Critical central sections of the cube

Abstract

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.

Differs from the published version in minor technical corrections due to degenerate cases

Cited by in corpus (3)

• Stability of polydisc slicing
• Non-diagonal critical central sections of the cube
• Estimates on the decay of the Laplace-Polya integral