The Pompeiu problem

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  $D\subset \R^n$, $n\ge 2$, 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.