Helly-type Theorems for Hollow Axis-aligned Boxes

Details Helly-type Theorems for Hollow Axis-aligned Boxes Helly-type Theorems for Hollow Axis-aligned Boxes

2009-09-23

arxiv.org/abs/0909.4244v1

Proceedings of the American Mathematical Society 127 (1999) 2155-2162

Mathematics

Combinatorics

7 pages. Old paper from 1999

Scientific paper

10.1090/S0002-9939-99-05220-X

A hollow axis-aligned box is the boundary of the cartesian product of $d$ compact intervals in 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 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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