N5EC364QE2JUOQT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:Z7LADLNTWB6N5EC364QE2JUOQT","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":"e3726f679dd4d957d768747f357a4b1f2e7b91537edfd1e1c7b9752648f545ab","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2013-11-14T22:47:25Z","title_canon_sha256":"bd9b2099e3ecf8de4a6aab39a9ffa5a079becbbfae43e4f4a80f9cec86b77d3f"},"schema_version":"1.0","source":{"id":"1311.3690","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.3690","created_at":"2026-05-18T03:07:03Z"},{"alias_kind":"arxiv_version","alias_value":"1311.3690v1","created_at":"2026-05-18T03:07:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.3690","created_at":"2026-05-18T03:07:03Z"},{"alias_kind":"pith_short_12","alias_value":"Z7LADLNTWB6N","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_16","alias_value":"Z7LADLNTWB6N5EC3","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_8","alias_value":"Z7LADLNT","created_at":"2026-05-18T12:28:09Z"}],"graph_snapshots":[{"event_id":"sha256:972eaaa8304a4214ca001cda3c0c88d0fa9f3632a472628686c247a455db80c7","target":"graph","created_at":"2026-05-18T03:07:03Z","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":"We obtain optimal inequalities for the volume of the polar of random sets, generated for instance by the convex hull of independent random vectors in Euclidean space. Extremizers are given by random vectors uniformly distributed in Euclidean balls. This provides a random extension of the Blaschke-Santalo inequality which, in turn, can be derived by the law of large numbers. The method involves generalized shadow systems, their connection to Busemann type inequalities, and how they interact with functional rearrangement inequalities.","authors_text":"Dario Cordero-Erausquin, Grigoris Paouris, Matthieu Fradelizi, Peter Pivovarov","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2013-11-14T22:47:25Z","title":"Volume of the polar of random sets and shadow systems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.3690","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":"sha256:30709172ec5ffe0b0b62c110724b70243a0310753546621fb187ff821f9f6265","target":"record","created_at":"2026-05-18T03:07:03Z","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":"e3726f679dd4d957d768747f357a4b1f2e7b91537edfd1e1c7b9752648f545ab","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2013-11-14T22:47:25Z","title_canon_sha256":"bd9b2099e3ecf8de4a6aab39a9ffa5a079becbbfae43e4f4a80f9cec86b77d3f"},"schema_version":"1.0","source":{"id":"1311.3690","kind":"arxiv","version":1}},"canonical_sha256":"cfd601adb3b07cde905bf7204d268e84dce809e27a0608b38b11d2ad46535598","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cfd601adb3b07cde905bf7204d268e84dce809e27a0608b38b11d2ad46535598","first_computed_at":"2026-05-18T03:07:03.258072Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:07:03.258072Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"m4j9d8+zL5giLHaf6Juc83AqiN6wQT1wKRuvuKRsB9/8cDckGcLsLXMdmshW49dS1XUVZn/s3ZbztBUWolA2Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:07:03.258590Z","signed_message":"canonical_sha256_bytes"},"source_id":"1311.3690","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:30709172ec5ffe0b0b62c110724b70243a0310753546621fb187ff821f9f6265","sha256:972eaaa8304a4214ca001cda3c0c88d0fa9f3632a472628686c247a455db80c7"],"state_sha256":"b70a67d87c08377917a14895ae76b2099d08b9c8ae341363bde2903e92f72989"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"S2jfo+tjG057NGFjmlxYUAyRkwMmdtfgjUqG6FPENl3eEeRaF2jYywAbNa4EebXdHVO8DEtU2b9MzlJMYX24Bw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T00:44:34.788212Z","bundle_sha256":"fa5f5bb0bc7c59ae2d706c38df1cff1fdf19782a59daa9de7277c52ee43909ad"}}