Spectrality of Polytopes and Equidecomposability by Translations

Let A be a polytope in ℝd (not necessarily convex or connected). We say that A is spectral if the space L2(A) has an orthogonal basis consisting of exponential functions. A result due to Kolountzakis and Papadimitrakis (2002) asserts that if A is a spectral polytope, then the total area of the (d-1)-dimensional faces of A on which the outward normal is pointing at a given direction, must coincide with the total area of those (d-1)-dimensional faces on which the outward normal is pointing at the opposite direction. In this paper, we prove an extension of this result to faces of all dimensions between 1 and d-1. As a consequence we obtain that any spectral polytope A can be dissected into a finite number of smaller polytopes, which can be rearranged using translations to form a cube.