  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.
