{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2008:IMVTWTQFE6IZU72DYBFP2HIN6L","short_pith_number":"pith:IMVTWTQF","canonical_record":{"source":{"id":"0804.0705","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.DG","submitted_at":"2008-04-04T12:06:19Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"bd0ceeb902bbced737403921c9e669e9f3bde8a7421ae129624b4c1cf872e5df","abstract_canon_sha256":"8803d70934415ac5cf6af0736da4cf68b1e0de0267b5485ef798ee03c8f47caf"},"schema_version":"1.0"},"canonical_sha256":"432b3b4e0527919a7f43c04afd1d0df2ef705654c7c2801d88e18e474d687c62","source":{"kind":"arxiv","id":"0804.0705","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0804.0705","created_at":"2026-05-18T02:15:54Z"},{"alias_kind":"arxiv_version","alias_value":"0804.0705v1","created_at":"2026-05-18T02:15:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0804.0705","created_at":"2026-05-18T02:15:54Z"},{"alias_kind":"pith_short_12","alias_value":"IMVTWTQFE6IZ","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_16","alias_value":"IMVTWTQFE6IZU72D","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_8","alias_value":"IMVTWTQF","created_at":"2026-05-18T12:25:57Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2008:IMVTWTQFE6IZU72DYBFP2HIN6L","target":"record","payload":{"canonical_record":{"source":{"id":"0804.0705","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.DG","submitted_at":"2008-04-04T12:06:19Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"bd0ceeb902bbced737403921c9e669e9f3bde8a7421ae129624b4c1cf872e5df","abstract_canon_sha256":"8803d70934415ac5cf6af0736da4cf68b1e0de0267b5485ef798ee03c8f47caf"},"schema_version":"1.0"},"canonical_sha256":"432b3b4e0527919a7f43c04afd1d0df2ef705654c7c2801d88e18e474d687c62","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:15:54.024625Z","signature_b64":"NdlvrB8O2epAMumSZ0OXPZKwFO4sOCH7wjvAhPiR4HK2P7ZsmLATxVvm2da6Gzn8YbIuccK0uBR+XudVCupQBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"432b3b4e0527919a7f43c04afd1d0df2ef705654c7c2801d88e18e474d687c62","last_reissued_at":"2026-05-18T02:15:54.023932Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:15:54.023932Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0804.0705","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":"2026-05-18T02:15:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2jNnkMIQ7BeZbWr3CgVtR6wwJBq2wXRRPkL4vy2CkHlBzL0MdkHjBzdBSIyKHnaEZVhAJtjgRplzXx4pjvVcAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T07:49:33.468802Z"},"content_sha256":"cfdedddfe29a09036e04889303ad8f28505f33f0d7134de4218cee1c501ceab5","schema_version":"1.0","event_id":"sha256:cfdedddfe29a09036e04889303ad8f28505f33f0d7134de4218cee1c501ceab5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2008:IMVTWTQFE6IZU72DYBFP2HIN6L","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Weak Finsler Strutures and the Funk Metric","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.DG","authors_text":"Athanase Papadopoulos, Marc Troyanov","submitted_at":"2008-04-04T12:06:19Z","abstract_excerpt":"We discuss general notions of metrics and of Finsler structures which we call weak metrics and weak Finsler structures. Any convex domain carries a canonical weak Finsler structure, which we call its tautological weak Finsler structure. We compute distances in the tautological weak Finsler structure of a domain and we show that these are given by the so-called Funk weak metric. We conclude the paper with a discussion of geodesics, of metric balls and of convexity properties of the Funk weak metric."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0804.0705","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":"2026-05-18T02:15:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iruAqvwqH3Plwxdf7HcRhwMzgLIJ9MuZEHOCcIvOAIWh67U0TFlN7lAEqWTunB6ORQfu3C0MRL6O4//lyXEqAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T07:49:33.469487Z"},"content_sha256":"fbe1c514733ff0bbbd54ad92f8f45b648bd7f2716e9d0ee4b075c91cd8529e43","schema_version":"1.0","event_id":"sha256:fbe1c514733ff0bbbd54ad92f8f45b648bd7f2716e9d0ee4b075c91cd8529e43"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IMVTWTQFE6IZU72DYBFP2HIN6L/bundle.json","state_url":"https://pith.science/pith/IMVTWTQFE6IZU72DYBFP2HIN6L/