{"paper":{"title":"Quantitative affine approximation for UMD targets","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.FA","authors_text":"Assaf Naor, Sean Li, Tuomas Hyt\\\"onen","submitted_at":"2015-10-01T15:09:20Z","abstract_excerpt":"It is shown here that if $(Y,\\|\\cdot\\|_Y)$ is a Banach space in which martingale differences are unconditional (a UMD Banach space) then there exists $c=c(Y)\\in (0,\\infty)$ with the following property. For every $n\\in \\mathbb{N}$ and $\\varepsilon\\in (0,1/2]$, if $(X,\\|\\cdot\\|_X)$ is an $n$-dimensional normed space with unit ball $B_X$ and $f:B_X\\to Y$ is a $1$-Lipschitz function then there exists an affine mapping $\\Lambda:X\\to Y$ and a sub-ball $B^*=y+\\rho B_X\\subseteq B_X$ of radius $\\rho\\ge \\exp(-(1/\\varepsilon)^{cn})$ such that $\\|f(x)-\\Lambda(x)\\|_Y\\le \\varepsilon \\rho$ for all $x\\in B^*$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.00276","kind":"arxiv","version":5},"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"}