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Helly-type theorems for hollow axis-aligned boxes

Authors:	Konrad J Swanepoel

Institution:	Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa

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

Keywords:	Helly-type theorem box cube hypercube

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