{"paper":{"title":"Random vectors in the isotropic position","license":"","headline":"","cross_cats":["math.FA"],"primary_cat":"math.MG","authors_text":"Mark Rudelson","submitted_at":"1996-08-19T00:00:00Z","abstract_excerpt":"Let $y$ be a random vector in \\rn, satisfying $$ \\Bbb E \\, \\tens{y} = id. $$ Let $M$ be a natural number and let $y_1 \\etc y_M$ be independent copies of $y$. We prove that for some absolute constant $C$ $$ \\enor{\\frac{1}{M} \\sum_i \\tens{y_i} - id} \\le C \\cdot \\frac{\\sqrt{\\log M}}{\\sqrt{M}} \\cdot \\left ( \\enor{y}^{\\log M} \\right )^{1/ \\log M}, $$ provided that the last expression is smaller than 1.\n  We apply this estimate to obtain a new proof of a result of Bourgain concerning the number of random points needed to bring a convex body into a nearly isotropic position."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9608208","kind":"arxiv","version":1},"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"}