A class of non-convex polytopes that admit no orthonormal basis of exponentials

Abstract

A conjecture of Fuglede states that a bounded measurable set $\Omega\subset\mathbb{R}^d$, of measure $1$, can tile $\mathbb{R}^d$ by translations if and only if the Hilbert space $L^2(\Omega)$ has an orthonormal basis consisting of exponentials $e_\lambda(x) = \exp \{2\pi i\langle{\lambda},{x}\rangle\}$. If $\Omega$ has the latter property it is called {\em spectral}. Let $\Omega$ be a polytope in $\mathbb{R}^d$ with the following property: there is a direction $\xi \in S^{d-1}$ such that, of all the polytope faces perpendicular to $\xi$, the total area of the faces pointing in the positive $\xi$ direction is more than the total area of the faces pointing in the negative $\xi$ direction. It is almost obvious that such a polytope $\Omega$ cannot tile space by translation. We prove in this paper that such a domain is also not spectral, which agrees with Fuglede's conjecture. As a corollary, we obtain a new proof of the fact that a convex body that is spectral is necessarily symmetric, in the case where the body is a polytope.