Analysis & PDE
- Anal. PDE
- Volume 10, Number 6 (2017), 1497-1538.
Fuglede's spectral set conjecture for convex polytopes
Let be a convex polytope in . We say that is spectral if the space admits an orthogonal basis consisting of exponential functions. There is a conjecture, which goes back to Fuglede (1974), that is spectral if and only if it can tile the space by translations. It is known that if tiles then it is spectral, but the converse was proved only in dimension , by Iosevich, Katz and Tao.
By a result due to Kolountzakis, if a convex polytope is spectral, then it must be centrally symmetric. We prove that also all the facets of are centrally symmetric. These conditions are necessary for to tile by translations.
We also develop an approach which allows us to prove that in dimension , any spectral convex polytope indeed tiles by translations. Thus we obtain that Fuglede’s conjecture is true for convex polytopes in .
Anal. PDE, Volume 10, Number 6 (2017), 1497-1538.
Received: 5 March 2017
Accepted: 29 May 2017
First available in Project Euclid: 16 November 2017
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Greenfeld, Rachel; Lev, Nir. Fuglede's spectral set conjecture for convex polytopes. Anal. PDE 10 (2017), no. 6, 1497--1538. doi:10.2140/apde.2017.10.1497. https://projecteuclid.org/euclid.apde/1510843530