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A class of non-convex polytopes that admit no orthonormal basis of exponentials. (English) Zbl 1047.52013
A bounded measurable set \(\Omega\subset\mathbb{R}^d\) of measure 1 is called spectral if the Hilbert space \(L^2(\Omega)\) has an orthonormal basis consisting of exponentials \(e_\lambda(x)= \exp(2\pi i\langle\lambda, x\rangle)\). A conjecture of B. Fuglede [J. Funct. Anal. 16, 101–121 (1974; Zbl 0279.47014)] claims that \(\Omega\) can tile \(\mathbb{R}^d\) by translation if and only if it is spectral. This conjecture has been verified in a number of cases, but in full generality it is open in both directions.
The authors show that, if \(\Omega\) is a not necessarily convex polytope, with the property that, for some direction \(\xi\in S^{d-1}\), the area of the facets of \(\Omega\) with outer normal \(\xi\) exceeds that of the facets with outer normal \(-\xi\), then \(\Omega\) is not spectral. This accords with the (almost obvious) fact that \(\Omega\) cannot tile \(\mathbb{R}^d\) by translation. As a consequence, a spectral convex polytope must be centrally symmetric; this is well known for a translation tile.

52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
42B05 Fourier series and coefficients in several variables
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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