线性空间子集上的transmittance数
Transversal numbers over subsets of linear spaces
摘要：
令$M$是$\mathbb{R}^k$的子集。在线性不等式理论中，估计最小数$h=h(M)$使得在$M$上不可行的每个线性不等式组都有一个至多$h$不等式的子系统，该子系统在$M.$上已经是不可行的。这个数$h(M)$被称为$M.$的Helly数。鉴于Helly定理，$h(\mathbb{R}^n)=n+1$，并根据Doignon，Bell和Scarf的定理，$h(\mathbb{Z}^d)=2^d.$我们给出了这些等式的一个共同的扩展，表明$h(\mathbb{R}^n \times \mathbb{Z}^d) = (n+1) 2^d.$我们证明了空间$M \subseteq \mathbb{R}^d$（具有由$\mathbb{R}^d$诱导的凸结构）的分数Helly数至多是$d+1$，只要$h(M)$是有限的。最后给出了混合整数空间的Radon数的估计。
英文摘要：
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.
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