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

Abstract

We study the null set $N({\cal P})$ of the Fourier–Laplace transform of a polytope ${\cal P} \subset {\mathbb{R}^d}$, and we find that $N({\cal P})$ does not contain (almost all) circles in ℝ^d. As a consequence, the null set does not contain the algebraic varieties {z ∈ ℂ^d ∣ z ²₁ + ⋯ + z ²_d = α²} for each fixed α ∈ ℂ, and hence we get an explicit proof that the Pompeiu property is true for all polytopes.

The original proof that polytopes (as well as some other bodies) possess the Pompeiu property was given by Brown, Schreiber, and Taylor [7] for dimension 2. Williams [14, p. 184] later observed that the same proof also works for d > 2 and, using eigenvalues of the Laplacian, also gave a proof (valid for d ≥ 2) that polytopes have the Pompeiu property.

Here we use the Brion–Barvinok theorem, which gives a concrete formulation for the Fourier–Laplace transform of a polytope. Hence our proof offers a more direct approach, requiring less machinery.