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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 .
Keywords
Fuglede's conjecture, spectral set, tiling, convex polytope
Mathematical Subject Classification 2010
Primary: 42B10, 52C22
Milestones
Received: 5 March 2017
Accepted: 29 May 2017
Published: 14 July 2017