{"paper":{"title":"On the Impossibility of Dimension Reduction for Doubling Subsets of $\\ell_p$, $p>2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Lee-Ad Gottlieb, Ofer Neiman, Yair Bartal","submitted_at":"2013-08-22T21:34:33Z","abstract_excerpt":"A major open problem in the field of metric embedding is the existence of dimension reduction for $n$-point subsets of Euclidean space, such that both distortion and dimension depend only on the {\\em doubling constant} of the pointset, and not on its cardinality. In this paper, we negate this possibility for $\\ell_p$ spaces with $p>2$. In particular, we introduce an $n$-point subset of $\\ell_p$ with doubling constant O(1), and demonstrate that any embedding of the set into $\\ell_p^d$ with distortion $D$ must have $D\\ge\\Omega\\left(\\left(\\frac{c\\log n}{d}\\right)^{\\frac{1}{2}-\\frac{1}{p}}\\right)$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.4996","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"}