Non-symmetric convex domains have no basis of exponentials

TL;DR

This paper proves that all non-symmetric convex domains in space cannot have an orthonormal basis of exponentials, thus not being spectral, which advances understanding of Fuglede's conjecture.

Contribution

It generalizes Fuglede's result by showing that every non-symmetric convex domain is not spectral, extending previous specific cases.

Findings

Non-symmetric convex domains are not spectral.

Triangles in the plane are not spectral.

Supports Fuglede's conjecture in convex geometry.

Abstract

A conjecture of Fuglede states that a bounded measurable set $\Omega$ in space, of measure 1, can tile space by translations if and only if the Hilbert space $L^2(\Omega)$ has an orthonormal basis consisting of exponentials. 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 is not spectral.