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IMVTWTQFE6IZU72DYBFP2HIN6L/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":"2026-06-07T07:49:33Z","links":{"resolver":"https://pith.science/pith/IMVTWTQFE6IZU72DYBFP2HIN6L","bundle":"https://pith.science/pith/IMVTWTQFE6IZU72DYBFP2HIN6L/bundle.json","state":"https://pith.science/pith/IMVTWTQFE6IZU72DYBFP2HIN6L/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IMVTWTQFE6IZU72DYBFP2HIN6L/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:IMVTWTQFE6IZU72DYBFP2HIN6L","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":"8803d70934415ac5cf6af0736da4cf68b1e0de0267b5485ef798ee03c8f47caf","cross_cats_sorted":["math.MG"],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.DG","submitted_at":"2008-04-04T12:06:19Z","title_canon_sha256":"bd0ceeb902bbced737403921c9e669e9f3bde8a7421ae129624b4c1cf872e5df"},"schema_version":"1.0","source":{"id":"0804.0705","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0804.0705","created_at":"2026-05-18T02:15:54Z"},{"alias_kind":"arxiv_version","alias_value":"0804.0705v1","created_at":"2026-05-18T02:15:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0804.0705","created_at":"2026-05-18T02:15:54Z"},{"alias_kind":"pith_short_12","alias_value":"IMVTWTQFE6IZ","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_16","alias_value":"IMVTWTQFE6IZU72D","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_8","alias_value":"IMVTWTQF","created_at":"2026-05-18T12:25:57Z"}],"graph_snapshots":[{"event_id":"sha256:fbe1c514733ff0bbbd54ad92f8f45b648bd7f2716e9d0ee4b075c91cd8529e43","target":"graph","created_at":"2026-05-18T02:15:54Z","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 discuss general notions of metrics and of Finsler structures which we call weak metrics and weak Finsler structures. Any convex domain carries a canonical weak Finsler structure, which we call its tautological weak Finsler structure. We compute distances in the tautological weak Finsler structure of a domain and we show that these are given by the so-called Funk weak metric. We conclude the paper with a discussion of geodesics, of metric balls and of convexity properties of the Funk weak metric.","authors_text":"Athanase Papadopoulos, Marc Troyanov","cross_cats":["math.MG"],"headline":"","license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.DG","submitted_at":"2008-04-04T12:06:19Z","title":"Weak Finsler Strutures and the Funk Metric"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0804.0705","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:cfdedddfe29a09036e04889303ad8f28505f33f0d7134de4218cee1c501ceab5","target":"record","created_at":"2026-05-18T02:15:54Z","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":"8803d70934415ac5cf6af0736da4cf68b1e0de0267b5485ef798ee03c8f47caf","cross_cats_sorted":["math.MG"],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.DG","submitted_at":"2008-04-04T12:06:19Z","title_canon_sha256":"bd0ceeb902bbced737403921c9e669e9f3bde8a7421ae129624b4c1cf872e5df"},"schema_version":"1.0","source":{"id":"0804.0705","kind":"arxiv","version":1}},"canonical_sha256":"432b3b4e0527919a7f43c04afd1d0df2ef705654c7c2801d88e18e474d687c62","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"432b3b4e0527919a7f43c04afd1d0df2ef705654c7c2801d88e18e474d687c62","first_computed_at":"2026-05-18T02:15:54.023932Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:15:54.023932Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NdlvrB8O2epAMumSZ0OXPZKwFO4sOCH7wjvAhPiR4HK2P7ZsmLATxVvm2da6Gzn8YbIuccK0uBR+XudVCupQBA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:15:54.024625Z","signed_message":"canonical_sha256_bytes"},"source_id":"0804.0705","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cfdedddfe29a09036e04889303ad8f28505f33f0d7134de4218cee1c501ceab5","sha256:fbe1c514733ff0bbbd54ad92f8f45b648bd7f2716e9d0ee4b075c91cd8529e43"],"state_sha256":"742009af2ccbc5caf428064f3708f0efde6798f79b7ec3f6734024487a3d732a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+X6/qxgdch1qW71+1Z3gJqzo2BnIjaywpEpbn7OeC4aCI6gY6HwM8++cGS4PTBmngXxv7C53CMXcgwKTPuscAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T07:49:33.472916Z","bundle_sha256":"fbcca1440c5fc6eb47caebbb562a0010cd0de13a5d5633f3d22b56a3807b930e"}}