{"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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.7670","kind":"arxiv","version":1},"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"}