Helly-type theorems for hollow axis-aligned boxes
Konrad J. Swanepoel · Proceedings of the American Mathematical Society · 1999
A hollow axis-aligned box is the boundary of the cartesian product of d d compact intervals in R d \mathbb {R}^d . We show that for d ≥ 3 d\geq 3 , if any 2 d 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 5 of a collection of hollow axis-aligned rectangles in R 2 \mathbb {R}^2 have non-empty intersection, then the whole collection has non-empty intersection. The values 2 d 2^d for d ≥ 3 d\geq 3 and 5 5 for d = 2 d=2 are the best possible in general. We also characterize the collections of hollow boxes which would be counterexamples if 2 d 2^d were lowered to 2 d − 1 2^d-1 , and 5 5 to 4 4 , respectively.