ON MAXIMAL S-FREE CONVEX SETS

Let S subset of Z(n) satisfy the property that conv(S) boolean AND Z(n) - S. Then a convex set K is called an S-free convex set if int(K) boolean AND S = empty set 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. [SIAM J. Discrete Math., 24 (2010), pp. 158-168] for the case where S is the set of integer points in a rational polyhedron and a result of Lovasz [Mathematical Programming: Recent Developments and Applications, M. Iri and K. Tanabe, eds., Kluwer, Dordrecht, 1989, pp. 177-210] and Basu et al. [Math. Oper. Res., 35 (2010), pp. 704-720] for the case where S is the set of integer points in some affine subspace of R-n.