| Full text | |
| Author(s): |
Machado, Fabricio Caluza
;
Robins, Sinai
Total Authors: 2
|
| Document type: | Journal article |
| Source: | JOURNAL D ANALYSE MATHEMATIQUE; v. 150, n. 2, p. 11-pg., 2023-07-18. |
| Abstract | |
We study the null set N(P) of the Fourier-Laplace transform of a polytope P subset of R-d, and we find that N(P) does not contain (almost all) circles in Rd. As a consequence, the null set does not contain the algebraic varieties {z is an element of C-d | z(1)(2) + center dot center dot center dot + z(d)(2) = alpha(2)} for each fixed alpha is an element of 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. (AU) | |
| FAPESP's process: | 17/25237-4 - Applications of harmonic analysis to Discrete Geometry |
| Grantee: | Fabrício Caluza Machado |
| Support Opportunities: | Scholarships in Brazil - Doctorate |