论文标题
带有盒子的离散定量helly型定理
Discrete quantitative Helly-type theorems with boxes
论文作者
论文摘要
在组合凸的几何形状中对Helly型定理的研究已使用Doignon定理的证人集和定量扩展产生了Helly定理的体积版本。本文结合了这些哲学，并列出了整数晶格的定量Helly型定理，并将轴平行的盒子作为证人集。我们的主要结果表明，虽然整数晶格的定量螺旋数在每个固定尺寸中多一部分生长，但它们的变体带有盒子，作为证人集均匀地界定。我们在该定理上证明了几种色彩差异和分数变化。我们还证明，即使$ a \ subseteq \ mathbb {r}^2 $，即使$ a \ a \ subseteq \ subseteq \ mathbb {z} $是一个综合集，也不需要有限。
Research on Helly-type theorems in combinatorial convex geometry has produced volumetric versions of Helly's theorem using witness sets and quantitative extensions of Doignon's theorem. This paper combines these philosophies and presents quantitative Helly-type theorems for the integer lattice with axis-parallel boxes as witness sets. Our main result shows that, while quantitative Helly numbers for the integer lattice grow polynomially in each fixed dimension, their variants with boxes as witness sets are uniformly bounded. We prove several colorful and fractional variations on this theorem. We also prove that the Helly number for $A \times A \subseteq \mathbb{R}^2$ need not be finite even when $A \subseteq \mathbb{Z}$ is a syndetic set.
