{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:NU2RQ5RB35UYF2OP7UFNVUYEGZ","short_pith_number":"pith:NU2RQ5RB","canonical_record":{"source":{"id":"1206.1965","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-06-09T19:21:19Z","cross_cats_sorted":[],"title_canon_sha256":"b09d13f3d52f1271b94a18c4ead316e6ab718bebc447b58af88a7e2fc4e805ea","abstract_canon_sha256":"3ad17576144d61aec95c5b2b68ad8ab9ae8e03586465f293c642310b7d81b098"},"schema_version":"1.0"},"canonical_sha256":"6d35187621df6982e9cffd0adad304367d69cf380dbb70d86eb9e85dae8b76b9","source":{"kind":"arxiv","id":"1206.1965","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.1965","created_at":"2026-05-18T03:50:24Z"},{"alias_kind":"arxiv_version","alias_value":"1206.1965v2","created_at":"2026-05-18T03:50:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.1965","created_at":"2026-05-18T03:50:24Z"},{"alias_kind":"pith_short_12","alias_value":"NU2RQ5RB35UY","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_16","alias_value":"NU2RQ5RB35UYF2OP","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_8","alias_value":"NU2RQ5RB","created_at":"2026-05-18T12:27:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:NU2RQ5RB35UYF2OP7UFNVUYEGZ","target":"record","payload":{"canonical_record":{"source":{"id":"1206.1965","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-06-09T19:21:19Z","cross_cats_sorted":[],"title_canon_sha256":"b09d13f3d52f1271b94a18c4ead316e6ab718bebc447b58af88a7e2fc4e805ea","abstract_canon_sha256":"3ad17576144d61aec95c5b2b68ad8ab9ae8e03586465f293c642310b7d81b098"},"schema_version":"1.0"},"canonical_sha256":"6d35187621df6982e9cffd0adad304367d69cf380dbb70d86eb9e85dae8b76b9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:50:24.977185Z","signature_b64":"wWTpsar5gbpeoqyQYPBHOzW8HZjGqC3FvIqHczVzDL4zhajEbhd2nbmGmbHid0uhvoZpNBPmzcg0MEzndOFjCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6d35187621df6982e9cffd0adad304367d69cf380dbb70d86eb9e85dae8b76b9","last_reissued_at":"2026-05-18T03:50:24.976437Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:50:24.976437Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1206.1965","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:50:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Hx0bylYoehGYCZr8TgHhhdkGH0FGVpoo4rZkY83625XctmWXUF2xq6TexiieKxcFSRmvhjIlW3ibbTxPZKjODA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-22T18:50:58.103149Z"},"content_sha256":"e27a87e92808ca3342b0b003a48cad33d578930492c83dbb70a77bc81bb2735c","schema_version":"1.0","event_id":"sha256:e27a87e92808ca3342b0b003a48cad33d578930492c83dbb70a77bc81bb2735c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:NU2RQ5RB35UYF2OP7UFNVUYEGZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Near equality in the two-dimensional Brunn-Minkowski inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Michael Christ","submitted_at":"2012-06-09T19:21:19Z","abstract_excerpt":"If a pair of subsets of two-dimensional Euclidean space nearly achieves equality in the Brunn-Minkowski inequality, in the sense that the measure of the associated sumset is nearly equal to the lower bound provided by the inequality, then these sets must nearly coincide with a pair of homothetic convex sets. The proof relies on a continuum analogue of a theorem of Freiman which characterizes finite sets of integers whose sumsets are of nearly minimal size.\n  Small corrections and clarifications have been made in this draft."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.1965","kind":"arxiv","version":2},"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":"2026-05-18T03:50:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OP8mBd7QQJ5PTIpUwg6wXq2lxUnm0rb/4/TIERXLh3ZgK+KUG22WxsqCt8XPWu/D6VcnMJHaoNYeqcA/ftNqAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-22T18:50:58.103609Z"},"content_sha256":"535acc4ef776b08dbcf00db28325551d8b71c3cc116453ffc169a00a0ec8318b","schema_version":"1.0","event_id":"sha256:535acc4ef776b08dbcf00db28325551d8b71c3cc116453ffc169a00a0ec8318b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NU2RQ5RB35UYF2OP7UFNVUYEGZ/bundle.json","state_url":"https://pith.science/pith/NU