## Covering with Euclidean boxes.(English)Zbl 0623.05041

The authors prove the following box cover theorem: For each dimension d there exists a constant c depending only on d such that any finite (or compact) set $$V\subset {\mathbb{R}}^ d$$ contains a subset S with $$| S| \leq c$$ and satisfying $$V\subset \cup_{p,q\in S}box(p,q)$$ (box(p,q) denotes the smallest box containing p and q and whose faces are parallel to the coordinate axes of $${\mathbb{R}}^ d)$$. For $$d=2$$, the result was shown, with $$c=4$$, by A. Gyárfás and the second author [Combinatorica 3, 351-358 (1983; Zbl 0534.05052)].
As a first consequence of the box cover theorem, it is shown that if H is a d-dimensional box-hypergraph then $$\rho$$ (H)$$\leq c_ 0\cdot \alpha (H)^ c$$, where $$\alpha$$ (H) and $$\rho$$ (H) denote, respectively, the stability and the covering number of H, and c and $$c_ 0$$ are constants depending only on d. Another consequence is the extension to the general case of boxes defined by arbitrary finite-cone-subdivisions of $${\mathbb{R}}^ d$$. Geometric variants of these results, with boxes replaced by angles, are also derived.
Reviewer: J.Weinstein

### MSC:

 05C65 Hypergraphs 52A37 Other problems of combinatorial convexity 51M20 Polyhedra and polytopes; regular figures, division of spaces

Zbl 0534.05052
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### References:

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