Helly-type theorems for hollow axis-aligned boxes

Abstract:

A hollow axis-aligned box is the boundary of the cartesian product of $d$ compact intervals in $\mathbb {R}^d$. 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 $\mathbb {R}^2$ 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.