Fuglede's spectral set conjecture for convex polytopes

Rachel Greenfeld, Nir Lev

Research output: Contribution to journalArticlepeer-review

29 Scopus citations


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.

Original languageEnglish
Pages (from-to)1497-1538
Number of pages42
JournalAnalysis and PDE
Issue number6
StatePublished - 2017

Bibliographical note

Publisher Copyright:
© 2017 Mathematical Sciences Publishers.


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


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