{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:3LDICW5RERMUVAPVZTHJNFYPKR","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":"224964ec36605d1764069d4b0837541ad843929a2cd7473e48eb288736106ec8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-09-13T09:44:29Z","title_canon_sha256":"af5d2dca9ef95ef19e18c7fc6c9b9b33d81a0c20d8536ba1e752d3928f73ec76"},"schema_version":"1.0","source":{"id":"1809.04859","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.04859","created_at":"2026-05-18T00:05:47Z"},{"alias_kind":"arxiv_version","alias_value":"1809.04859v1","created_at":"2026-05-18T00:05:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.04859","created_at":"2026-05-18T00:05:47Z"},{"alias_kind":"pith_short_12","alias_value":"3LDICW5RERMU","created_at":"2026-05-18T12:32:02Z"},{"alias_kind":"pith_short_16","alias_value":"3LDICW5RERMUVAPV","created_at":"2026-05-18T12:32:02Z"},{"alias_kind":"pith_short_8","alias_value":"3LDICW5R","created_at":"2026-05-18T12:32:02Z"}],"graph_snapshots":[{"event_id":"sha256:bbda74068fb67a848c2b6413153d966162ac3a10ff42e49593845eb85538ad97","target":"graph","created_at":"2026-05-18T00:05:47Z","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":"The scope of this note is to make a self-contained survey of the recent developments and achievements of the theory of L1-Optimal Transportation on metric measure spaces. Among the results proved in the recent papers [20, 21] where the author, together with A. Mondino, proved a series of sharp (and in some cases rigid) geometric and functional inequalities in the setting of metric measure spaces enjoying a weak form of Ricci curvature lower bound, we review the proof of the L\\'evy-Gromov isoperimetric inequality.","authors_text":"Fabio Cavalletti","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-09-13T09:44:29Z","title":"An overview of L1 Optimal Transportation on metric measure spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.04859","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":"sha256:52413a774e70739c421e2a5264cdbb83c0d5c196d90a2e70426be4946431a498","target":"record","created_at":"2026-05-18T00:05:47Z","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":"224964ec36605d1764069d4b0837541ad843929a2cd7473e48eb288736106ec8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-09-13T09:44:29Z","title_canon_sha256":"af5d2dca9ef95ef19e18c7fc6c9b9b33d81a0c20d8536ba1e752d3928f73ec76"},"schema_version":"1.0","source":{"id":"1809.04859","kind":"arxiv","version":1}},"canonical_sha256":"dac6815bb124594a81f5ccce96970f544ee95a7b24cf915ae0f48a3ab628ab9d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dac6815bb124594a81f5ccce96970f544ee95a7b24cf915ae0f48a3ab628ab9d","first_computed_at":"2026-05-18T00:05:47.281206Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:05:47.281206Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"F5/7ZtNS45EAXasDz+T1fgPUKliRYT0D3gCpe2XjbSSzpH+qPXZwbP/8qSeC1G8FmP5GXN0EobyLzoBY2TuRAg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:05:47.281870Z","signed_message":"canonical_sha256_bytes"},"source_id":"1809.04859","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:52413a774e70739c421e2a5264cdbb83c0d5c196d90a2e70426be4946431a498","sha256:bbda74068fb67a848c2b6413153d966162ac3a10ff42e49593845eb85538ad97"],"state_sha256":"c20709f3a905f0276623960c7d97bb52569a4c317c76758d9fbd0ee568db5cea"}