# $\mathcal{H}$-polyhedron under a linear map

Let $P = \{ x \in \mathbb{R}^n \mid Ax \leq b \}$ be a (bounded) polyhedron for $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^m$, $n,m > 0$.

Moreover, let $M \colon \mathbb{R}^n \to \mathbb{R}^p$ be a linear map for $p \leq n$.

I'm interested in computing a $\mathcal{H}$-representation of $M \cdot P = \{Cx \mid x \in P\}.$ Are there any known algorithms, besides the "Quantifier elimination for linear arithmetic" as pointed out in 1?

Yes, by the Motzkin Double Description Method you compute the $V$-representation of $P,$ hence the $V$-representation of $M P,$ then, by the double description method (again) the $H$-representation of $M P.$