{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:CCQLXHN7IJKHDIVBBEDQEVSVUI","short_pith_number":"pith:CCQLXHN7","canonical_record":{"source":{"id":"1312.6702","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-12-23T21:18:17Z","cross_cats_sorted":[],"title_canon_sha256":"794675d107d1bd023f4a645a094d0dd1c406c19b23a94dd4e9fead166b845306","abstract_canon_sha256":"570339e00cc6164c6bbc1b761d7d64dac2c60b003e29cd96ba21a8c85bca28d0"},"schema_version":"1.0"},"canonical_sha256":"10a0bb9dbf425471a2a10907025655a23d1dc2a75adaecd334fa97e2c821265a","source":{"kind":"arxiv","id":"1312.6702","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.6702","created_at":"2026-05-18T01:07:57Z"},{"alias_kind":"arxiv_version","alias_value":"1312.6702v2","created_at":"2026-05-18T01:07:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.6702","created_at":"2026-05-18T01:07:57Z"},{"alias_kind":"pith_short_12","alias_value":"CCQLXHN7IJKH","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_16","alias_value":"CCQLXHN7IJKHDIVB","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_8","alias_value":"CCQLXHN7","created_at":"2026-05-18T12:27:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:CCQLXHN7IJKHDIVBBEDQEVSVUI","target":"record","payload":{"canonical_record":{"source":{"id":"1312.6702","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-12-23T21:18:17Z","cross_cats_sorted":[],"title_canon_sha256":"794675d107d1bd023f4a645a094d0dd1c406c19b23a94dd4e9fead166b845306","abstract_canon_sha256":"570339e00cc6164c6bbc1b761d7d64dac2c60b003e29cd96ba21a8c85bca28d0"},"schema_version":"1.0"},"canonical_sha256":"10a0bb9dbf425471a2a10907025655a23d1dc2a75adaecd334fa97e2c821265a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:07:57.166004Z","signature_b64":"EdA43BP+NIN0clI+63z7TKtY+hffavUNsrNs5g/XuJMWMTjcQ16iZy6M3pAoDMUY1UpdqZfjMtUgK3byEOCBCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"10a0bb9dbf425471a2a10907025655a23d1dc2a75adaecd334fa97e2c821265a","last_reissued_at":"2026-05-18T01:07:57.165578Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:07:57.165578Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1312.6702","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":"2026-05-18T01:07:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bEr1L8jBG94ihKdWJbEashpSt75LGd3u7cYLfU28OVjOU+HAhLKMeF1GLSKfOJU1RmKM+7HmpbDCurq6YAL3Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T04:54:31.681913Z"},"content_sha256":"ab1eb64b02f64429abd2631ca8213130d1df876ada78ad051fd5dd3bd0cd0dd3","schema_version":"1.0","event_id":"sha256:ab1eb64b02f64429abd2631ca8213130d1df876ada78ad051fd5dd3bd0cd0dd3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:CCQLXHN7IJKHDIVBBEDQEVSVUI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Metric measure spaces supporting Gagliardo-Nirenberg inequalities: volume non-collapsing and rigidities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alexandru Krist\\'aly","submitted_at":"2013-12-23T21:18:17Z","abstract_excerpt":"Let $({M},\\textsf{d},\\textsf{m})$ be a metric measure space which satisfies the Lott-Sturm-Villani curvature-dimension condition $\\textsf{CD}(K,n)$ for some $K\\geq 0$ and $n\\geq 2$, and a lower $n-$density assumption at some point of $M$. We prove that if $({M},\\textsf{d},\\textsf{m})$ supports the Gagliardo-Nirenberg inequality or any of its limit cases ($L^p-$logarithmic Sobolev inequality or Faber-Krahn-type inequality), then a global non-collapsing $n-$dimensional volume growth holds, i.e., there exists a universal constant $C_0>0$ such that $\\textsf{m}( B_x(\\rho))\\geq C_0 \\rho^n$ for all $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.6702","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":"2026-05-18T01:07:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"A8S62fvSaYJeCrG+/pnRR/puP0rr1rcja8SwbUdNhpEO6ygIMHxZAdFzkyYo8fqRpsUEQMlpDDnyAFMODJvcAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T04:54:31.682635Z"},"content_sha256":"b2d6d1fd1fcd3deba463cd1f52a58149f902410bb2df4a5ea73da091f8031e2e","schema_version":"1.0","event_id":"sha256:b2d6d1fd1fcd3deba463cd1f52a58149f902410bb2df4a5ea73da091f8031e2e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CCQLXHN7IJKHDIVBBEDQEVSVUI/bundle.json","state_url":"https://pith.