{"paper":{"title":"Metric and geometric relaxations of self-contracted curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Aris Daniilidis, Estibalitz Durand Cartagena, Robert Deville","submitted_at":"2018-02-26T23:18:53Z","abstract_excerpt":"Self-contractedness (or self-expandedness, depending on the orientation) is hereby extended in two natural ways giving rise, for any $\\lambda\\in\\lbrack-1,1)$, to the metric notion of $\\lambda $-curve and the (weaker) geometric notion of $\\lambda$-cone property ($\\lambda$-eel). In the Euclidean space $\\mathbb{R}^{d}$ it is established that for $\\lambda\\in\\lbrack-1,1/d)$ bounded $\\lambda$-curves have finite length. For $\\lambda\\geq 1/\\sqrt{5}$ it is always possible to construct bounded curves of infinite length in ${\\mathbb{R}}^{3}$ which do satisfy the $\\lambda $-cone property. This can never h"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.09637","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"}