{"paper":{"title":"Self-contracted curves have finite length","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Eugene Stepanov, Yana Teplitskaya","submitted_at":"2017-07-16T17:46:48Z","abstract_excerpt":"A curve $\\theta$: $I\\to E$ in a metric space $E$ equipped with the distance $d$, where $I\\subset \\R$ is a (possibly unbounded) interval, is called self-contracted, if for any triple of instances of time $\\{t_i\\}_{i=1}^3\\subset I$ with $t_1\\leq t_2\\leq t_3$ one has $d(\\theta(t_3),\\theta(t_2))\\leq d(\\theta(t_3),\\theta(t_1))$. We prove that if $E$ is a finite-dimensional normed space with an arbitrary norm, the trace of $\\theta$ is bounded, then $\\theta$ has finite length, i.e. is rectifiable, thus answering positively the question raised in~\\cite{Lemenant16sc-rectif}."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.04922","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"}