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The Pompeiu problem

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Author(s): Alexander G. Ramm

Journal: Global Journal of Mathematical Analysis
ISSN 2307-9002

Volume: 1;
Issue: 1;
Start page: 1;
Date: 2013;
Original page

ABSTRACT
Let $f in L_{loc}^1 (R^n)cap mathcal{S}'$,$mathcal{S}'$ is the Schwartz class of distributions,  and$$int_{sigma (D)} f(x) dx = 0 quad forall sigma in G, qquad (*)$$where  $Dsubset R^n$, $nge 2$, is a bounded domain, the closure $ar{D}$ ofwhich is $C^1-$diffeomorphic to a closed ball. Then the complement of $ar{D}$is connected and path connected.Here  $G$ denotes the group of all rigid motions in $R^n$.This groupconsists of all translations and rotations.It is conjectured that if $feq 0$ and $(*)$ holds, then $D$ is aball. Other two conjectures, equivalent to the above one, are formulatedand discussed. Three additional conjectures are formulated.Several new short proofs are given for various results.
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