{"paper":{"title":"$M$-estimates for isotropic convex bodies and their $L_q$-centroid bodies","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Apostolos Giannopoulos, Emanuel Milman","submitted_at":"2014-02-04T22:36:08Z","abstract_excerpt":"Let $K$ be a centrally-symmetric convex body in $\\mathbb{R}^n$ and let $\\|\\cdot\\|$ be its induced norm on ${\\mathbb R}^n$. We show that if $K \\supseteq r B_2^n$ then: \\[ \\sqrt{n} M(K) \\leqslant C \\sum_{k=1}^{n} \\frac{1}{\\sqrt{k}} \\min\\left(\\frac{1}{r} , \\frac{n}{k} \\log\\Big(e + \\frac{n}{k}\\Big) \\frac{1}{v_{k}^{-}(K)}\\right) . \\] where $M(K)=\\int_{S^{n-1}} \\|x\\|\\, d\\sigma(x)$ is the mean-norm, $C>0$ is a universal constant, and $v^{-}_k(K)$ denotes the minimal volume-radius of a $k$-dimensional orthogonal projection of $K$. We apply this result to the study of the mean-norm of an isotropic conv"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.0904","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"}