{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:YAX23QVGGVZ6J3YH6BBPEXFPBZ","short_pith_number":"pith:YAX23QVG","canonical_record":{"source":{"id":"2507.06759","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2025-07-09T11:36:29Z","cross_cats_sorted":["math.MG","math.PR"],"title_canon_sha256":"b8b3340ebe064435e4159c0a2814080fbc72da99e45cf05d6f9b9def3bd718e4","abstract_canon_sha256":"69adc87deaf97950c4bcff7cbbdf0d765aa0d70b9dc4b478ba5b1c054b1fa286"},"schema_version":"1.0"},"canonical_sha256":"c02fadc2a63573e4ef07f042f25caf0e621ca277327e570d02fb5443da21104c","source":{"kind":"arxiv","id":"2507.06759","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.06759","created_at":"2026-07-05T11:38:09Z"},{"alias_kind":"arxiv_version","alias_value":"2507.06759v3","created_at":"2026-07-05T11:38:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.06759","created_at":"2026-07-05T11:38:09Z"},{"alias_kind":"pith_short_12","alias_value":"YAX23QVGGVZ6","created_at":"2026-07-05T11:38:09Z"},{"alias_kind":"pith_short_16","alias_value":"YAX23QVGGVZ6J3YH","created_at":"2026-07-05T11:38:09Z"},{"alias_kind":"pith_short_8","alias_value":"YAX23QVG","created_at":"2026-07-05T11:38:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:YAX23QVGGVZ6J3YH6BBPEXFPBZ","target":"record","payload":{"canonical_record":{"source":{"id":"2507.06759","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2025-07-09T11:36:29Z","cross_cats_sorted":["math.MG","math.PR"],"title_canon_sha256":"b8b3340ebe064435e4159c0a2814080fbc72da99e45cf05d6f9b9def3bd718e4","abstract_canon_sha256":"69adc87deaf97950c4bcff7cbbdf0d765aa0d70b9dc4b478ba5b1c054b1fa286"},"schema_version":"1.0"},"canonical_sha256":"c02fadc2a63573e4ef07f042f25caf0e621ca277327e570d02fb5443da21104c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:38:09.013583Z","signature_b64":"6ZMlTJ1tv2it2pEtgekLPRbQu+G3Q04Ckrh05Z4GH5xOzGpjsDo8C6+qwQhOiNUuDkLtgwSa43of2WU8hW6WAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c02fadc2a63573e4ef07f042f25caf0e621ca277327e570d02fb5443da21104c","last_reissued_at":"2026-07-05T11:38:09.013050Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:38:09.013050Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2507.06759","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T11:38:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6hNCgg1NX33e/LoOxqSM4KXGo58duk1CJbAs8ByqHSHv7EJGsXO5ZoV28EnRkwmehXWmdmeDsuv4tYOfs8vjAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T06:19:28.626139Z"},"content_sha256":"9e5512129c55cd3d8a02bb182dc6c869cba8cb56d5ddf1a149d2395c3b5513df","schema_version":"1.0","event_id":"sha256:9e5512129c55cd3d8a02bb182dc6c869cba8cb56d5ddf1a149d2395c3b5513df"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:YAX23QVGGVZ6J3YH6BBPEXFPBZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Gr\\\"unbaum's inequality for Gaussian and convex probability measures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG","math.PR"],"primary_cat":"math.FA","authors_text":"Dylan Langharst, Francisco Mar\\'in Sola, Jiaqian Liu, Matthieu Fradelizi, Shengyu Tang","submitted_at":"2025-07-09T11:36:29Z","abstract_excerpt":"A celebrated result in convex geometry is Gr\\\"unbaum's inequality, which quantifies how much volume of a convex body can be cut off by a hyperplane passing through its barycenter. In this work, we establish a series of sharp Gr\\\"unbaum-type inequalities - with equality characterizations - for probability measures under certain concavity assumptions. As an application, we apply the renowned Ehrhard inequality and deduce an ``Ehrhard-Gr\\\"unbaum'' inequality for the Gaussian measure on $\\mathbb{R}^n$, which improves upon the bound derived from its log-concavity.\n  For $s$-concave Radon measures, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.06759","kind":"arxiv","version":3},"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/2507.06759/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":"2026-07-05T11:38:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yYmUzExg1or7fYopWbSn/Dh/h77LGWKwutPNx4nDXMC43dDbrC+ncIjjqn0esahTUloQyVwZBU/wtfmmloaDAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T06:19:28.626732Z"},"content_sha256":"9ad48a29bcce1564e48dbfc12f2ea8f5a7a6b49b5d5cb27c07e5511bcad6d135","schema_version":"1.0","event_id":"sha256:9ad48a29bcce1564e48dbfc12f2ea8f5a7a6b49b5d5cb27c07e5511bcad6d135"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YAX23QVGGVZ6J3YH6BBPEXFPBZ/bundle.json","state_url":"https://pith.science/pith/YAX23