A new result on the Pompeiu problem

A nonempty bounded open set $\Omega \subset {\mathbb{R}}^{n}$ ($n \geq 2$) is said to have the Pompeiu property if and only if the only continuous function $f$ on ${\mathbb{R}}^{n}$ for which the integral of $f$ over $\sigma (\Omega )$ is zero for all rigid motions $\sigma$ of ${\mathbb{R}}^{n}$ is $f \equiv 0$. We consider a nonempty bounded open set $\Omega \subset {\mathbb{R}}^{n}$ $(n \geq 2)$ with Lipschitz boundary and we assume that the complement of $\overline{\Omega }$ is connected. We show that the failure of the Pompeiu property for $\Omega$ implies some geometric conditions. Using these conditions we prove that a special kind of solid tori in ${\mathbb{R}}^{n}$, $n \geq 3$, has the Pompeiu property. So far the result was proved only for solid tori in ${\mathbb{R}}^{4}$. We also examine the case of planar domains. Finally we extend the example of solid tori to domains in ${\mathbb{R}}^{n}$ bounded by hypersurfaces of revolution.