On the volume of sections of the cube

The problem of volume extrema of the intersection of the standard $n$-dimensional cube $\square^n=[-1,1]^n$ with a $k$-dimensional linear subspace $H$ has been studied intensively. The celebrated Vaaler theorem says that only the coordinate subspaces are the volume minimizers. Using the Brascamb-Lieb inequality, K. Ball proved two upper bounds which are tight for some $k$ and $n$. Typically, methods of functional analysis or some tricky inequalities for measures are used in such problems. In this talk, we will discuss a 'naive' variational principle for the problem of volume extrema of $\square^n \cap H$ and some geometrical consequences of this principle. Particularly, we will sketch how to find all planar maximizers ($k=2$). Planar maximizers were unknown for all odd $k$ starting with $5$.

Joint work with Igor Tsiutsiurupa.