Products of polytopes and the normals of their facets I need to compute the normals of the facets of certain polytopes that can be represented as products of polytopes in smaller dimensions.
Searching the bibliography I found that the facets of the product of two polytopes $P_1$ and $P_2$ are all the products of one of the two polytopes with the facets of the other (this means that if $P_1$ has $m_1$ facets and $P_2$ has $m_2$ facets then the total number of facets is $m_1+m_2$).
Intuitively, I think that the normals of the facets of $P_1 \times P_2$ are of the form $(\mathbf{p_1}, \mathbf{0})$ and $( \mathbf{0},\mathbf{p_2})$ where $\mathbf{p_1}$ and $\mathbf{p_2}$ are normals of the facets of $P_1$ and $P_2$ respectively. 
Is this right?
Where can I find in the bibliography a formal proof for this statement?
 A: If $facets(P)=\{P_i\}_i$ and $facets(Q)=\{Q_j\}_j$, then 
$$facets(P\times Q)=\{P_i\times Q\}_i\cup\{P\times Q_j\}_j$$
$$ $$
If $\vec{n}_{P_i}$ is the outer normal vector of $P_i$ wrt. $span(P)$ (similar for $Q$), if 
$\vec{n}_{P_i\times Q}$ is the outer normal vector of $P_i\times Q$ wrt. $span(P\times Q)$ (similar for $P\times Q_j$), and if $\vec{0}_P$ is the origin of $span(P)$ (similar for $Q$), then those facet normals are mutually perpendicular, i.e.
$$\vec{n}_{P_i\times Q}\cdot\vec{n}_{P\times Q_j}=0\ \ \text{for each}\ i,j$$
and hence indeed
$$\vec{n}_{P_i\times Q}=\left({\vec{n}_{P_i}}^T,\ {\vec{0}_Q}^T\right)^T$$
(and similar for $P\times Q_j$).
--- rk
A: I would say this statement is sufficiently elementary, so that it does not require an explicit source, or can be proved in a few lines (see below).

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A (non-empty) face of a polytope is the set of points that maximize some linear functional $\<n,\cdot\,\>$, and then $n$ is a normal of that face. If the face is a facet, this normal is unique (up to scaling).
A point $(p_1,p_2)\in P_1\times P_2\subset\Bbb R^{d_1+d_2}$ maximizes the functional $\<n,\cdot \,\>, n=(n_1,n_2)\in\Bbb R^{d_1+d_2}$ on $P_1\times P_2$ if and only if $p_1$ maximizes $\<n_1,\cdot\,\>$ on $P_1$ and $p_2$ maximizes $\<n_2,\cdot\,\>$ on $P_2$.
This shows that the faces of the product are the products of faces of the factors.
So any facet of $P_1\times P_2$ is a product of, say, $P_1$ and a facet $F\subset P_2$.
The normal $(n_1,n_2)$ of that facet must be so, so that $n_1$ is maximized on all of $P_1$ (hence must be zero) and $n_2$ is maximized on all of $F$ (hence must be a normal of $F$).
So the normal of any facet is indeed $(n_1,0)$ or $(0,n_2)$, where $n_1,n_2$ are normals of facets respectively.
