Discrete quantitative Helly-type theorems with boxes

TL;DR

This paper introduces new quantitative Helly-type theorems for integer lattices using axis-parallel boxes, showing bounded Helly numbers in fixed dimensions and exploring variants and counterexamples.

Contribution

It combines volumetric and combinatorial approaches to establish bounded Helly numbers with boxes as witness sets and extends the theory with colorful, fractional, and counterexample results.

Findings

Helly numbers for integer lattices grow polynomially in fixed dimensions.

Helly numbers with boxes as witness sets are uniformly bounded.

Counterexamples show Helly numbers can be infinite for certain sets.

Abstract

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.