  We study the volume of central hyperplane sections of the cube. Using Fourier
analytic and variational methods, we retrieve a geometric condition
characterizing critical sections which, by entirely different methods, was
recently proven by Ivanov and Tsiutsiurupa. Using this characterization result,
we prove that critical central hyperplane sections in the 3-dimensional case
are all diagonal to a (possibly lower dimensional) face of the cube, while in
the 4-dimensional case, they are either diagonal to a face, or, up to permuting
the coordinates and sign changes, perpendicular to the vector $(1,1,2,2)$. This
shows the existence of non-diagonal critical central sections.
