# Facet

A face of a convex set, not equal to the set, of maximal dimension. If the set is polyhedral, say $LaTeX: P = \{x: Ax \le b\}$, where the defining inequalities are irredundant, then the facets are in one-to-one correspondence with $LaTeX: \{x \in P: A_{i,:} x = b_i\}$ for $LaTeX: i$ such that the equality is not forced – i.e., there exists $LaTeX: x$ in $LaTeX: P$ for which $LaTeX: A_{i,:} x < b_i$. Here $LaTeX: A_{i,:}$ is the i-th row of $LaTeX: A$.