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 to an arbitrary set S \subseteq R^d, where d \in 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 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 = (Z^d \times R^n) \cap C, where n \ge 0 and C \subseteq R^{d+n} is convex. The presented material generalizes and unifies results of various authors, including the result h(Z^d) = 2^d of Doignon, the related result f(Z^d)=2^d of Lov\'asz and the inequality f(Z^d \cap C) \le 2^d, which has recently been proved for every convex set C \subseteq R^d by Dey & Mor\'an.