On maximal $S$-free sets and the Helly number for the family of $S$-convex sets

Details On maximal $S$-free sets and the Helly number for the family of $S$-convex sets On maximal $S$-free sets and the Helly number for the family of $S$-convex sets

2011-09-28

arxiv.org/abs/1109.6179v2

Mathematics

Optimization and Control

Scientific paper

Let $S$ be a subset of $\mathbb{R}^d$. A subset $K$ of $\mathbb{R}^d$ is said to be $S$-free if $K$ is closed, convex and the interior of $K$ is disjoint with $S$. An $S$-free set $K$ is said to be maximal if $K$ is not properly contained in another $S$-free set. We present a condition on $S$ which guarantees that every maximal $S$-free set is a polyhedron with at most $f$ facets, where the bound $f$ depends only on $S$. This condition on $S$ is formulated in terms of the Helly number for the family of $S$-convex sets. The presented result yields corollaries related to the cutting-plane theory from integer and mixed-integer optimization.

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