{"paper":{"title":"Average-distance problem for parameterized curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Dejan Slep\\v{c}ev, Xin Yang Lu","submitted_at":"2014-11-11T00:50:06Z","abstract_excerpt":"We consider approximating a measure by a parameterized curve subject to length penalization. That is for a given finite positive compactly supported measure $\\mu$, for $p \\geq 1$ and $\\lambda>0$ we consider the functional \\[ E(\\gamma) = \\int_{\\mathbb{R}^d} d(x, \\Gamma_\\gamma)^p d\\mu(x) + \\lambda \\,\\textrm{Length}(\\gamma) \\] where $\\gamma:I \\to \\mathbb{R}^d$, $I$ is an interval in $\\mathbb{R}$, $\\Gamma_\\gamma = \\gamma(I)$, and $d(x, \\Gamma_\\gamma)$ is the distance of $x$ to $\\Gamma_\\gamma$.\n  The problem is closely related to the average-distance problem, where the admissible class are the conn"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.2673","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"}