{"paper":{"title":"Wasserstein Distance and the Rectifiability of Doubling Measures: Part I","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Guy David, Jonas Azzam, Tatiana Toro","submitted_at":"2014-08-28T08:18:54Z","abstract_excerpt":"Let $\\mu$ be a doubling measure in $\\mathbb{R}^n$. We investigate quantitative relations between the rectifiability of $\\mu$ and its distance to flat measures. More precisely, for $x$ in the support $\\Sigma$ of $\\mu$ and $r > 0$, we introduce a number $\\alpha(x,r)\\in (0,1]$ that measures, in terms of a variant of the $L^1$-Wasserstein distance, the minimal distance between the restriction of $\\mu$ to $B(x,r)$ and a multiple of the Lebesgue measure on an affine subspace that meets $B(x,r/2)$. We show that the set of points of $\\Sigma$ where $\\int_0^1 \\alpha(x,r) \\frac{dr}{r} < \\infty$ can be de"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.6645","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"}