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A conjecture of Fuglede states that a bounded measurable set ⊂ R d , of measure 1, can tile R d by translations if and only if the Hilbert space L 2 () has an orthonormal basis consisting of exponentials e(x) = exp 2πih λ, xi . If has the latter property it is called spectral. Let be a polytope in R d with the following property: there is a direction ξ ∈ S d 1 such that, of all the polytope faces perpendicular to ξ, the total area of the faces pointing in the positive ξ direction is more than the total area of the faces pointing in the negative ξ direction. It is almost obvious that such a polytope cannot tile space by translation. We pr ove in this paper that such a domain is also not spectral, which agrees with Fuglede’s conjecture. Let be a measurable subset of R d , which we take for convenience to be of measure 1. Let also Λ be a discrete subset of R d . We write eλ(x) = exp 2πih λ, xi , (λ, x ∈ R d ), E = {eλ : λ ∈ Λ} ⊂ L 2 () . The inner product and norm on L 2 () are

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