{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:Y55XLQ7KNGE323SABFYYVZ3Q3N","short_pith_number":"pith:Y55XLQ7K","schema_version":"1.0","canonical_sha256":"c77b75c3ea6989bd6e4009718ae770db6e56314bfde32b5c735f2eb0a9ff4974","source":{"kind":"arxiv","id":"1002.0373","version":3},"attestation_state":"computed","paper":{"title":"A Generalization of Caffarelli's Contraction Theorem via (reverse) Heat Flow","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Emanuel Milman, Young-Heon Kim","submitted_at":"2010-02-02T03:55:11Z","abstract_excerpt":"A theorem of L. Caffarelli implies the existence of a map pushing forward a source Gaussian measure to a target measure which is more log-concave than the source one, which contracts Euclidean distance (in fact, Caffarelli showed that the optimal-transport Brenier map $T_{opt}$ is a contraction in this case). We generalize this result to more general source and target measures, using a condition on the third derivative of the potential, using two different proofs. The first uses a map $T$, whose inverse is constructed as a flow along an advection field associated to an appropriate heat-diffusi"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1002.0373","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-02-02T03:55:11Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"3599f555ae7fbb0984535fdebdefcdc7262ebda3c679f0f55f7b3208bfc9a920","abstract_canon_sha256":"67297a83751ecba411d8ae75db83fc05907d44e29c8975988ca659b60f75401f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:17:25.170991Z","signature_b64":"uC6+FB+jm9muR+lWAGsh5k6d4qiMjCwsp/Te5bWDxGGk1EJug59L47Tj28AwhVeys2tE8Bdz1aGNozl+Mom4Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c77b75c3ea6989bd6e4009718ae770db6e56314bfde32b5c735f2eb0a9ff4974","last_reissued_at":"2026-05-18T04:17:25.170526Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:17:25.170526Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Generalization of Caffarelli's Contraction Theorem via (reverse) Heat Flow","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Emanuel Milman, Young-Heon Kim","submitted_at":"2010-02-02T03:55:11Z","abstract_excerpt":"A theorem of L. Caffarelli implies the existence of a map pushing forward a source Gaussian measure to a target measure which is more log-concave than the source one, which contracts Euclidean distance (in fact, Caffarelli showed that the optimal-transport Brenier map $T_{opt}$ is a contraction in this case). We generalize this result to more general source and target measures, using a condition on the third derivative of the potential, using two different proofs. The first uses a map $T$, whose inverse is constructed as a flow along an advection field associated to an appropriate heat-diffusi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1002.0373","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":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1002.0373","created_at":"2026-05-18T04:17:25.170595+00:00"},{"alias_kind":"arxiv_version","alias_value":"1002.0373v3","created_at":"2026-05-18T04:17:25.170595+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1002.0373","created_at":"2026-05-18T04:17:25.170595+00:00"},{"alias_kind":"pith_short_12","alias_value":"Y55XLQ7KNGE3","created_at":"2026-05-18T12:26:17.028572+00:00"},{"alias_kind":"pith_short_16","alias_value":"Y55XLQ7KNGE323SA","created_at":"2026-05-18T12:26:17.028572+00:00"},{"alias_kind":"pith_short_8","alias_value":"Y55XLQ7K","created_at":"2026-05-18T12:26:17.028572+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/Y55XLQ7KNGE323SABFYYVZ3Q3N","json":"https://pith.science/pith/Y55XLQ7KNGE323SABFYYVZ3Q3N.json","graph_json":"https://pith.science/api/pith-number/Y55XLQ7KNGE323SABFYYVZ3Q3N/graph.json","events_json":"https://pith.science/api/pith-number/Y55XLQ7KNGE323SABFYYVZ3Q3N/events.json","paper":"https://pith.science/paper/Y55XLQ7K"},"agent_actions":{"view_html":"https://pith.science/pith/Y55XLQ7KNGE323SABFYYVZ3Q3N","download_json":"https://pith.science/pith/Y55XLQ7KNGE323SABFYYVZ3Q3N.json","view_paper":"https://pith.science/paper/Y55XLQ7K","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1002.0373&json=true","fetch_graph":"https://pith.science/api/pith-number/Y55XLQ7KNGE323SABFYYVZ3Q3N/graph.json","fetch_events":"https://pith.science/api/pith-number/Y55XLQ7KNGE323SABFYYVZ3Q3N/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Y55XLQ7KNGE323SABFYYVZ3Q3N/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Y55XLQ7KNGE323SABFYYVZ3Q3N/action/storage_attestation","attest_author":"https://pith.science/pith/Y55XLQ7KNGE323SABFYYVZ3Q3N/action/author_attestation","sign_citation":"https://pith.science/pith/Y55XLQ7KNGE323SABFYYVZ3Q3N/action/citation_signature","submit_replication":"https://pith.science/pith/Y55XLQ7KNGE323SABFYYVZ3Q3N/action/replication_record"}},"created_at":"2026-05-18T04:17:25.170595+00:00","updated_at":"2026-05-18T04:17:25.170595+00:00"}