Helly-type Theorems for Hollow Axis-aligned Boxes

TL;DR

This paper establishes Helly-type intersection theorems for hollow axis-aligned boxes in various dimensions, identifying optimal intersection thresholds and characterizing potential counterexamples.

Contribution

It proves sharp Helly-type theorems for hollow boxes in higher dimensions and characterizes collections that could serve as counterexamples if thresholds are lowered.

Findings

For d ≥ 3, 2^d boxes ensure intersection if every 2^d intersect.

In R^2, 5 rectangles guarantee intersection if every 5 intersect.

The thresholds 2^d and 5 are proven to be optimal.

Abstract

A hollow axis-aligned box is the boundary of the cartesian product of $d$ compact intervals in R^d. We show that for d\geq 3, if any 2^d of a collection of hollow axis-aligned boxes have non-empty intersection, then the whole collection has non-empty intersection; and if any 5 of a collection of hollow axis-aligned rectangles in R^2 have non-empty intersection, then the whole collection has non-empty intersection. The values 2^d for d\geq 3 and 5 for d=2 are the best possible in general. We also characterize the collections of hollow boxes which would be counterexamples if 2^d were lowered to 2^d-1, and 5 to 4, respectively.