Definition of Hypercube in Mathematics
Definition
The set $[0, 1]^{n}$ obtained by taking the closed interval $[0, 1] \subset \mathbb{R}$’s Cartesian product with itself $n$ times is called the $n$-dimensional unit hypercube.
Description
When a theorem or technique assumes a hypercube, one usually thinks of it as a compact, convex subset of Euclidean space. In particular, in data science it is often sufficient to regard it as each independent component having a prescribed range. You do not have to consider infinity (bounded), you do not need to worry about endpoints (closed), and the set is not unexpectedly hollow in the middle (convex) — it is a well-behaved set.
$n$-cell
If the $k = 1 , \cdots , n$-th interval is replaced by $\left[ a_{k} , b_{k} \right] \subset \mathbb{R}$ instead of $[0, 1]$, the result is called a $n$-cell. Since there exists a linear bijection between such a cell and the unit hypercube, they are essentially the same concept.
$n$-unit disk
One can imagine that if a hypercube is a boxy shape, then the $n$-disk is the corresponding rounded shape.