The Pompeiu problem

Abstract

Let $f\in L_{loc}^1({\mathbb{R}}^n)\cap {\mathcal{S}}^{\prime }$, where ${\mathcal{S}}^{\prime }$ is the Schwartz class of distributions, and $${\int }_{\sigma (D)}f(x)dx=0\forall \sigma \in G,(\ast )$$ where $D\subset {\mathbb{R}}^n$ 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 ${\mathbb{R}}^n$. This group consists of all translations and rotations. It is conjectured that if $f\ne 0$ 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.