Transversal numbers over subsets of linear spaces

Abstract

Let $M$ be a subset of $\mathbb{R}^k$. It is an important question in the theory of linear inequalities to estimate the minimal number $h=h(M)$ such that every system of linear inequalities which is infeasible over $M$ has a subsystem of at most $h$ inequalities which is already infeasible over $M.$ This number $h(M)$ is said to be the Helly number of $M.$ In view of Helly's theorem, $h(\mathbb{R}^n)=n+1$ and, by the theorem due to Doignon, Bell and Scarf, $h(\mathbb{Z}^d)=2^d.$ We give a common extension of these equalities showing that $h(\mathbb{R}^n \times \mathbb{Z}^d) = (n+1) 2^d.$ We show that the fractional Helly number of the space $M \subseteq \mathbb{R}^d$ (with the convexity structure induced by $\mathbb{R}^d$) is at most $d+1$ as long as $h(M)$ is finite. Finally we give estimates for the Radon number of mixed integer spaces.