How do we calculate the Cartesian product between $n$ polygons? Is it always a polytope? If so, can we say anything about its faces?
For example, $n$ squares: $\{S_1:[a_1, a_2] \times [b_1, b_2], \ldots, S_n:[a_k, a_{k+1}] \times [b_{k}, b_{k+1}]\}$ so, $S_1 \times \ldots \times S_n \in \mathcal{R}^n$ forms a $n-$dimensional box.
Cartesian products of polytopes $P,Q$ are indeed polytopes again. The vertices are pairs of vertices $(p,q)$ where $p$ is a vertex of $P$, and $q$ is a vertex of $Q$.
In fact, if you have two polytopes, say $P \subset \mathbb{R}^n$ and $Q \subset \mathbb{R}^m$, then all faces of $P \times Q$ are just the Cartesian products of faces of $P$ and $Q$. You can see this by using the hyperplane descriptions: the hyperplanes used to define $P \times Q$ are the same as those for $P$ and $Q$, but now they are using disjoint sets of variables. So, if $F \subset P \times Q$ is a face, then the first $n$ coordinates of every point in $F$ must satisfy equalities for some face of $P$ and the last $m$ coordinates of every point in $F$ must satisfy equalities for some face of $Q$.