{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:IEJYMKSSG7DVFHKRDI7ALGZKXA","short_pith_number":"pith:IEJYMKSS","schema_version":"1.0","canonical_sha256":"4113862a5237c7529d511a3e059b2ab8225e4490cad867332b699a3dc7961086","source":{"kind":"arxiv","id":"1210.7670","version":1},"attestation_state":"computed","paper":{"title":"The Pompeiu problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"A. G. Ramm","submitted_at":"2012-10-29T14:06:24Z","abstract_excerpt":"Let $f \\in L_{loc}^1 (\\R^n)\\cap \\mathcal{S}'$, where $\\mathcal{S}'$ is the Schwartz class of distributions, and $$\\int_{\\sigma (D)} f(x) dx = 0 \\quad \\forall \\sigma \\in G, \\qquad (*)$$ where $D\\subset \\R^n$ is a bounded domain, the closure $\\bar{D}$ of which is diffeomorphic to a closed ball. Then the complement of $\\bar{D}$ is connected and path connected. Here $G$ denotes the group of all rigid motions in $\\R^n$. This group consists of all translations and rotations.\n  It is conjectured that if $f\\neq 0$ and (*) holds, then $D$ is a ball. Other conjectures, equivalent to the above one, are f"},"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":"1210.7670","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-10-29T14:06:24Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"de097bea699bc105ed36d1913d4c5560eb8802080c41de849845a376da93829e","abstract_canon_sha256":"5c0cac9a1761c7f3967984ec903abaa8f29a94836a42e959270f457ebd79b592"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:42:09.135292Z","signature_b64":"fykiQJmizGTmaJAZvRtDPZ71ivcQjo5PyfsKSKFWBwZtBlYipCsI/9ADkfT6tnbnod99XBwVa809dLrbKi4mDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4113862a5237c7529d511a3e059b2ab8225e4490cad867332b699a3dc7961086","last_reissued_at":"2026-05-18T03:42:09.134619Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:42:09.134619Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Pompeiu problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"A. G. Ramm","submitted_at":"2012-10-29T14:06:24Z","abstract_excerpt":"Let $f \\in L_{loc}^1 (\\R^n)\\cap \\mathcal{S}'$, where $\\mathcal{S}'$ is the Schwartz class of distributions, and $$\\int_{\\sigma (D)} f(x) dx = 0 \\quad \\forall \\sigma \\in G, \\qquad (*)$$ where $D\\subset \\R^n$ is a bounded domain, the closure $\\bar{D}$ of which is diffeomorphic to a closed ball. Then the complement of $\\bar{D}$ is connected and path connected. Here $G$ denotes the group of all rigid motions in $\\R^n$. This group consists of all translations and rotations.\n  It is conjectured that if $f\\neq 0$ and (*) holds, then $D$ is a ball. 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