## Analysis & PDE

• Anal. PDE
• Volume 10, Number 6 (2017), 1497-1538.

### Fuglede's spectral set conjecture for convex polytopes

#### Abstract

Let $Ω$ be a convex polytope in $ℝd$. We say that $Ω$ is spectral if the space $L2(Ω)$ admits an orthogonal basis consisting of exponential functions. There is a conjecture, which goes back to Fuglede (1974), that $Ω$ is spectral if and only if it can tile the space by translations. It is known that if $Ω$ tiles then it is spectral, but the converse was proved only in dimension $d = 2$, by Iosevich, Katz and Tao.

By a result due to Kolountzakis, if a convex polytope $Ω ⊂ ℝd$ is spectral, then it must be centrally symmetric. We prove that also all the facets of $Ω$ are centrally symmetric. These conditions are necessary for $Ω$ to tile by translations.

We also develop an approach which allows us to prove that in dimension $d = 3$, any spectral convex polytope $Ω$ indeed tiles by translations. Thus we obtain that Fuglede’s conjecture is true for convex polytopes in $ℝ3$.

#### Article information

Source
Anal. PDE, Volume 10, Number 6 (2017), 1497-1538.

Dates
Received: 5 March 2017
Accepted: 29 May 2017
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1510843530

Digital Object Identifier
doi:10.2140/apde.2017.10.1497

Mathematical Reviews number (MathSciNet)
MR3678495

Zentralblatt MATH identifier
1377.42014

#### Citation

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

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