Proving the Facet Property of Dual Polytopes with a Vertex in the Interior

[Combinatorics]

Homework Statement

Let v be a vertex of a d-polytope P such that $0 \in intP$ .

Prove that $P^* \cap \{y \in \mathbb{R}^d \mid\left < y, v\right>=1\ \}$ is a facet of $P^{*}$.

Thanks

Homework Equations

The definitions are:

$P^*=\{ y\in\mathbb{R}^{d}\mid\left < x, y\right>\leq 1\ \forall x\in P\}$

and a face of P is the empty set, P itself, or an intersection of P with a supporting hyperplane (i.e.- a hyperplane, such that P is located in one of the halfspaces it determines).

A facet is a face of maximal degree

The Attempt at a Solution

I tried showing that if it isn't a facet (the fact that it's a face is obvious), we can delete one of the vertices that form $P^{*}$ but without any success.

 

[lippyka]

I also tried using the fact that 0\in intP in order to find a contradiction. Any help would be appreciated.