Non-symmetric convex domains have no 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 spectral. We generalize a result of Fuglede, that a triangle in the plane is not spectral, proving that every non-symmetric convex domain in $\mathbb{R}^{d}$ is not spectral.