A new result on the Pompeiu problem
Résumé
A nonempty bounded open set $\Omega \subset \text{R}^{n}$ (n≥ 2) is said to have the Pompeiu property if and only if the only continuous function f on Rn for which the integral of f over σ (Ω) is zero for all rigid motions σ of Rn is $f\equiv 0$. We consider a nonempty bounded open set $\Omega \subset \text{R}^{n}$ (n≥ 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 Ω implies some geometric conditions. Using these conditions we prove that a special kind of solid tori in Rn, n≥ 3, has the Pompeiu property. So far the result was proved only for solid tori in R4. We also examine the case of planar domains. Finally we extend the example of solid tori to domains in Rn bounded by hypersurfaces of revolution.