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.


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