On the Rank of Disjunctive Cuts

Let ℒ be a family of lattice-free polyhedra in ℝ^m containing the splits. Given a polyhedron P in ℝ^{m + n}, we characterize when a valid inequality for P ∩ (ℤ^m × ℝⁿ) can be obtained with a finite number of disjunctive cuts corresponding to the polyhedra in ℒ. We also characterize the lattice-free polyhedra M such that all the disjunctive cuts corresponding to M can be obtained with a finite number of disjunctive cuts corresponding to the polyhedra in ℒ for every polyhedron P. Our results imply interesting consequences, related to split rank and to integral lattice-free polyhedra, that extend recent research findings.