On Maximal $S$-Free Sets and the Helly Number for the Family of $S$-Convex Sets

Abstract

We study two combinatorial parameters, which we denote by $f(S)$ and $h(S)$, associated with an arbitrary set $S \subseteq \mathbb{R}^d$, where $d \in \mathbb{N}$. In the nondegenerate situation, $f(S)$ is the largest possible number of facets of a $d$-dimensional polyhedron $L$ such that the interior of $L$ is disjoint with $S$ and $L$ is inclusion-maximal with respect to this property. The parameter $h(S)$ is the Helly number of the family of all sets that can be given as the intersection of $S$ with a convex subset of $\mathbb{R}^d$. We obtain the inequality $f(S) \le h(S)$ for an arbitrary $S$, and the equality $f(S)=h(S)$ for every discrete $S$. Furthermore, motivated by research in integer and mixed-integer optimization, we show that $2^d$ is the sharp upper bound on $f(S)$ in the case $S = (\mathbb{Z}^d \times \mathbb{R}^n) \cap C$, where $n \ge 0$ and $C \subseteq \mathbb{R}^{d+n}$ is convex. The presented material generalizes and unifies results of various authors, including the result $h(\mathbb{Z}^d) = 2^d$ of Doignon, the related result $f(\mathbb{Z}^d)=2^d$ of Lovász, and the inequality $f(\mathbb{Z}^d \cap C) \le 2^d$, which has recently been proved for every convex set $C \subseteq \mathbb{R}^d$ by Morán and Dey.