Stabbing boxes with finitely many axis-parallel lines and flats

In this short note, we provide the necessary and sufficient condition for an infinite collection of axis-parallel boxes in \(\mathbb{R}^{d}\) to be pierceable by finitely many axis-parallel \(k\)-flats, where \(0 \leq k < d\). We also consider colorful generalizations of the above result and esta...

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Bibliographic Details
Published in:	arXiv.org
Main Authors:	Chakraborty, Sutanoya, Ghosh, Arijit, Nandi, Soumi
Format:	Paper
Language:	English
Published:	Ithaca Cornell University Library, arXiv.org 11.12.2024
Subjects:	Boxes
ISSN:	2331-8422
Online Access:	Get full text
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Summary:	In this short note, we provide the necessary and sufficient condition for an infinite collection of axis-parallel boxes in \(\mathbb{R}^{d}\) to be pierceable by finitely many axis-parallel \(k\)-flats, where \(0 \leq k < d\). We also consider colorful generalizations of the above result and establish their feasibility. The problem considered in this paper is an infinite variant of the Hadwiger-Debrunner \((p, q)\)-problem.
Bibliography:	SourceType-Working Papers-1 ObjectType-Working Paper/Pre-Print-1 content type line 50
ISSN:	2331-8422
DOI:	10.48550/arxiv.2308.10479