A note on the intersection property for flat boxes and boxicity in $\mathbb{R}^d$

TL;DR

This paper extends the concept of boxicity and a Helly-type intersection result for boxes in Euclidean space, reducing the dimension of the boxes involved and broadening the understanding of intersection properties.

Contribution

It introduces an extended definition of boxicity and generalizes a Helly-type theorem for 2-piercings in higher-dimensional Euclidean spaces.

Findings

Extended the definition of boxicity.

Generalized Helly-type intersection result for boxes.

Lowered the dimension requirement for intersection properties.

Abstract

By extending the definition of boxicity, we extend a Helly-type result given by Danzer and Grumbaum on 2-piercings of family of boxes in $d$-dimensional Euclidian space by lowering the dimension of the boxes in the ambient space.