The Fuglede conjecture for convex domains is true in all dimensions

Nir Lev, Máté Matolcsi

Published 2019-04-28Version 1

Let $\Omega$ be a convex body in $\mathbb{R}^d$. We say that $\Omega$ is spectral if the space $L^2(\Omega)$ has an orthogonal basis of exponential functions. There is a conjecture going back to Fuglede (1974) which states that $\Omega$ is spectral if and only if it can tile the space by translations. It has long been known that if a convex body $\Omega$ tiles then it must be a polytope, and it is also spectral. The converse, however, was proved only in dimensions $d \leq 3$ and under the a priori assumption that $\Omega$ is a polytope. In this paper we prove that for every dimension $d$, if a convex body $\Omega \subset \mathbb{R}^d$ is spectral then it must be a polytope, and it can tile the space by translations. The result thus settles Fuglede's conjecture for convex bodies in the affirmative. Our approach involves a construction from crystallographic diffraction theory, that allows us to establish a geometric "weak tiling" condition necessary for the spectrality of $\Omega$.