{"paper":{"title":"Dichotomies, structure, and concentration in normed spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.FA","authors_text":"Grigoris Paouris, Petros Valettas","submitted_at":"2017-08-17T06:42:42Z","abstract_excerpt":"We use probabilistic, topological and combinatorial methods to establish the following deviation inequality: For any normed space $X=(\\mathbb R^n ,\\|\\cdot\\| )$ there exists an invertible linear map $T:\\mathbb R^n \\to \\mathbb R^n$ with \\[ \\mathbb P\\left( \\big| \\|TG\\| -\\mathbb E\\|TG\\| \\big| > \\varepsilon \\mathbb E\\|TG\\| \\right) \\leq C\\exp \\left( -c\\max\\{ \\varepsilon^2, \\varepsilon \\} \\log n \\right),\\quad \\varepsilon>0, \\] where $G$ is the standard $n$-dimensional Gaussian vector and $C,c>0$ are universal constants. It follows that for every $\\varepsilon\\in (0,1)$ and for every normed space $X=(\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.05149","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"}