Let $P$ be a polyhedron in $\mathbb{R}^n$ and $\omega \in \mathbb{R}^n$, viewed as a linear functional
$\text{face}_{\omega}= \{ u \in P : \omega\cdot u \geq \omega\cdot v\mbox{ for all }v \in P \}$.
How does this definition match with general notion of the face of a polyhedron?