Pompeiu problem

Let $n \ge 2$, and let $M(n)$ denote the group of rigid motions of $\mathbb{R}^n$, i.e. the maps $\sigma(x) = \rho x + b$ with $\rho \in O(n)$ and $b \in \mathbb{R}^n$.

Let $K \subset \mathbb{R}^n$ be a bounded, simply connected Lipschitz domain, and suppose there exists a continuous function $f : \mathbb{R}^n \to \mathbb{C}$, not identically zero, such that

$$\int_{\sigma(K)} f(x)\, dx = 0 \qquad \text{for every } \sigma \in M(n),$$

that is, the integral of $f$ vanishes over every congruent copy of $K$. Then $K$ is a ball.

In the setting of overdetermined boundary value problems, the corresponding assertion is the following. Let $\Omega \subset \mathbb{R}^n$ be a bounded, simply connected domain with smooth (indeed real analytic) boundary $\partial \Omega$, and let $\partial/\partial n$ denote differentiation along the outward unit normal to $\partial \Omega$. Suppose there exist $\lambda > 0$, a constant $c$, and a function $u \in C^2(\overline{\Omega})$ not identically zero satisfying

$$\begin{cases}\Delta u + \lambda u = 0 & \text{in } \Omega,\\[2pt] \dfrac{\partial u}{\partial n} = 0 & \text{on } \partial \Omega,\\[2pt] u = c & \text{on } \partial \Omega.\end{cases}$$

Then $\Omega$ is a ball.