On Maximal S-free Convex Sets

Let S ⊆ Zn satisfy the property that conv(S) ∩ Zn = S. Then a convex set K is called an S-free convex set if int(K) ∩ S = ∅. A maximal S-free convex set is an S-free convex set that is not properly contained in any S-free convex set. We show that maximal S-free convex sets are polyhedra. This result generalizes a result of Basu et al. [6] for the case where S is the set of integer points in a rational polyhedron and a result of Lovász [18] and Basu et al. [5] for the case where S is the set of integer points in some affine subspace of Rn.