The Pompeiu problem

  • Authors

    • Alexander G. Ramm Mathematics Department, Kansas State University, CW 207, Manhattan, KS 66506-2602, USA
    2013-03-14
    https://doi.org/10.14419/gjma.v1i1.728
  • 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  $D\subset \R^n$, $n\ge 2$, is a bounded domain, the closure $\bar{D}$ ofwhich is $C^1-$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 groupconsists of all translations and rotations.
    It is conjectured that if $f\neq 0$ 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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  • How to Cite

    Ramm, A. G. (2013). The Pompeiu problem. Global Journal of Mathematical Analysis, 1(1), 1-10. https://doi.org/10.14419/gjma.v1i1.728