{"paper":{"title":"Ultrametric skeletons","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.PR"],"primary_cat":"math.MG","authors_text":"Assaf Naor, Manor Mendel","submitted_at":"2011-12-15T03:41:12Z","abstract_excerpt":"We prove that for every $\\epsilon\\in (0,1)$ there exists $C_\\epsilon\\in (0,\\infty)$ with the following property. If $(X,d)$ is a compact metric space and $\\mu$ is a Borel probability measure on $X$ then there exists a compact subset $S\\subseteq X$ that embeds into an ultrametric space with distortion $O(1/\\epsilon)$, and a probability measure $\\nu$ supported on $S$ satisfying $\\nu(B_d(x,r))\\le (\\mu(B_d(x,C_\\epsilon r))^{1-\\epsilon}$ for all $x\\in X$ and $r\\in (0,\\infty)$. The dependence of the distortion on $\\epsilon$ is sharp. We discuss an extension of this statement to multiple measures, as"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.3416","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"}