# Face

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A convex subset, say $LaTeX: S$, of a convex set, say $LaTeX: C$, such that for any $LaTeX: x$ and $LaTeX: y$ in $LaTeX: C$

$LaTeX: \{(1 - \alpha) x + \alpha y : 0 \le \alpha \le 1 \} \cap \mbox{ri}(S) \neq \emptyset \Rightarrow x, \; y \in S$.

The set $LaTeX: C$ is itself a face of $LaTeX: C$, and most authors consider the empty set a face. The faces of zero dimension are the extreme points of $LaTeX: C$. When $LaTeX: C$ is a polyhedron, i.e. $LaTeX: \{x : Ax \le b\}$, the faces are the subsystems with some inequalities holding with equality: $LaTeX: \{x: Bx = c, \; Dx \le d\}$, where $LaTeX: A = [B \; D]$ and $LaTeX: b = [c^T \; d^T]^T$.