On the volume of non-central sections of a cube

TL;DR

This paper establishes a lower bound for the volume of non-central sections of high-dimensional cubes, showing the bound depends only on the codimension, with specific results for hyperplanes and a complex analogue.

Contribution

It provides a dimension-independent lower bound for the volume of non-central cube sections, including a specific bound for hyperplanes and an extension to complex polydiscs.

Findings

Lower bound c(d) depends only on codimension d

For hyperplanes, c(1) = 1/17 is achievable

Extension to complex polydisc sections

Abstract

Let $Q_n$ be the cube of side length one centered at the origin in $\mathbb{R}^n$, and let $F$ be an affine $(n-d)$-dimensional subspace of $\mathbb{R}^n$ having distance to the origin less than or equal to $\frac 1 2$, where $0<d<n$. We show that the $(n-d)$-dimensional volume of the section $Q_n \cap F$ is bounded below by a value $c(d)$ depending only on the codimension $d$ but not on the ambient dimension $n$ or a particular subspace $F$. In the case of hyperplanes, $d=1$, we show that $c(1) = \frac{1}{17}$ is a possible choice. We also consider a complex analogue of this problem for a hyperplane section of the polydisc.