The null set of a polytope, and the Pompeiu property for polytopes

We study the null set $N(\mathcal{P})$ of the Fourier-Laplace transform of a polytope $\mathcal{P} \subset \mathbb{R}^d$, and we find that $N(\mathcal{P})$ does not contain (almost all) circles in $\mathbb{R}^d$. As a consequence, the null set does not contain the algebraic varieties $\{z \in \mathbb{C}^d \mid z_1^2 + \dots + z_d^2 = \alpha^2\}$ for each fixed $\alpha \in \mathbb{C}$, and hence we get an explicit proof that the Pompeiu property is true for all polytopes. Our proof uses the Brion-Barvinok theorem, which gives a concrete formulation for the Fourier-Laplace transform of a polytope, and it also uses properties of Bessel functions. The original proof that polytopes (as well as other bodies) possess the Pompeiu property was given by Brown, Schreiber, and Taylor (1973) for dimension 2. Williams (1976) later observed that the same proof also works for $d>2$ and, using eigenvalues of the Laplacian, gave another proof valid for $d \geq 2$ that polytopes have the Pompeiu property.