{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:VBNZSICRAZJ254IOTKKL6CA7RA","short_pith_number":"pith:VBNZSICR","canonical_record":{"source":{"id":"1712.09699","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2017-12-27T22:51:54Z","cross_cats_sorted":[],"title_canon_sha256":"4dff5cf5c508d14ae6e11300fd6f7b488e1b427d03466f0f2c6e21fa5f6cb9ba","abstract_canon_sha256":"1e4aedbe4d86f6d8912e770bdc1d674b56bc0e70e1cfe0a51449f0917e008a2c"},"schema_version":"1.0"},"canonical_sha256":"a85b9920510653aef10e9a94bf081f8825d257f0599ca0bc4fc964ea4cf91d4d","source":{"kind":"arxiv","id":"1712.09699","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1712.09699","created_at":"2026-05-18T00:27:07Z"},{"alias_kind":"arxiv_version","alias_value":"1712.09699v1","created_at":"2026-05-18T00:27:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.09699","created_at":"2026-05-18T00:27:07Z"},{"alias_kind":"pith_short_12","alias_value":"VBNZSICRAZJ2","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_16","alias_value":"VBNZSICRAZJ254IO","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_8","alias_value":"VBNZSICR","created_at":"2026-05-18T12:31:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:VBNZSICRAZJ254IOTKKL6CA7RA","target":"record","payload":{"canonical_record":{"source":{"id":"1712.09699","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2017-12-27T22:51:54Z","cross_cats_sorted":[],"title_canon_sha256":"4dff5cf5c508d14ae6e11300fd6f7b488e1b427d03466f0f2c6e21fa5f6cb9ba","abstract_canon_sha256":"1e4aedbe4d86f6d8912e770bdc1d674b56bc0e70e1cfe0a51449f0917e008a2c"},"schema_version":"1.0"},"canonical_sha256":"a85b9920510653aef10e9a94bf081f8825d257f0599ca0bc4fc964ea4cf91d4d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:27:07.580572Z","signature_b64":"FUueU9Rk1i6jY6JwOOngzKkFA3waOij2hPHNyY29/KyEm9AZunL3hj3sOG/U7isQgATCgcb3ZDWOnMGi1SIiBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a85b9920510653aef10e9a94bf081f8825d257f0599ca0bc4fc964ea4cf91d4d","last_reissued_at":"2026-05-18T00:27:07.579816Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:27:07.579816Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1712.09699","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":"2026-05-18T00:27:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tOEvyFNwKSb3kP+M4E80NCFUQGpjIRP5b5LX+x1KahpQB7ae9Gbw5DOgWdKDnJrZllsABpNNmB3+1DBkba9aBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T09:45:13.917186Z"},"content_sha256":"ccf4f96c05c8cf696de7c387e0155045ee4bb114bcb30692c5b4019160008220","schema_version":"1.0","event_id":"sha256:ccf4f96c05c8cf696de7c387e0155045ee4bb114bcb30692c5b4019160008220"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:VBNZSICRAZJ254IOTKKL6CA7RA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Integral geometric formulae for Minkowski tensors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Daniel Hug, Jan A. Weis","submitted_at":"2017-12-27T22:51:54Z","abstract_excerpt":"The Minkowski tensors are the natural tensor-valued generalizations of the intrinsic volumes of convex bodies. We prove two complete sets of integral geometric formulae, so called kinematic and Crofton formulae, for these Minkowski tensors. These formulae express the integral mean of the Minkowski tensors of the intersection of a given convex body with a second geometric object (another convex body in the kinematic case and an affine subspace in the Crofton case) which is uniformly moved by a proper rigid motion, in terms of linear combinations of the Minkowski tensors of the given geometric o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.09699","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":"2026-05-18T00:27:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VKTVU89Iw6SoNzLIEKXtzJ0vPCw5IwDlRju75xS4O3rOv4P6jqftRGlgEZ5L/WnRn2I6UJLzH27LAg9JUJucDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T09:45:13.917545Z"},"content_sha256":"7db5ec8eac85569a4c22158aac2566d540d10182bd23065ca56c9c6b71f7f3b1","schema_version":"1.0","event_id":"sha256:7db5ec8eac85569a4c22158aac2566d540d10182bd23065ca56c9c6b71f7f3b1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VBNZSICRAZJ254IOTKKL6CA7RA/bundle.json","state_url":"https://pith.science