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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 $"},"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":"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1312.6702","created_at":"2026-05-18T01:07:57.165651+00:00"},{"alias_kind":"arxiv_version","alias_value":"1312.6702v2","created_at":"2026-05-18T01:07:57.165651+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.6702","created_at":"2026-05-18T01:07:57.165651+00:00"},{"alias_kind":"pith_short_12","alias_value":"CCQLXHN7IJKH","created_at":"2026-05-18T12:27:40.988391+00:00"},{"alias_kind":"pith_short_16","alias_value":"CCQLXHN7IJKHDIVB","created_at":"2026-05-18T12:27:40.988391+00:00"},{"alias_kind":"pith_short_8","alias_value":"CCQLXHN7","created_at":"2026-05-18T12:27:40.988391+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/CCQLXHN7IJKHDIVBBEDQEVSVUI","json":"https://pith.science/pith/CCQLXHN7IJKHDIVBBEDQEVSVUI.json","graph_json":"https://pith.science/api/pith-number/CCQLXHN7IJKHDIVBBEDQEVSVUI/graph.json","events_json":"https://pith.science/api/pith-number/CCQLXHN7IJKHDIVBBEDQEVSVUI/events.json","paper":"https://pith.science/paper/CCQLXHN7"},"agent_actions":{"view_html":"https://pith.science/pith/CCQLXHN7IJKHDIVBBEDQEVSVUI","download_json":"https://pith.science/pith/CCQLXHN7IJKHDIVBBEDQEVSVUI.json","view_paper":"https://pith.science/paper/CCQLXHN7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1312.6702&json=true","fetch_graph":"https://pith.science/api/pith-number/CCQLXHN7IJKHDIVBBEDQEVSVUI/graph.json","fetch_events":"https://pith.science/api/pith-number/CCQLXHN7IJKHDIVBBEDQEVSVUI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CCQLXHN7IJKHDIVBBEDQEVSVUI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CCQLXHN7IJKHDIVBBEDQEVSVUI/action/storage_attestation","attest_author":"https://pith.science/pith/CCQLXHN7IJKHDIVBBEDQEVSVUI/action/author_attestation","sign_citation":"https://pith.science/pith/CCQLXHN7IJKHDIVBBEDQEVSVUI/action/citation_signature","submit_replication":"https://pith.science/pith/CCQLXHN7IJKHDIVBBEDQEVSVUI/action/replication_record"}},"created_at":"2026-05-18T01:07:57.165651+00:00","updated_at":"2026-05-18T01:07:57.165651+00:00"}