{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:YZ6HDQVN5EAFXPSEOBU27GSCRY","short_pith_number":"pith:YZ6HDQVN","canonical_record":{"source":{"id":"1301.1782","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2013-01-09T09:04:54Z","cross_cats_sorted":["math.AP","math.PR"],"title_canon_sha256":"3725f9140c558d39429057f0c1d32d88e71b032bc78700963e7e8fa8984c22f9","abstract_canon_sha256":"101357cd16f08fb019b66aa0b6b9728d017a85e26b689a3b2de7400ce9de11c1"},"schema_version":"1.0"},"canonical_sha256":"c67c71c2ade9005bbe447069af9a428e37789a8abfc4ad3c80ed0f8e333f7001","source":{"kind":"arxiv","id":"1301.1782","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"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:YZ6HDQVN5EAFXPSEOBU27GSCRY","target":"record","payload":{"canonical_record":{"source":{"id":"1301.1782","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2013-01-09T09:04:54Z","cross_cats_sorted":["math.AP","math.PR"],"title_canon_sha256":"3725f9140c558d39429057f0c1d32d88e71b032bc78700963e7e8fa8984c22f9","abstract_canon_sha256":"101357cd16f08fb019b66aa0b6b9728d017a85e26b689a3b2de7400ce9de11c1"},"schema_version":"1.0"},"canonical_sha256":"c67c71c2ade9005bbe447069af9a428e37789a8abfc4ad3c80ed0f8e333f7001","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:45:58.383860Z","signature_b64":"U4tCytbnBkF4N4VabBxjD1OwXFa6NnY6l3hxy1FaCFwKZw1nCybswU0G+XphN1zSsXxR/yl2byzWpTF5YM9DAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c67c71c2ade9005bbe447069af9a428e37789a8abfc4ad3c80ed0f8e333f7001","last_reissued_at":"2026-05-18T02:45:58.383462Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:45:58.383462Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1301.1782","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:45:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Zin+xlBPoAi4R5ET9wV010zcjZQyDoMcD9mmFozLKB3oDLHsk67YGhYQ3fdlDLdc9l6mIljYED4OvVGmE/9CDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-21T20:55:13.118427Z"},"content_sha256":"5feecb8c6daa97177452d81579f9a4956352434ae5ae1285948670216791fea5","schema_version":"1.0","event_id":"sha256:5feecb8c6daa97177452d81579f9a4956352434ae5ae1285948670216791fea5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:YZ6HDQVN5EAFXPSEOBU27GSCRY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Existence and uniqueness of optimal transport maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.PR"],"primary_cat":"math.MG","authors_text":"Fabio Cavalletti, Martin Huesmann","submitted_at":"2013-01-09T09:04:54Z","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)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.1782","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"2026-05-18T02:45:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eACtYWz7lTvbE0X25vAo7PCl1CMexqaZSwwYtOkMuKrIVbpFc8hFbgcAI08ypF/hKyOR0bnyS5lNIdB5NQsPBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-21T20:55:13.118855Z"},"content_sha256":"bb0e5922d95617c4e394f2d73de4508b0c3fb647ac47b6f3d926fc1bd0065f6e","schema_version":"1.0","event_id":"sha256:bb0e5922d95617c4e394f2d73de4508b0c3fb647ac47b6f3d926fc1bd0065f6e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YZ6HDQVN5EAFXPSEOBU27GSCRY/bundle.json","state_url":"https://pith.science/pith/YZ6HDQVN5EAFXPSEOBU27GSCRY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YZ6HDQVN5EAFXPSEOBU27GSCRY/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":"2026-05-21T20:55:13Z","links":{"resolver":"https://pith.science/pith/YZ6HDQVN5EAFXPSEOBU27GSCRY","bundle":"https://pith.science/pith/YZ6HDQVN5EAFXPSEOBU27GSCRY/bundle.json","state":"https://pith.science/pith/YZ6HDQVN5EAFXPSEOBU27GSCRY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YZ6HDQVN5EAFXPSEOBU27GSCRY/bundle.json"},"state":{"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"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uwaKTRVGT3JUUBQGwUBPQZGgWea5EGFjsO19c/loO9+wz5v8U59jzunhzZJy/uTz5/+mROLQLxJ9E5KhA1DCBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-21T20:55:13.121162Z","bundle_sha256":"d3dc16bd8c0fa4c5d0fdaf48b1b7d421546eba0ad60ea7ef7fedbd3872070738"}}