{"paper":{"title":"On the optimality of gluing over scales","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Alexander Jaffe, James R. Lee, Mohammad Moharrami","submitted_at":"2009-10-01T21:45:27Z","abstract_excerpt":"We show that for every $\\alpha > 0$, there exist $n$-point metric spaces (X,d) where every \"scale\" admits a Euclidean embedding with distortion at most $\\alpha$, but the whole space requires distortion at least $\\Omega(\\sqrt{\\alpha \\log n})$. This shows that the scale-gluing lemma [Lee, SODA 2005] is tight, and disproves a conjecture stated there. This matching upper bound was known to be tight at both endpoints, i.e. when $\\alpha = \\Theta(1)$ and $\\alpha = \\Theta(\\log n)$, but nowhere in between.\n  More specifically, we exhibit $n$-point spaces with doubling constant $\\lambda$ requiring Eucli"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.0277","kind":"arxiv","version":3},"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"}