On finitely generated closures in the theory of cutting planes

Let $P$ be a rational polyhedron in $\mathbb{R}^d$ and let $\mathcal{L}$ be a class of $d$-dimensional maximal lattice-free rational polyhedra in $\mathbb{R}^d$. For $L \in \mathcal{L}$ by $R_L(P)$ 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 $\bigcap_{L\in \mathcal{L}} R_L(P)$ 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.