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. It is conjectured that if $f\neq 0$ and (*) holds, then $D$ is a ball. Other conjectures, equivalent to the above one, are formulated and discussed. Several new short proofs are given for the earlier proved results.
Categories: math.AP math.FA