# Piercing axis-parallel boxes

## TL;DR

This paper proves new bounds on piercing axis-parallel boxes in high-dimensional space under specific intersection conditions, showing that a small number of points suffices to intersect all boxes.

## Contribution

It introduces novel geometric conditions under which axis-parallel boxes can be pierced by a linear or near-linear number of points, extending previous results.

## Key findings

- Families with no $k+1$ disjoint boxes can be pierced by $O(k)$ points under certain intersection conditions.
- In 2D with bounded side ratios, $O(k)$ points suffice for piercing.
- Special cases with nested boxes or cubes require $O(k ext{ log log } k)$ or $O(k)$ points respectively.

## Abstract

Let $\F$ be a finite family of axis-parallel boxes in $\R^d$ such that $\F$ contains no $k+1$ pairwise disjoint boxes. We prove that if $\F$ contains a subfamily $\M$ of $k$ pairwise disjoint boxes with the property that for every $F\in \F$ and $M\in \M$ with $F \cap M \neq \emptyset$, either $F$ contains a corner of $M$ or $M$ contains $2^{d-1}$ corners of $F$, then $\F$ can be pierced by $O(k)$ points. One consequence of this result is that if $d=2$ and the ratio between any of the side lengths of any box is bounded by a constant, then $\F$ can be pierced by $O(k)$ points. We further show that if for each two intersecting boxes in $\F$ a corner of one is contained in the other, then $\F$ can be pierced by at most $O(k\log\log(k))$ points, and in the special case where $\F$ contains only cubes this bound improves to $O(k)$.