{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:RYR5FVTIOB7LF7NUN57ND6QWRC","short_pith_number":"pith:RYR5FVTI","canonical_record":{"source":{"id":"1706.08092","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-06-25T12:50:13Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"41a47bd1df57bb4bb5c81daf02d52a835a8c4ef27270935aaa7e789221f28e2e","abstract_canon_sha256":"f721df94da76d07fc3813894f89650c63e2ed5b745d1c4b009bcf7fc280ad8f9"},"schema_version":"1.0"},"canonical_sha256":"8e23d2d668707eb2fdb46f7ed1fa1688aa3d11750d633bbdacb87d8d7d7add60","source":{"kind":"arxiv","id":"1706.08092","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1706.08092","created_at":"2026-05-18T00:41:43Z"},{"alias_kind":"arxiv_version","alias_value":"1706.08092v1","created_at":"2026-05-18T00:41:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.08092","created_at":"2026-05-18T00:41:43Z"},{"alias_kind":"pith_short_12","alias_value":"RYR5FVTIOB7L","created_at":"2026-05-18T12:31:43Z"},{"alias_kind":"pith_short_16","alias_value":"RYR5FVTIOB7LF7NU","created_at":"2026-05-18T12:31:43Z"},{"alias_kind":"pith_short_8","alias_value":"RYR5FVTI","created_at":"2026-05-18T12:31:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:RYR5FVTIOB7LF7NUN57ND6QWRC","target":"record","payload":{"canonical_record":{"source":{"id":"1706.08092","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-06-25T12:50:13Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"41a47bd1df57bb4bb5c81daf02d52a835a8c4ef27270935aaa7e789221f28e2e","abstract_canon_sha256":"f721df94da76d07fc3813894f89650c63e2ed5b745d1c4b009bcf7fc280ad8f9"},"schema_version":"1.0"},"canonical_sha256":"8e23d2d668707eb2fdb46f7ed1fa1688aa3d11750d633bbdacb87d8d7d7add60","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:41:43.568549Z","signature_b64":"wDoGmiMUmXxfpMYKvT8M6eJ/E9WUbPFVUf3sA/OeVY2AY1jHK8Je9fmBAFd9+wZpZrn5khKtn15vr5qROcDYAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8e23d2d668707eb2fdb46f7ed1fa1688aa3d11750d633bbdacb87d8d7d7add60","last_reissued_at":"2026-05-18T00:41:43.567945Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:41:43.567945Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1706.08092","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":"2026-05-18T00:41:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HcT0BSgDhp/QnuI95b7WR/4zJdqtZL7hTP8i4k5wcdjwYCgCR7VxdlAVv4NkTNiVKh+JxDzFgxENg1i7JiP+Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T17:01:46.752432Z"},"content_sha256":"5a9950c911f23b59e01e12cb699d0abb835172b5535a5757bda9d3a3fef21259","schema_version":"1.0","event_id":"sha256:5a9950c911f23b59e01e12cb699d0abb835172b5535a5757bda9d3a3fef21259"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:RYR5FVTIOB7LF7NUN57ND6QWRC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Expected volumes of Gaussian polytopes, external angles, and multiple order statistics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.PR","authors_text":"Dmitry Zaporozhets, Zakhar Kabluchko","submitted_at":"2017-06-25T12:50:13Z","abstract_excerpt":"Let $X_1,\\ldots,X_n$ be a standard normal sample in $\\mathbb R^d$. We compute exactly the expected volume of the Gaussian polytope $\\mathrm{conv}[X_1,\\ldots,X_n]$, the symmetric Gaussian polytope $\\mathrm{conv}[\\pm X_1,\\ldots,\\pm X_n]$, and the Gaussian zonotope $[0,X_1]+\\ldots+[0,X_n]$ by exploiting their connection to the regular simplex, the regular crosspolytope, and the cube with the aid of Tsirelson's formula. The expected volumes of these random polytopes are given by essentially the same expressions as the intrinsic volumes and external angles of the regular polytopes. For all these qu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.08092","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":"2026-05-18T00:41:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KGTfr1x9aOyqQY8f+ysYmnzXATuZfCX0XtT0Lwhafn5YkZG27fzquafpEdJW1m43krrK6oX4pt+i/B1EdxbJCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T17:01:46.752793Z"},"content_sha256":"e2280426d305e77f35475921111df33469374ae598302971a8f9ca669f98b445","schema_version":"1.0","event_id":"sha256:e2280426d305e77f35475921111df33469374ae598302971a8f9ca669f98b445"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RYR5FVTIOB7LF7NUN57ND6QWRC/bundle.json","state_url":"https