A solution to the Pompeiu problem

Let $f \in L_{loc}^1 (\R^n)\cap \mathcal{S}$, where $\mathcal{S}$ is the Schwartz class of distributions, and $\int_{σ(D)} f(x) dx = 0 \quad \forall σ\in G, \qquad (*)$ where $D\subset \R^n$ is a bounded domain, the closure $\bar{D}$ of which is diffeomorphic to a closed ball, and $S$ is its boundary. Then the comp$is connected and path connected. By$G$the group of all rigid motions of$\R^n$ is denoted. This group consists of all translations and rotations. A proof of the following theorem is given. Theorem 1. {\it Assume that $n=2$,$f\not\equiv 0$, and (*) holds. Then$D$ is a ball.} Corollary. {\it If the problem $(\nabla^2+k^2)u=0$in$D$,$u_N|_S=0$,$u|_S=const\neq 0$has a solution, then$D$ is a ball.} Here $N$is the outer unit normal to$S$.

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