Cubes in Rn

Subjects: Vector Analysis
Def: A cube in ${\mathbb{R}}^n$ is a rectangle $C\subseteq {\mathbb{R}}^n$ with all of its dimensions being equal, i.e. $C=\prod _{i=1}^n[a_i,b_i]$, then for any $1\le i,j\le n$, $(b_j-a_j)=(b_i-a_i)$. The value $s=b_i-a_i$ is called the cube’s dimension.

Prop: If $C=\prod _{i=1}^n[a_i,b_i]$ is a cube with dimension $s$ and center $c=(c_i{)}_{i=1}^n\in C$, then $c_i=a_i+\frac{s}{2}$ for all $i$ , and the diagonal of $C$ has length of $s\sqrt{n}$, i.e. $d(C)=s\sqrt{n}$

Prop: Let $R=\prod _{i=1}^n[a_i,b_i]$ be a rectangle with rational dimensions, then there’s a partition of $R$ of cubes, with the same rational dimension.

This can be made stronger into a version where the dimension of the cubes can be arbitrarily small dimension, while preserving the rationality of the dimension.

Lemma: Let $R_1,\dots ,R_k\subseteq {\mathbb{R}}^n$ are rectangles, for any $\epsilon >0$ exists a finite amount of cubes $C_1,\dots ,C_l$ with rational dimensions that

and