{"paper":{"title":"Heat flow and quantitative differentiation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.FA","authors_text":"Assaf Naor, Tuomas Hyt\\\"onen","submitted_at":"2016-08-05T15:23:58Z","abstract_excerpt":"For every Banach space $(Y,\\|\\cdot\\|_Y)$ that admits an equivalent uniformly convex norm we prove that there exists $c=c(Y)\\in (0,\\infty)$ with the following property. Suppose that $n\\in \\mathbb{N}$ and that $X$ is an $n$-dimensional normed space with unit ball $B_X$. Then for every $1$-Lipschitz function $f:B_X\\to Y$ and for every $\\varepsilon\\in (0,1/2]$ there exists a radius $r\\ge\\exp(-1/\\varepsilon^{cn})$, a point $x\\in B_X$ with $x+rB_X\\subset B_X$, and an affine mapping $\\Lambda:X\\to Y$ such that $\\|f(y)-\\Lambda(y)\\|_Y\\le \\varepsilon r$ for every $y\\in x+rB_X$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.01915","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"}