Relative interior

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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 \}.

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