Fuglede's spectral set conjecture for convex polytopes

Abstract

Let $\Omega$ be a convex polytope in ${ℝ}^d$. We say that $\Omega$ is spectral if the space $L^2\left(\Omega \right)$ admits an orthogonal basis consisting of exponential functions. There is a conjecture, which goes back to Fuglede (1974), that $\Omega$ is spectral if and only if it can tile the space by translations. It is known that if $\Omega$ 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 $\Omega \subset {ℝ}^d$ is spectral, then it must be centrally symmetric. We prove that also all the facets of $\Omega$ are centrally symmetric. These conditions are necessary for $\Omega$ to tile by translations.

We also develop an approach which allows us to prove that in dimension $d=3$, any spectral convex polytope $\Omega$ indeed tiles by translations. Thus we obtain that Fuglede’s conjecture is true for convex polytopes in ${ℝ}^3$.