Optimal face of a linear program ($\{x\in P:c^{T}x=z^{\ast }\}$)

The set of minimizers of a linear objective on a linear polyhedron. It is a face: if a strict convex combination of two feasible points minimizes the objective, both points must also minimize it. Hence an extreme point of an optimal face is an extreme point of the original polyhedron. Successively minimizing further linear objectives restricts to nested faces, whose vertices remain original vertices. This preserves uniform rational denominator bounds in exact feasibility-oracle optimization.