Abstract

Given a subset $S$ of $\mathbb{R}^d$, the Helly number $h(S)$ is the largest size of an inclusionwise minimal family of convex sets whose intersection is disjoint from $S$. A convex set is $S$-free if its interior contains no point of $S$. The parameter $f(S)$ is the largest number of maximal faces in an inclusionwise maximal $S$-free convex set. We study the relation between the parameters $h(S)$ and $f(S)$. Our main result is that $h(S)\le (d+1)f(S)$ for every nonempty proper closed subset $S$ of $\mathbb{R}^d$. We also study the Helly number of the Cartesian product of two discrete sets.

Keywords

  1. $S$-free convex sets
  2. Helly number
  3. cut-generating functions

MSC codes

  1. 52C07
  2. 90C10

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Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 2206 - 2216
ISSN (online): 1095-7146

History

Submitted: 29 February 2016
Accepted: 23 September 2016
Published online: 29 November 2016

Keywords

  1. $S$-free convex sets
  2. Helly number
  3. cut-generating functions

MSC codes

  1. 52C07
  2. 90C10

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Progetto di Ateneo 2013

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