On finitely generated closures in the theory of cutting planes

Let PP be a rational polyhedron in RdRd and let LL be a class of dd-dimensional maximal lattice-free rational polyhedra in RdRd. For L∈LL∈L by RL(P)RL(P) we denote the convex hull of points belonging to PP but not to the interior of LL. Andersen, Louveaux and Weismantel showed that if the so-called max-facet-width of all L∈LL∈L is bounded from above by a constant independent of LL, then ⋂L∈LRL(P)⋂L∈LRL(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 LL. The presented results are motivated by applications in cutting-plane theory from mixed-integer optimization.