{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:YZ6HDQVN5EAFXPSEOBU27GSCRY","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":"101357cd16f08fb019b66aa0b6b9728d017a85e26b689a3b2de7400ce9de11c1","cross_cats_sorted":["math.AP","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2013-01-09T09:04:54Z","title_canon_sha256":"3725f9140c558d39429057f0c1d32d88e71b032bc78700963e7e8fa8984c22f9"},"schema_version":"1.0","source":{"id":"1301.1782","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1301.1782","created_at":"2026-05-18T02:45:58Z"},{"alias_kind":"arxiv_version","alias_value":"1301.1782v2","created_at":"2026-05-18T02:45:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1301.1782","created_at":"2026-05-18T02:45:58Z"},{"alias_kind":"pith_short_12","alias_value":"YZ6HDQVN5EAF","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_16","alias_value":"YZ6HDQVN5EAFXPSE","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_8","alias_value":"YZ6HDQVN","created_at":"2026-05-18T12:28:09Z"}],"graph_snapshots":[{"event_id":"sha256:bb0e5922d95617c4e394f2d73de4508b0c3fb647ac47b6f3d926fc1bd0065f6e","target":"graph","created_at":"2026-05-18T02:45:58Z","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":"Let $(X,d,m)$ be a proper, non-branching, metric measure space. We show existence and uniqueness of optimal transport maps for cost written as non-decreasing and strictly convex functions of the distance, provided $(X,d,m)$ satisfies a new weak property concerning the behavior of $m$ under the shrinking of sets to points, see Assumption 1. This in particular covers spaces satisfying the measure contraction property.\n  We also prove a stability property for Assumption 1: If $(X,d,m)$ satisfies Assumption 1 and $\\tilde m = g\\cdot m$, for some continuous function $g >0$, then also $(X,d,\\tilde m)","authors_text":"Fabio Cavalletti, Martin Huesmann","cross_cats":["math.AP","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2013-01-09T09:04:54Z","title":"Existence and uniqueness of optimal transport maps"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.1782","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":"sha256:5feecb8c6daa97177452d81579f9a4956352434ae5ae1285948670216791fea5","target":"record","created_at":"2026-05-18T02:45:58Z","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":"101357cd16f08fb019b66aa0b6b9728d017a85e26b689a3b2de7400ce9de11c1","cross_cats_sorted":["math.AP","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2013-01-09T09:04:54Z","title_canon_sha256":"3725f9140c558d39429057f0c1d32d88e71b032bc78700963e7e8fa8984c22f9"},"schema_version":"1.0","source":{"id":"1301.1782","kind":"arxiv","version":2}},"canonical_sha256":"c67c71c2ade9005bbe447069af9a428e37789a8abfc4ad3c80ed0f8e333f7001","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c67c71c2ade9005bbe447069af9a428e37789a8abfc4ad3c80ed0f8e333f7001","first_computed_at":"2026-05-18T02:45:58.383462Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:45:58.383462Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"U4tCytbnBkF4N4VabBxjD1OwXFa6NnY6l3hxy1FaCFwKZw1nCybswU0G+XphN1zSsXxR/yl2byzWpTF5YM9DAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:45:58.383860Z","signed_message":"canonical_sha256_bytes"},"source_id":"1301.1782","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5feecb8c6daa97177452d81579f9a4956352434ae5ae1285948670216791fea5","sha256:bb0e5922d95617c4e394f2d73de4508b0c3fb647ac47b6f3d926fc1bd0065f6e"],"state_sha256":"d29e11773411f8aa6e1674f24db4250e15eb07c72434ca0b783419a2c72ba8de"}