On Maximal S-Free Convex Sets

Abstract

Let $S\subseteq\mathbb{Z}^n$ satisfy the property that $\mathrm{conv}(S)\cap\mathbb{Z}^n=S$. Then a convex set K is called an S-free convex set if $\mathrm{int}(K)\cap S=\emptyset$. 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 Lovász [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 $\mathbb{R}^n$.