{"paper":{"title":"On the behavior of $1$-Laplacian Ratio Cuts on nearly rectangular domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Hau-Tieng Wu, Jeremy L. Marzuola, Wesley Hamilton","submitted_at":"2020-01-06T14:56:41Z","abstract_excerpt":"Given a connected set $\\Omega_0 \\subset \\mathbb{R}^2$, define a sequence of sets $(\\Omega_n)_{n=0}^{\\infty}$ where $\\Omega_{n+1}$ is the subset of $\\Omega_n$ where the first eigenfunction of the (properly normalized) Neumann $p-$Laplacian $ -\\Delta^{(p)} \\phi = \\lambda_1 |\\phi|^{p-2} \\phi$ is positive (or negative). For $p=1$, this is also referred to as the Ratio Cut of the domain. We conjecture that, unless $\\Omega_0$ is an isosceles right triangle, these sets converge to the set of rectangles with eccentricity bounded by 2 in the Gromov-Hausdorff distance as long as they have a certain dist"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2001.01615","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2001.01615/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}