On Maximal $S$-Free Convex Sets

• Authors: Diego A. Morán R.;Santanu S. Dey
• Venue: SIAM Journal on Discrete Mathematics
• Year: 2011

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 $\mathbb{R}^n$.