Faces number of convex polyhedra

Question

I have set of vectors $v_1, \dots v_n$in $\mathbb{R}^n$.

Let $P = P(v_1, \ldots, v_n) = \{x \in \mathbb{R^n} \ |\ (v_i,x)\geq0 \ \forall i = 1, \ldots ,k\}$.

Let $P^* = \{z \in \mathbb{R^n} \ | \ (z, w) \geq 0 \ \forall w\in P \}$.

For all vectors $d \in P^*$ define face $S_d = \{w \in P \ | \ (w, d) = 0\}$ $\subset P$.

How to prove that set of such faces for $P$ is finite?

Answer 1

Hint: Here $P$ is a polyhedral and in any polyhedral the number of faces is finite, since the number of its hyperplane is finite ! and $S_d$ is indeed a face

Answer 2

Show first that $P$ is the convex hull of a finite number of points.

Then show that every face is the convex hull of a subset of those points.