{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:RZPVR33GPGHGPN6YSEHY24WDHG","short_pith_number":"pith:RZPVR33G","canonical_record":{"source":{"id":"2407.10932","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-07-15T17:32:21Z","cross_cats_sorted":["math.CO","math.MG"],"title_canon_sha256":"b6f43cb848427087e61502244ba5d9af9f1121d1751e7d6403c71e3393e45fc7","abstract_canon_sha256":"0f5f97c75f1b8043ca21add41eee5748ed6f36a48978adceda4e8b17b9df6d2a"},"schema_version":"1.0"},"canonical_sha256":"8e5f58ef66798e67b7d8910f8d72c33999d260d1ccd2965e9136c584088a12f1","source":{"kind":"arxiv","id":"2407.10932","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2407.10932","created_at":"2026-07-05T08:44:06Z"},{"alias_kind":"arxiv_version","alias_value":"2407.10932v1","created_at":"2026-07-05T08:44:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2407.10932","created_at":"2026-07-05T08:44:06Z"},{"alias_kind":"pith_short_12","alias_value":"RZPVR33GPGHG","created_at":"2026-07-05T08:44:06Z"},{"alias_kind":"pith_short_16","alias_value":"RZPVR33GPGHGPN6Y","created_at":"2026-07-05T08:44:06Z"},{"alias_kind":"pith_short_8","alias_value":"RZPVR33G","created_at":"2026-07-05T08:44:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:RZPVR33GPGHGPN6YSEHY24WDHG","target":"record","payload":{"canonical_record":{"source":{"id":"2407.10932","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-07-15T17:32:21Z","cross_cats_sorted":["math.CO","math.MG"],"title_canon_sha256":"b6f43cb848427087e61502244ba5d9af9f1121d1751e7d6403c71e3393e45fc7","abstract_canon_sha256":"0f5f97c75f1b8043ca21add41eee5748ed6f36a48978adceda4e8b17b9df6d2a"},"schema_version":"1.0"},"canonical_sha256":"8e5f58ef66798e67b7d8910f8d72c33999d260d1ccd2965e9136c584088a12f1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:44:06.335609Z","signature_b64":"oCPvLpRs9GoQcvk58tWFuM1dZmbdAxWQaXlP9e/Du0/hsjBsTKooDxrpbyGkjJbOnHKWqy34Zpb8D0QJGILbDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8e5f58ef66798e67b7d8910f8d72c33999d260d1ccd2965e9136c584088a12f1","last_reissued_at":"2026-07-05T08:44:06.335132Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:44:06.335132Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2407.10932","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":"2026-07-05T08:44:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hV2j/9pofCP+OlqsVUH3efIaJPMYuXayh94aQWP1Et6QQTCv98wuF+pl/IOynPnhjqoJu1C8cey6Y3pLLQ6SDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T21:36:30.390962Z"},"content_sha256":"26dc630f0783da4dd0e692d90adb0d679489b702d834ff26895e75a39dcd9ca5","schema_version":"1.0","event_id":"sha256:26dc630f0783da4dd0e692d90adb0d679489b702d834ff26895e75a39dcd9ca5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:RZPVR33GPGHGPN6YSEHY24WDHG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sharp stability of the Brunn-Minkowski inequality via optimal mass transportation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.MG"],"primary_cat":"math.AP","authors_text":"Alessio Figalli, Marius Tiba, Peter van Hintum","submitted_at":"2024-07-15T17:32:21Z","abstract_excerpt":"The Brunn-Minkowski inequality, applicable to bounded measurable sets $A$ and $B$ in $\\mathbb{R}^d$, states that $|A+B|^{1/d} \\geq |A|^{1/d}+|B|^{1/d}$. Equality is achieved if and only if $A$ and $B$ are convex and homothetic sets in $\\mathbb{R}^d$. The concept of stability in this context concerns how, when approaching equality, sets $A$ and $B$ are close to homothetic convex sets. In a recent breakthrough [FvHT23], the authors of this paper proved the following folklore conjectures on the sharp stability for the Brunn-Minkowski inequality:\n  (1) A linear stability result concerning the dist"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.10932","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2407.10932/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T08:44:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"R2aeOVTHmUIRBZmWL7eahmR73Gb4GThvK7sJApelNopSduayYmlGRzu/UtVSzr4KacZ85Uy2gbuO0TQsTI0+Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T21:36:30.391454Z"},"content_sha256":"2015d23cb3f2372c9b6126eeba75051707bdc9b8ab80e8ec19793a91f009e62d","schema_version":"1.0","event_id":"sha256:2015d23cb3f2372c9b6126eeba75051707bdc9b8ab80e8ec19793a91f009e62d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RZPVR33GPGHGPN6YSEHY24WDHG/bundle.json","state_url":"https://pith.science/pith/RZPVR33