Non-symmetric convex domains have no basis of exponentials

Abstract

A conjecture of Fuglede states that a bounded measurable set Ω ⊂ ℝd, of measure 1, can tile ℝd by translations if and only if the Hilbert space L2(Ω) has an orthonormal basis consisting of exponentials eλ(x) = exp 2πi(λ, x). If Ω 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 ℝd is not spectral.