Base normal inductive dimension $\mathrm I$ of cubes

It is shown that $\{1,\infty\}$ is the set of possible base normal inductive dimensions $\mathrm I$ of the segment $I=[0,1]$ and $\{n,n+1,\dots,\infty\}$ is the set of possible base normal inductive dimensions $\mathrm I$ of the $n$-dimensional cubes $I^n$ for $n\geq2$.