How to find an optimal covering number of a rectangle $A$ in $\mathbb{R}^{m^2+m}$ where the $m^2$ sides have length $2$ and the $m$ sides have length $r>0$?
An optimal covering number is defined as the minimum number of balls of radius $\epsilon>0$, where a ball at a point $x$ is given by $$\mathbb{B}_{\epsilon}(x)=\{y\in\mathbb{R}^{m^2+m}~|~\|x-y\|<\epsilon \}.$$
How to even think of(or define) such a rectangle in $\mathbb{R}^{m^2+m}$? There are some standard results in the case of $m=1$, i.e., in $\mathbb{R}^{2}$. There are some results on bounds of minimal covering number of a rectangle by the circles of radius $\epsilon$.
Please let me know your thoughts, or any references I should be reading to tackle even for the case of $n=2$, i.e., $A\in \mathbb{R}^{4+2}$?