UC Davis

We study S-convex sets, which are the geometric objects obtained as the intersection of the usual convex sets in Rd with a proper subset S ? Rd, and contribute new results about their S-Helly numbers. We extend prior work for S = Rd, Zd, and Zd-k × Rk, and give some sharp bounds for several new cases: low-dimensional situations, sets that have some algebraic structure, in particular when S is an arbitrary subgroup of Rd or when S is the difference between a lattice and some of its sublattices. By abstracting the ingredients of Lovász method we obtain colorful versions of many monochromatic Helly-Type results, including several colorful versions of our own results.