://pith.science/pith/RYR5FVTIOB7LF7NUN57ND6QWRC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RYR5FVTIOB7LF7NUN57ND6QWRC/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":"2026-05-26T17:01:46Z","links":{"resolver":"https://pith.science/pith/RYR5FVTIOB7LF7NUN57ND6QWRC","bundle":"https://pith.science/pith/RYR5FVTIOB7LF7NUN57ND6QWRC/bundle.json","state":"https://pith.science/pith/RYR5FVTIOB7LF7NUN57ND6QWRC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RYR5FVTIOB7LF7NUN57ND6QWRC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:RYR5FVTIOB7LF7NUN57ND6QWRC","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":"f721df94da76d07fc3813894f89650c63e2ed5b745d1c4b009bcf7fc280ad8f9","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-06-25T12:50:13Z","title_canon_sha256":"41a47bd1df57bb4bb5c81daf02d52a835a8c4ef27270935aaa7e789221f28e2e"},"schema_version":"1.0","source":{"id":"1706.08092","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1706.08092","created_at":"2026-05-18T00:41:43Z"},{"alias_kind":"arxiv_version","alias_value":"1706.08092v1","created_at":"2026-05-18T00:41:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.08092","created_at":"2026-05-18T00:41:43Z"},{"alias_kind":"pith_short_12","alias_value":"RYR5FVTIOB7L","created_at":"2026-05-18T12:31:43Z"},{"alias_kind":"pith_short_16","alias_value":"RYR5FVTIOB7LF7NU","created_at":"2026-05-18T12:31:43Z"},{"alias_kind":"pith_short_8","alias_value":"RYR5FVTI","created_at":"2026-05-18T12:31:43Z"}],"graph_snapshots":[{"event_id":"sha256:e2280426d305e77f35475921111df33469374ae598302971a8f9ca669f98b445","target":"graph","created_at":"2026-05-18T00:41:43Z","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 $X_1,\\ldots,X_n$ be a standard normal sample in $\\mathbb R^d$. We compute exactly the expected volume of the Gaussian polytope $\\mathrm{conv}[X_1,\\ldots,X_n]$, the symmetric Gaussian polytope $\\mathrm{conv}[\\pm X_1,\\ldots,\\pm X_n]$, and the Gaussian zonotope $[0,X_1]+\\ldots+[0,X_n]$ by exploiting their connection to the regular simplex, the regular crosspolytope, and the cube with the aid of Tsirelson's formula. The expected volumes of these random polytopes are given by essentially the same expressions as the intrinsic volumes and external angles of the regular polytopes. For all these qu","authors_text":"Dmitry Zaporozhets, Zakhar Kabluchko","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-06-25T12:50:13Z","title":"Expected volumes of Gaussian polytopes, external angles, and multiple order statistics"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.08092","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":"sha256:5a9950c911f23b59e01e12cb699d0abb835172b5535a5757bda9d3a3fef21259","target":"record","created_at":"2026-05-18T00:41:43Z","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":"f721df94da76d07fc3813894f89650c63e2ed5b745d1c4b009bcf7fc280ad8f9","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-06-25T12:50:13Z","title_canon_sha256":"41a47bd1df57bb4bb5c81daf02d52a835a8c4ef27270935aaa7e789221f28e2e"},"schema_version":"1.0","source":{"id":"1706.08092","kind":"arxiv","version":1}},"canonical_sha256":"8e23d2d668707eb2fdb46f7ed1fa1688aa3d11750d633bbdacb87d8d7d7add60","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8e23d2d668707eb2fdb46f7ed1fa1688aa3d11750d633bbdacb87d8d7d7add60","first_computed_at":"2026-05-18T00:41:43.567945Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:41:43.567945Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wDoGmiMUmXxfpMYKvT8M6eJ/E9WUbPFVUf3sA/OeVY2AY1jHK8Je9fmBAFd9+wZpZrn5khKtn15vr5qROcDYAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:41:43.568549Z","signed_message":"canonical_sha256_bytes"},"source_id":"1706.08092","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5a9950c911f23b59e01e12cb699d0abb835172b5535a5757bda9d3a3fef21259","sha256:e2280426d305e77f35475921111df33469374ae598302971a8f9ca669f98b445"],"state_sha256":"f3f76f5fcc232800d0e1730be2b4355798fad0e4be62ab51d2ec57f2f1bb2ab7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rvOTpLVcD8KtYygEJ22mXbPdwjHbUwTwTCHnoBxUyjs/e3dTLk5QXZ29Zf+ud23bwl59GA2bAOq93pS4eRxlAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T17:01:46.755027Z","bundle_sha256":"c0d6b73106c7bb9c71c38a6c76426426ed4b6928d4aa21dc51f9928924b5142e"}}