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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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
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ISSN:2331-8422
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Abstract 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.
AbstractList 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.
Author Ghosh, Arijit
Nandi, Soumi
Chakraborty, Sutanoya
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