Non-diagonal critical central sections of the cube

We study the (n -1)-dimensional volume of central hyperplane sections of the n-dimensional cube Q n .Our main goal is two-fold: first, we provide an alternative, simpler argument for proving that the volume of the section perpendicular to the main diagonal of the cube is strictly locally maximal for every n ≥ 4, which was shown before by L. Pournin [27].Then, we prove that non-diagonal critical central sections of Q n exist in all dimensions at least 4. The crux of both proofs is an estimate on the rate of decay of the Laplace-Pólya integral J n (r ) = ∞ -∞ sinc n t • cos(r t ) dt that is achieved by combinatorial means.This also yields improved bounds for Eulerian numbers of the first kind.