Critical central sections of the cube

TL;DR

This paper characterizes critical central hyperplane sections of the cube in three and four dimensions, revealing new geometric structures and the existence of non-diagonal critical sections using Fourier and variational methods.

Contribution

It introduces a new geometric characterization of critical sections and identifies their specific forms in 3D and 4D, including non-diagonal cases.

Findings

Critical sections in 3D are diagonal to a face.

In 4D, critical sections are either diagonal or perpendicular to (1,1,2,2).

Existence of non-diagonal critical sections is demonstrated.

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.