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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Helly-type theorems
for hollow axis-aligned boxes


Author: Konrad J. Swanepoel
Journal: Proc. Amer. Math. Soc. 127 (1999), 2155-2162
MSC (1991): Primary 52A35
Published electronically: March 3, 1999
MathSciNet review: 1646208
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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.


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Additional Information

Konrad J. Swanepoel
Affiliation: Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa
Email: konrad@math.up.ac.za

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05220-X
PII: S 0002-9939(99)05220-X
Keywords: Helly-type theorem, box, cube, hypercube
Received by editor(s): October 15, 1997
Published electronically: March 3, 1999
Communicated by: Jeffry N. Kahn
Article copyright: © Copyright 1999 American Mathematical Society