Helly numbers of subsets of ℝᵈ and sampling techniques in optimization

We present Helly-type theorems where the convex sets are required to intersect a subset S of R d . This is a continuation of prior work for S = R d , Z d , and Z d k R k (motivated by mixed-integer optimization). We are particularly interested in the case when S has some algebraic structure, in particular when S is a subgroup or the dierence between a lattice and some sublattices. We give sharp bounds on the Helly numbers for S in several cases. By abstracting the ingredients of a general method we obtain colorful versions of many monochro- matic Helly-type results, including several of our results. In the second part of the article we discuss a notion of S-optimization that generalizes continuous, integral, and mixed-integer optimization and show that two well-known random- ized sampling algorithms, Clarkson's and Calaore-Campi's algorithms, can be extended to work with more sophisticated variables over S as long as the S-Helly number h(S) is nite.