Let $A \in \mathbb{R}^{m \times n}$, $b \in \mathbb{R}^{m}$. Consider the Polyhedron $\mathcal{P}$ given by
$$ \mathcal{P} = \{x \in \mathbb{R}^{n} \mid Ax \leq b\}. $$
Is it true that $\mathcal{P}$ has only finitely many vertices? If not, under which condition it has and is there an upper bound for the numbers of vertices of a convex polyhedron depent on $m$ and $n$?