2015/08/10 by Jesús A. De Loera, R. N. La Haye, De Loera, J. A. +5
Computer Science · Mathematics · #52A35 (Primary) #52C07 (Secondary) #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1508.02380
openalex publication_date 2015/08/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study S-convex sets, which are the geometric objects obtained as the intersection of the usual convex sets in \mathbb Rd with a proper subset S⊂ \mathbb Rd. We contribute new results about their S-Helly numbers. We extend prior work for S=\mathbb Rd, \mathbb Zd, and \mathbb Zd-k×\mathbb Rk; we give sharp bounds on the S-Helly numbers in several new cases. We considered the situation for low-dimensional S and for sets S that have some algebraic structure, in particular when S is an arbitrary subgroup of \mathbb 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.