{"paper":{"title":"Geometric properties of $\\varphi$-uniform domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Matti Vuorinen, Peter H\\\"ast\\\"o, Riku Kl\\'en, Swadesh Kumar Sahoo","submitted_at":"2015-10-20T12:06:09Z","abstract_excerpt":"We consider proper subdomains $G$ of $\\mathbb{R}^n$ and their images $G'=f(G)$ under quasiconformal mappings $f$ of $\\mathbb{R}^n$. We compare the distance ratio metrics of $G$ and $G'$; as an application we show that $\\varphi$-uniform domains are preserved under quasiconformal mappings of $\\mathbb{R}^n$. A sufficient condition for $\\varphi$-uniformity is obtained in terms of the quasi-symmetry condition. We give a geometric condition for uniformity: If $G\\subset\\mathbb{R}^n$ is $\\phi$-uniform and satisfies the twisted cone condition, then it is uniform. We also construct a planar $\\phi$-unifo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.05847","kind":"arxiv","version":4},"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"}