I have a full dimensional polyhedron $P_1 \subseteq \mathbb{R}^d.$ Now i define another polyhedron as follows:
$$P_2 = AP_1 \oplus B$$ with $A \in \mathbb{R}^{(d-1) \times d}, \,\, B \in \mathbb{R}^{d-1}.$
I wonder if one can say something about the number of facets of $P_1$ and $P_2,$ denoted as $\mathcal{H}(P_1)$ and $\mathcal{H}(P_2),$ e.g.
$$\mathcal{H}(P_1) \geq \mathcal{H}(P_2).$$
Sincerely.