Helly-type Theorems for Hollow Axis-aligned Boxes

A hollow axis-aligned box is the boundary of the cartesian product of d compact intervals in Rd. We show that for d ≥ 3, if any 2d 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 R2 have non-empty intersection, then the whole collection has non-empty intersection. The values 2d for d ≥ 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 2d were lowered to 2d − 1, and 5 to 4, respectively. 1. General notation and definitions We denote the cardinality of a set S by #S. Let Π(S, k) denote the property that any subcollection of S of at most k sets has non-empty intersection (where k is any positive integer), and Π(S) the property that S has non-empty intersection. For any set S ⊆ R, we denote the convex hull, interior and boundary by co S, intS and bd S, respectively. An axis-aligned box in R is the cartesian product of d compact intervals, i.e. a set of the form