Maximal $S$-Free Convex Sets and the Helly Number

Given a subset S of R-d, the Helly number h(S) is the largest size of an inclusionwise minimal family of convex sets whose intersection is disjoint from S. A convex set is S-free if its interior contains no point of S. The parameter f(S) is the largest number of maximal faces in an inclusionwise maximal S-free convex set. We study the relation between the parameters h(S) and f(S). Our main result is that h(S) <= (d + 1) f(S) for every nonempty proper closed subset S of R-d. We also study the Helly number of the Cartesian product of two discrete sets.