{"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"}