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
.