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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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1111.4169","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-11-17T18:10:36Z","cross_cats_sorted":[],"title_canon_sha256":"29ef88bedebcc0650618c58faa7ee63b7bdee660766c3526f2b7800c67d668fd","abstract_canon_sha256":"7f66e8c7dc09e14ab20eaf693666361a361d12918e1f25e6558f859bad12862f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:05:47.587631Z","signature_b64":"7kmFPOgb21TaR1a9uTCqLKLAajsLl0AuGUMyfaiqEDoEmDjzUJGliOtJ6bxb9PwuhLJ3h3KGBuYKTPPuJshyCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e26963f729bb10f7e627f5720fbfa0b0847ef7657720509e136b021da5bee330","last_reissued_at":"2026-05-18T04:05:47.586872Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:05:47.586872Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $A$. 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