QVGGVZ6J3YH6BBPEXFPBZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YAX23QVGGVZ6J3YH6BBPEXFPBZ/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":"2026-08-06T06:19:28Z","links":{"resolver":"https://pith.science/pith/YAX23QVGGVZ6J3YH6BBPEXFPBZ","bundle":"https://pith.science/pith/YAX23QVGGVZ6J3YH6BBPEXFPBZ/bundle.json","state":"https://pith.science/pith/YAX23QVGGVZ6J3YH6BBPEXFPBZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YAX23QVGGVZ6J3YH6BBPEXFPBZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:YAX23QVGGVZ6J3YH6BBPEXFPBZ","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":"69adc87deaf97950c4bcff7cbbdf0d765aa0d70b9dc4b478ba5b1c054b1fa286","cross_cats_sorted":["math.MG","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2025-07-09T11:36:29Z","title_canon_sha256":"b8b3340ebe064435e4159c0a2814080fbc72da99e45cf05d6f9b9def3bd718e4"},"schema_version":"1.0","source":{"id":"2507.06759","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.06759","created_at":"2026-07-05T11:38:09Z"},{"alias_kind":"arxiv_version","alias_value":"2507.06759v3","created_at":"2026-07-05T11:38:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.06759","created_at":"2026-07-05T11:38:09Z"},{"alias_kind":"pith_short_12","alias_value":"YAX23QVGGVZ6","created_at":"2026-07-05T11:38:09Z"},{"alias_kind":"pith_short_16","alias_value":"YAX23QVGGVZ6J3YH","created_at":"2026-07-05T11:38:09Z"},{"alias_kind":"pith_short_8","alias_value":"YAX23QVG","created_at":"2026-07-05T11:38:09Z"}],"graph_snapshots":[{"event_id":"sha256:9ad48a29bcce1564e48dbfc12f2ea8f5a7a6b49b5d5cb27c07e5511bcad6d135","target":"graph","created_at":"2026-07-05T11:38:09Z","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/2507.06759/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A celebrated result in convex geometry is Gr\\\"unbaum's inequality, which quantifies how much volume of a convex body can be cut off by a hyperplane passing through its barycenter. In this work, we establish a series of sharp Gr\\\"unbaum-type inequalities - with equality characterizations - for probability measures under certain concavity assumptions. As an application, we apply the renowned Ehrhard inequality and deduce an ``Ehrhard-Gr\\\"unbaum'' inequality for the Gaussian measure on $\\mathbb{R}^n$, which improves upon the bound derived from its log-concavity.\n  For $s$-concave Radon measures, ","authors_text":"Dylan Langharst, Francisco Mar\\'in Sola, Jiaqian Liu, Matthieu Fradelizi, Shengyu Tang","cross_cats":["math.MG","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2025-07-09T11:36:29Z","title":"Gr\\\"unbaum's inequality for Gaussian and convex probability measures"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.06759","kind":"arxiv","version":3},"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":"sha256:9e5512129c55cd3d8a02bb182dc6c869cba8cb56d5ddf1a149d2395c3b5513df","target":"record","created_at":"2026-07-05T11:38:09Z","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":"69adc87deaf97950c4bcff7cbbdf0d765aa0d70b9dc4b478ba5b1c054b1fa286","cross_cats_sorted":["math.MG","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2025-07-09T11:36:29Z","title_canon_sha256":"b8b3340ebe064435e4159c0a2814080fbc72da99e45cf05d6f9b9def3bd718e4"},"schema_version":"1.0","source":{"id":"2507.06759","kind":"arxiv","version":3}},"canonical_sha256":"c02fadc2a63573e4ef07f042f25caf0e621ca277327e570d02fb5443da21104c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c02fadc2a63573e4ef07f042f25caf0e621ca277327e570d02fb5443da21104c","first_computed_at":"2026-07-05T11:38:09.013050Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:38:09.013050Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6ZMlTJ1tv2it2pEtgekLPRbQu+G3Q04Ckrh05Z4GH5xOzGpjsDo8C6+qwQhOiNUuDkLtgwSa43of2WU8hW6WAA==","signature_status":"signed_v1","signed_at":"2026-07-05T11:38:09.013583Z","signed_message":"canonical_sha256_bytes"},"source_id":"2507.06759","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9e5512129c55cd3d8a02bb182dc6c869cba8cb56d5ddf1a149d2395c3b5513df","sha256:9ad48a29bcce1564e48dbfc12f2ea8f5a7a6b49b5d5cb27c07e5511bcad6d135"],"state_sha256":"850b9597376f79ac58af3fe67d3f0d0547eb93bdeab8261d12f030c64f2c1b5d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tBCTcovEcGzKnusmr2c1EFpU7tjXRPhtKff77B/krapu6VoXrLfenqC7GLkGZlxVu2TbrNZpQ+O80Re3+XFlCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-06T06:19:28.631533Z","bundle_sha256":"a8167e6fab381ffe3732bc6cece1fdd4472f9d64f8a7918773add164ac2a6817"}}