/pith/VBNZSICRAZJ254IOTKKL6CA7RA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VBNZSICRAZJ254IOTKKL6CA7RA/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":"2026-06-09T09:45:13Z","links":{"resolver":"https://pith.science/pith/VBNZSICRAZJ254IOTKKL6CA7RA","bundle":"https://pith.science/pith/VBNZSICRAZJ254IOTKKL6CA7RA/bundle.json","state":"https://pith.science/pith/VBNZSICRAZJ254IOTKKL6CA7RA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VBNZSICRAZJ254IOTKKL6CA7RA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:VBNZSICRAZJ254IOTKKL6CA7RA","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":"1e4aedbe4d86f6d8912e770bdc1d674b56bc0e70e1cfe0a51449f0917e008a2c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2017-12-27T22:51:54Z","title_canon_sha256":"4dff5cf5c508d14ae6e11300fd6f7b488e1b427d03466f0f2c6e21fa5f6cb9ba"},"schema_version":"1.0","source":{"id":"1712.09699","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1712.09699","created_at":"2026-05-18T00:27:07Z"},{"alias_kind":"arxiv_version","alias_value":"1712.09699v1","created_at":"2026-05-18T00:27:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.09699","created_at":"2026-05-18T00:27:07Z"},{"alias_kind":"pith_short_12","alias_value":"VBNZSICRAZJ2","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_16","alias_value":"VBNZSICRAZJ254IO","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_8","alias_value":"VBNZSICR","created_at":"2026-05-18T12:31:49Z"}],"graph_snapshots":[{"event_id":"sha256:7db5ec8eac85569a4c22158aac2566d540d10182bd23065ca56c9c6b71f7f3b1","target":"graph","created_at":"2026-05-18T00:27:07Z","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":"The Minkowski tensors are the natural tensor-valued generalizations of the intrinsic volumes of convex bodies. We prove two complete sets of integral geometric formulae, so called kinematic and Crofton formulae, for these Minkowski tensors. These formulae express the integral mean of the Minkowski tensors of the intersection of a given convex body with a second geometric object (another convex body in the kinematic case and an affine subspace in the Crofton case) which is uniformly moved by a proper rigid motion, in terms of linear combinations of the Minkowski tensors of the given geometric o","authors_text":"Daniel Hug, Jan A. Weis","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2017-12-27T22:51:54Z","title":"Integral geometric formulae for Minkowski tensors"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.09699","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:ccf4f96c05c8cf696de7c387e0155045ee4bb114bcb30692c5b4019160008220","target":"record","created_at":"2026-05-18T00:27:07Z","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":"1e4aedbe4d86f6d8912e770bdc1d674b56bc0e70e1cfe0a51449f0917e008a2c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2017-12-27T22:51:54Z","title_canon_sha256":"4dff5cf5c508d14ae6e11300fd6f7b488e1b427d03466f0f2c6e21fa5f6cb9ba"},"schema_version":"1.0","source":{"id":"1712.09699","kind":"arxiv","version":1}},"canonical_sha256":"a85b9920510653aef10e9a94bf081f8825d257f0599ca0bc4fc964ea4cf91d4d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a85b9920510653aef10e9a94bf081f8825d257f0599ca0bc4fc964ea4cf91d4d","first_computed_at":"2026-05-18T00:27:07.579816Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:27:07.579816Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FUueU9Rk1i6jY6JwOOngzKkFA3waOij2hPHNyY29/KyEm9AZunL3hj3sOG/U7isQgATCgcb3ZDWOnMGi1SIiBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:27:07.580572Z","signed_message":"canonical_sha256_bytes"},"source_id":"1712.09699","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ccf4f96c05c8cf696de7c387e0155045ee4bb114bcb30692c5b4019160008220","sha256:7db5ec8eac85569a4c22158aac2566d540d10182bd23065ca56c9c6b71f7f3b1"],"state_sha256":"81cb2c0d816ca10cf5c03b1820603b4aa2555a4a9f03d120e1067b8b2d4e47a3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tnlOw2K7BeVqcKdgc0TSIVpXuDPQR0eu7c/JDSwj7jYkdJ9iQZmCWg3VkxH7sPjGNYcKe9lN90sbV7nOmfB3Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T09:45:13.919721Z","bundle_sha256":"ec2bba67b02e0feeff8efc576fe8b08e9d93e64e6313d7e1b4612b269852356a"}}