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 ℝd. As a consequence, the null set does not contain the algebraic varieties {z ∈ ℂd ∣ z 21 + ⋯ + z 2d = 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.
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F. C. M was supported by grant #2017/25237-4, from the São Paulo Research Foundation (FAPESP). This work was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico—CNPq (Proc. 423833/2018-9) and by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001.
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Machado, F.C., Robins, S. The null set of a polytope, and the Pompeiu property for polytopes. JAMA 150, 673–683 (2023). https://doi.org/10.1007/s11854-023-0290-3
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DOI: https://doi.org/10.1007/s11854-023-0290-3