{"paper":{"title":"How to take shortcuts in Euclidean space: making a given set into a short quasi-convex set","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.MG","authors_text":"Jonas Azzam, Raanan Schul","submitted_at":"2009-12-07T22:05:36Z","abstract_excerpt":"For a given connected set $\\Gamma$ in $d-$dimensional Euclidean space, we construct a connected set $\\tilde\\Gamma\\supset \\Gamma$ such that the two sets have comparable Hausdorff length, and the set $\\tilde\\Gamma$ has the property that it is quasiconvex, i.e. any two points $x$ and $y$ in $\\tilde\\Gamma$ can be connected via a path, all of which is in $\\tilde\\Gamma$, which has length bounded by a fixed constant multiple of the Euclidean distance between $x$ and $y$. Thus, for any set $K$ in $d-$dimensional Euclidean space we have a set $\\tilde\\Gamma$ as above such that $\\tilde\\Gamma$ has compara"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0912.1356","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"}