# Relative interior

The interior of a set when considered in the smallest subspace containing it. In particular, the polyhedron, $LaTeX: \left \{x: Ax \ge b \right \},$ can be defined as the intersection of the inequalities that are forced, say $LaTeX: \textstyle \left \{x: Qx = q \right \},$ and the others, say $LaTeX: \textstyle \left \{x: Px \ge p \right \}$ (so $LaTeX: \textstyle A = \begin{bmatrix} P & Q \end{bmatrix}$ and $LaTeX: \textstyle b = \begin{bmatrix} p & q \end{bmatrix}$ ). Then, the relative interior of the original polyhedron is $LaTeX: \textstyle \left \{x: Qx = q \mbox{ and } Px > p \right \}.$