## Abstract

Let Ω be a convex polytope in ℝ^{d}. We say that Ω is spectral if the space L^{2}(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 ⊂ ℝ^{d} 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 ℝ^{3}.

Original language | English |
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Pages (from-to) | 1497-1538 |

Number of pages | 42 |

Journal | Analysis and PDE |

Volume | 10 |

Issue number | 6 |

DOIs | |

State | Published - 2017 |

### Bibliographical note

Publisher Copyright:© 2017 Mathematical Sciences Publishers.

## Keywords

- Convex polytope
- Fuglede's conjecture
- Spectral set
- Tiling