science/pith/CCQLXHN7IJKHDIVBBEDQEVSVUI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CCQLXHN7IJKHDIVBBEDQEVSVUI/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":"2026-06-08T04:54:31Z","links":{"resolver":"https://pith.science/pith/CCQLXHN7IJKHDIVBBEDQEVSVUI","bundle":"https://pith.science/pith/CCQLXHN7IJKHDIVBBEDQEVSVUI/bundle.json","state":"https://pith.science/pith/CCQLXHN7IJKHDIVBBEDQEVSVUI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CCQLXHN7IJKHDIVBBEDQEVSVUI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:CCQLXHN7IJKHDIVBBEDQEVSVUI","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":"570339e00cc6164c6bbc1b761d7d64dac2c60b003e29cd96ba21a8c85bca28d0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-12-23T21:18:17Z","title_canon_sha256":"794675d107d1bd023f4a645a094d0dd1c406c19b23a94dd4e9fead166b845306"},"schema_version":"1.0","source":{"id":"1312.6702","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.6702","created_at":"2026-05-18T01:07:57Z"},{"alias_kind":"arxiv_version","alias_value":"1312.6702v2","created_at":"2026-05-18T01:07:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.6702","created_at":"2026-05-18T01:07:57Z"},{"alias_kind":"pith_short_12","alias_value":"CCQLXHN7IJKH","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_16","alias_value":"CCQLXHN7IJKHDIVB","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_8","alias_value":"CCQLXHN7","created_at":"2026-05-18T12:27:40Z"}],"graph_snapshots":[{"event_id":"sha256:b2d6d1fd1fcd3deba463cd1f52a58149f902410bb2df4a5ea73da091f8031e2e","target":"graph","created_at":"2026-05-18T01:07:57Z","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 $({M},\\textsf{d},\\textsf{m})$ be a metric measure space which satisfies the Lott-Sturm-Villani curvature-dimension condition $\\textsf{CD}(K,n)$ for some $K\\geq 0$ and $n\\geq 2$, and a lower $n-$density assumption at some point of $M$. We prove that if $({M},\\textsf{d},\\textsf{m})$ supports the Gagliardo-Nirenberg inequality or any of its limit cases ($L^p-$logarithmic Sobolev inequality or Faber-Krahn-type inequality), then a global non-collapsing $n-$dimensional volume growth holds, i.e., there exists a universal constant $C_0>0$ such that $\\textsf{m}( B_x(\\rho))\\geq C_0 \\rho^n$ for all $","authors_text":"Alexandru Krist\\'aly","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-12-23T21:18:17Z","title":"Metric measure spaces supporting Gagliardo-Nirenberg inequalities: volume non-collapsing and rigidities"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.6702","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:ab1eb64b02f64429abd2631ca8213130d1df876ada78ad051fd5dd3bd0cd0dd3","target":"record","created_at":"2026-05-18T01:07:57Z","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":"570339e00cc6164c6bbc1b761d7d64dac2c60b003e29cd96ba21a8c85bca28d0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-12-23T21:18:17Z","title_canon_sha256":"794675d107d1bd023f4a645a094d0dd1c406c19b23a94dd4e9fead166b845306"},"schema_version":"1.0","source":{"id":"1312.6702","kind":"arxiv","version":2}},"canonical_sha256":"10a0bb9dbf425471a2a10907025655a23d1dc2a75adaecd334fa97e2c821265a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"10a0bb9dbf425471a2a10907025655a23d1dc2a75adaecd334fa97e2c821265a","first_computed_at":"2026-05-18T01:07:57.165578Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:07:57.165578Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EdA43BP+NIN0clI+63z7TKtY+hffavUNsrNs5g/XuJMWMTjcQ16iZy6M3pAoDMUY1UpdqZfjMtUgK3byEOCBCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:07:57.166004Z","signed_message":"canonical_sha256_bytes"},"source_id":"1312.6702","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ab1eb64b02f64429abd2631ca8213130d1df876ada78ad051fd5dd3bd0cd0dd3","sha256:b2d6d1fd1fcd3deba463cd1f52a58149f902410bb2df4a5ea73da091f8031e2e"],"state_sha256":"c14c5ea49c5c4e691befc31270ab54843b8ec1f5d9b08f18bcf2e22cde15f5c8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pjnPIVqKW6yPkxw/+iw0voFNAvrujJqPsxOh9W5FCbzzVtHvZ8qwFQW+XMhr7+xxCpXy+UhjED8rsZgAEwi7Bg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T04:54:31.686360Z","bundle_sha256":"a28bc488b93eb124ea2417825ecd0dbc760b144e948284cb31e52d9a98188869"}}