GPGHGPN6YSEHY24WDHG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RZPVR33GPGHGPN6YSEHY24WDHG/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":"2026-08-08T21:36:30Z","links":{"resolver":"https://pith.science/pith/RZPVR33GPGHGPN6YSEHY24WDHG","bundle":"https://pith.science/pith/RZPVR33GPGHGPN6YSEHY24WDHG/bundle.json","state":"https://pith.science/pith/RZPVR33GPGHGPN6YSEHY24WDHG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RZPVR33GPGHGPN6YSEHY24WDHG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:RZPVR33GPGHGPN6YSEHY24WDHG","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":"0f5f97c75f1b8043ca21add41eee5748ed6f36a48978adceda4e8b17b9df6d2a","cross_cats_sorted":["math.CO","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-07-15T17:32:21Z","title_canon_sha256":"b6f43cb848427087e61502244ba5d9af9f1121d1751e7d6403c71e3393e45fc7"},"schema_version":"1.0","source":{"id":"2407.10932","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2407.10932","created_at":"2026-07-05T08:44:06Z"},{"alias_kind":"arxiv_version","alias_value":"2407.10932v1","created_at":"2026-07-05T08:44:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2407.10932","created_at":"2026-07-05T08:44:06Z"},{"alias_kind":"pith_short_12","alias_value":"RZPVR33GPGHG","created_at":"2026-07-05T08:44:06Z"},{"alias_kind":"pith_short_16","alias_value":"RZPVR33GPGHGPN6Y","created_at":"2026-07-05T08:44:06Z"},{"alias_kind":"pith_short_8","alias_value":"RZPVR33G","created_at":"2026-07-05T08:44:06Z"}],"graph_snapshots":[{"event_id":"sha256:2015d23cb3f2372c9b6126eeba75051707bdc9b8ab80e8ec19793a91f009e62d","target":"graph","created_at":"2026-07-05T08:44:06Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2407.10932/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The Brunn-Minkowski inequality, applicable to bounded measurable sets $A$ and $B$ in $\\mathbb{R}^d$, states that $|A+B|^{1/d} \\geq |A|^{1/d}+|B|^{1/d}$. Equality is achieved if and only if $A$ and $B$ are convex and homothetic sets in $\\mathbb{R}^d$. The concept of stability in this context concerns how, when approaching equality, sets $A$ and $B$ are close to homothetic convex sets. In a recent breakthrough [FvHT23], the authors of this paper proved the following folklore conjectures on the sharp stability for the Brunn-Minkowski inequality:\n  (1) A linear stability result concerning the dist","authors_text":"Alessio Figalli, Marius Tiba, Peter van Hintum","cross_cats":["math.CO","math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-07-15T17:32:21Z","title":"Sharp stability of the Brunn-Minkowski inequality via optimal mass transportation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.10932","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":"sha256:26dc630f0783da4dd0e692d90adb0d679489b702d834ff26895e75a39dcd9ca5","target":"record","created_at":"2026-07-05T08:44:06Z","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":"0f5f97c75f1b8043ca21add41eee5748ed6f36a48978adceda4e8b17b9df6d2a","cross_cats_sorted":["math.CO","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-07-15T17:32:21Z","title_canon_sha256":"b6f43cb848427087e61502244ba5d9af9f1121d1751e7d6403c71e3393e45fc7"},"schema_version":"1.0","source":{"id":"2407.10932","kind":"arxiv","version":1}},"canonical_sha256":"8e5f58ef66798e67b7d8910f8d72c33999d260d1ccd2965e9136c584088a12f1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8e5f58ef66798e67b7d8910f8d72c33999d260d1ccd2965e9136c584088a12f1","first_computed_at":"2026-07-05T08:44:06.335132Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:44:06.335132Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"oCPvLpRs9GoQcvk58tWFuM1dZmbdAxWQaXlP9e/Du0/hsjBsTKooDxrpbyGkjJbOnHKWqy34Zpb8D0QJGILbDA==","signature_status":"signed_v1","signed_at":"2026-07-05T08:44:06.335609Z","signed_message":"canonical_sha256_bytes"},"source_id":"2407.10932","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:26dc630f0783da4dd0e692d90adb0d679489b702d834ff26895e75a39dcd9ca5","sha256:2015d23cb3f2372c9b6126eeba75051707bdc9b8ab80e8ec19793a91f009e62d"],"state_sha256":"a551372748a932cbcce12167561c4f734133b15bfef0906777bf105b71565c28"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5X1zcAYBzlppXVvaWiu9NkFknwrax3k3Y0Nd093fxrHhkZddXkCqn0viNIe1PxL+/67Iok3JK3yZY1YfFcVeDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T21:36:30.394816Z","bundle_sha256":"22e25ec8233728c676b32496f8412680c6c87f9365d90ba976b93ca79194c63b"}}