{"paper":{"title":"A solution to the Pompeiu problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"A.G.Ramm","submitted_at":"2013-04-08T18:29:34Z","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, and $S$ is its boundary. Then the comp$ is connected and path connected. By $G$ the group of all rigid motions of $\\R^n$ is denoted. This group consists of all translations and rotations. A proof of the following theorem is given.\n  Theorem 1. {\\it Assume that $n=2$, $f\\not\\equiv 0$, and (*) holds. Then $D$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.2297","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"}