{"paper":{"title":"$(\\beta)$-distortion of some infinite graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.MG","authors_text":"Florent P. Baudier, Sheng Zhang","submitted_at":"2015-04-16T14:31:44Z","abstract_excerpt":"A distortion lower bound of $\\Omega(\\log(h)^{1/p})$ is proven for embedding the complete countably branching hyperbolic tree of height $h$ into a Banach space admitting an equivalent norm satisfying property $(\\beta)$ of Rolewicz with modulus of power type $p\\in(1,\\infty)$ (in short property ($\\beta_p$)). Also it is shown that a distortion lower bound of $\\Omega(\\ell^{1/p})$ is incurred when embedding the parasol graph with $\\ell$ levels into a Banach space with an equivalent norm with property ($\\beta_p$). The tightness of the lower bound for trees is shown adjusting a construction of Matou\\v"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.04250","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"}