Definition (Valid Inequality)
Let \(P\subseteq\mathbb{R}^d\) be a convex polytope. A linear inequality
\[
cx\leq c_0
\]
is called a valid inequality for \(P\) if it is satisfied for all points \(x\in P\).
Definition (Face)
Let \(P\subseteq\mathbb{R}^d\) be a convex polytope. A face of \(P\) is any set of the form
\[
F=P\cap\{x\in\mathbb{R}^d:cx=c_0\}
\]
where \(cx\leq c_0\) is a valid inequality for \(P\).
Note that first we must have a valid inequality for \(P\). Once we have a valid inequality \(cx\leq c_0\), the equality part intersection \(P\) is the part of \(P\) that touches the supporting hyperplane \(cx=c_0\).
Definition (Dimension of a Face)
The dimension of a face \(F\) is the dimension of its affine hull:
\[
\dim(F):=\dim(\operatorname{aff}(F)).
\]
The polytope \(P\) itself is also a face of \(P\). Moreover, a face \(F\) of a polytope \(P\) is called a proper face if \(F\subsetneq P\). The faces of dimensions \(0\), \(1\), \(\dim(P)-2\), and \(\dim(P)-1\) are called vertices, edges, ridges, and facets, respectively. So a vertex of a polytope \(P\) is a face of dimension \(P\). Equivalently, vertices are the minimal nonempty faces of \(P\). The set of all vertices of \(P\) is denoted by \(\operatorname{vert}(P)\)