Skip to main content
eScholarship
Open Access Publications from the University of California

Helly numbers of algebraic subsets of Rdand an extension of doignon's theorem

  • Author(s): De Loera, JA
  • La Haye, RN
  • Oliveros, D
  • Roldán-Pensado, E
  • et al.
Abstract

© 2017 by Walter de Gruyter Berlin/Boston 2017. We study S-convex sets, which are the geometric objects obtained as the intersection of the usual convex sets in Rdwith 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 Rdor 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.

Many UC-authored scholarly publications are freely available on this site because of the UC Academic Senate's Open Access Policy. Let us know how this access is important for you.

Main Content
Current View