On finitely generated closures in the theory of cutting planes

Let $P$ be a rational polyhedron in ${‐{R}}^d$ and let $\mathcal{L}$ be a class of $d$ ‐dimensional maximal lattice‐free rational polyhedra in ${‐{R}}^d$ . For $L\in \mathcal{L}$ by $R_L\left(P\right)$ we denote the convex hull of points belonging to $P$ but not to the interior of $L$ . Andersen, Louveaux and Weismantel showed that if the so‐called max‐facet‐width of all $L\in \mathcal{L}$ is bounded from above by a constant independent of $L$ , then ${\cap }_{L\in \mathcal{L}}{R}_L\left(P\right)$ is a rational polyhedron. We give a short proof of a generalization of this result. We also give a characterization for the boundedness of the max‐facet‐width on $\mathcal{L}$ . The presented results are motivated by applications in cutting‐plane theory from mixed‐integer optimization.