{"paper":{"title":"Set Reconstruction by Voronoi cells","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Dmitry Zaporozhets, Evgeny Spodarev, Matthias Reitzner","submitted_at":"2011-11-17T18:10:36Z","abstract_excerpt":"For a Borel set $A$ and a homogeneous Poisson point process $\\eta$ in $\\R^d$ of intensity $\\lambda >0$, define the Poisson--Voronoi approximation $ A_\\eta$ of $A$ as a union of all Voronoi cells with nuclei from $\\eta$ lying in $A$. If $A$ has a finite volume and perimeter we find an exact asymptotic of $\\E\\Vol(A\\Delta A_\\eta)$ as $\\lambda\\to\\infty$ where $\\Vol$ is the Lebesgue measure. Estimates for all moments of $\\Vol(A_\\eta)$ and $\\Vol(A\\Delta A_\\eta)$ together with their asymptotics for large $\\lambda$ are obtained as well."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.4169","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"}