{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:FVAJ3FYSF6AQZFPNWZVIO6MATK","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":"49a6ecde7a914867a40590569778ce0c398719c54322a3123a87459063acb29d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-04-19T19:57:25Z","title_canon_sha256":"ce3ea6c740ec3b40f0f92661f826e8b79f7e484b334b3ba2aa4dfcddcc27e823"},"schema_version":"1.0","source":{"id":"1504.04878","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.04878","created_at":"2026-05-18T01:37:07Z"},{"alias_kind":"arxiv_version","alias_value":"1504.04878v3","created_at":"2026-05-18T01:37:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.04878","created_at":"2026-05-18T01:37:07Z"},{"alias_kind":"pith_short_12","alias_value":"FVAJ3FYSF6AQ","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_16","alias_value":"FVAJ3FYSF6AQZFPN","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_8","alias_value":"FVAJ3FYS","created_at":"2026-05-18T12:29:22Z"}],"graph_snapshots":[{"event_id":"sha256:9d5e3ee8c0530b244f1b9b018478992c97b3a1a1153541538852a2b8629bb57b","target":"graph","created_at":"2026-05-18T01:37:07Z","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":"In this paper we present new versions of the classical Brunn-Minkowski inequality for different classes of measures and sets. We show that the inequality \\[\n  \\mu(\\lambda A + (1-\\lambda)B)^{1/n} \\geq \\lambda \\mu(A)^{1/n} + (1-\\lambda)\\mu(B)^{1/n} \\] holds true for an unconditional product measure $\\mu$ with decreasing density and a pair of unconditional convex bodies $A,B \\subset \\mathbb{R}^n$. We also show that the above inequality is true for any unconditional $\\log$-concave measure $\\mu$ and unconditional convex bodies $A,B \\subset \\mathbb{R}^n$. Finally, we prove that the inequality is tru","authors_text":"Arnaud Marsiglietti, Artem Zvavitch, Galyna Livshyts, Piotr Nayar","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-04-19T19:57:25Z","title":"On the Brunn-Minkowski inequality for general measures with applications to new isoperimetric-type inequalities"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.04878","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:b040727c05708d6e08948ff687c20d34e1247df04de0f1e089bca57f4fb2fbb1","target":"record","created_at":"2026-05-18T01:37:07Z","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":"49a6ecde7a914867a40590569778ce0c398719c54322a3123a87459063acb29d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-04-19T19:57:25Z","title_canon_sha256":"ce3ea6c740ec3b40f0f92661f826e8b79f7e484b334b3ba2aa4dfcddcc27e823"},"schema_version":"1.0","source":{"id":"1504.04878","kind":"arxiv","version":3}},"canonical_sha256":"2d409d97122f810c95edb66a8779809a89e271ebcc633f695915bfc0cde7ec78","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2d409d97122f810c95edb66a8779809a89e271ebcc633f695915bfc0cde7ec78","first_computed_at":"2026-05-18T01:37:07.375805Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:37:07.375805Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GJ9TN1xvi8ErWpHnypTrcPhhFNxPT9GhW6rukKTu8TjSGzKGaIObHpOaMKnQnPyqKhN+agy76VfEziIGJ/ziAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:37:07.376260Z","signed_message":"canonical_sha256_bytes"},"source_id":"1504.04878","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b040727c05708d6e08948ff687c20d34e1247df04de0f1e089bca57f4fb2fbb1","sha256:9d5e3ee8c0530b244f1b9b018478992c97b3a1a1153541538852a2b8629bb57b"],"state_sha256":"bac1cc452f11ddd8b87099961c52ab453c49602c4ba2ccd15821c0c141c695df"}