2RQ5RB35UYF2OP7UFNVUYEGZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NU2RQ5RB35UYF2OP7UFNVUYEGZ/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":"2026-08-22T18:50:58Z","links":{"resolver":"https://pith.science/pith/NU2RQ5RB35UYF2OP7UFNVUYEGZ","bundle":"https://pith.science/pith/NU2RQ5RB35UYF2OP7UFNVUYEGZ/bundle.json","state":"https://pith.science/pith/NU2RQ5RB35UYF2OP7UFNVUYEGZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NU2RQ5RB35UYF2OP7UFNVUYEGZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:NU2RQ5RB35UYF2OP7UFNVUYEGZ","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":"3ad17576144d61aec95c5b2b68ad8ab9ae8e03586465f293c642310b7d81b098","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-06-09T19:21:19Z","title_canon_sha256":"b09d13f3d52f1271b94a18c4ead316e6ab718bebc447b58af88a7e2fc4e805ea"},"schema_version":"1.0","source":{"id":"1206.1965","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.1965","created_at":"2026-05-18T03:50:24Z"},{"alias_kind":"arxiv_version","alias_value":"1206.1965v2","created_at":"2026-05-18T03:50:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.1965","created_at":"2026-05-18T03:50:24Z"},{"alias_kind":"pith_short_12","alias_value":"NU2RQ5RB35UY","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_16","alias_value":"NU2RQ5RB35UYF2OP","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_8","alias_value":"NU2RQ5RB","created_at":"2026-05-18T12:27:16Z"}],"graph_snapshots":[{"event_id":"sha256:535acc4ef776b08dbcf00db28325551d8b71c3cc116453ffc169a00a0ec8318b","target":"graph","created_at":"2026-05-18T03:50:24Z","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":"If a pair of subsets of two-dimensional Euclidean space nearly achieves equality in the Brunn-Minkowski inequality, in the sense that the measure of the associated sumset is nearly equal to the lower bound provided by the inequality, then these sets must nearly coincide with a pair of homothetic convex sets. The proof relies on a continuum analogue of a theorem of Freiman which characterizes finite sets of integers whose sumsets are of nearly minimal size.\n  Small corrections and clarifications have been made in this draft.","authors_text":"Michael Christ","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-06-09T19:21:19Z","title":"Near equality in the two-dimensional Brunn-Minkowski inequality"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.1965","kind":"arxiv","version":2},"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:e27a87e92808ca3342b0b003a48cad33d578930492c83dbb70a77bc81bb2735c","target":"record","created_at":"2026-05-18T03:50:24Z","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":"3ad17576144d61aec95c5b2b68ad8ab9ae8e03586465f293c642310b7d81b098","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-06-09T19:21:19Z","title_canon_sha256":"b09d13f3d52f1271b94a18c4ead316e6ab718bebc447b58af88a7e2fc4e805ea"},"schema_version":"1.0","source":{"id":"1206.1965","kind":"arxiv","version":2}},"canonical_sha256":"6d35187621df6982e9cffd0adad304367d69cf380dbb70d86eb9e85dae8b76b9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6d35187621df6982e9cffd0adad304367d69cf380dbb70d86eb9e85dae8b76b9","first_computed_at":"2026-05-18T03:50:24.976437Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:50:24.976437Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wWTpsar5gbpeoqyQYPBHOzW8HZjGqC3FvIqHczVzDL4zhajEbhd2nbmGmbHid0uhvoZpNBPmzcg0MEzndOFjCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:50:24.977185Z","signed_message":"canonical_sha256_bytes"},"source_id":"1206.1965","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e27a87e92808ca3342b0b003a48cad33d578930492c83dbb70a77bc81bb2735c","sha256:535acc4ef776b08dbcf00db28325551d8b71c3cc116453ffc169a00a0ec8318b"],"state_sha256":"425163beb5dd6552518a43d765bfd49a232222104af6e34018879b933806e992"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8g58zxR4iHPsHxDid3PB5kvsEndoN+vFi6IQeO9Di2uuJcSte3E2kxgmdBip1rL9/lX1GeCLC0bQCE456kjkCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-22T18:50:58.107774Z","bundle_sha256":"ed9c6e6b2e310ea47f7172e2a17755444715d2d169d4748ebc7b78c35df44115"}}