Helly numbers of algebraic subsets of ℝ d and an extension of Doignon’s Theorem
Jesús A. De Loera, Reuben N. La Haye, Déborah Oliveros, Edgardo Roldán-Pensado · Advances in Geometry · 2017
Abstract We study S-convex sets, which are the geometric objects obtained as the intersection of the usual convex sets in ℝ d with a proper subset S ⊂ ℝ d , and contribute new results about their S-Helly numbers. We extend prior work for S = ℝ d , ℤ d , and ℤ d−k × ℝ k , 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 ℝ d 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.