Fuglede’s spectral set conjecture for convex polytopes

Abstract

Let [math] be a convex polytope in [math] . We say that [math] is spectral if the space [math] admits an orthogonal basis consisting of exponential functions. There is a conjecture, which goes back to Fuglede (1974), that [math] is spectral if and only if it can tile the space by translations. It is known that if [math] tiles then it is spectral, but the converse was proved only in dimension [math] , by Iosevich, Katz and Tao. ¶ By a result due to Kolountzakis, if a convex polytope [math] is spectral, then it must be centrally symmetric. We prove that also all the facets of [math] are centrally symmetric. These conditions are necessary for [math] to tile by translations. ¶ We also develop an approach which allows us to prove that in dimension [math] , any spectral convex polytope [math] indeed tiles by translations. Thus we obtain that Fuglede’s conjecture is true for convex polytopes in [math] .