Transversal numbers over subsets of linear spaces

TL;DR

This paper extends classical Helly and Radon theorems to mixed integer spaces, providing bounds on Helly and fractional Helly numbers, and estimates for Radon numbers, thus advancing the understanding of linear inequalities over such spaces.

Contribution

It generalizes Helly's theorem to mixed spaces and establishes bounds on fractional Helly and Radon numbers for these spaces, unifying previous results.

Findings

Helly number for mixed spaces: (n+1) 2^d

Fractional Helly number is at most d+1 for spaces with finite Helly number

Provides estimates for Radon numbers in mixed integer 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.