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**Intersections and unions of orthogonal polygons starshaped via staircase \(n\)-paths.**
*(English)*
Zbl 1134.52007

This paper contains two new Helly-type theorems for orthogonal polygons and staircase paths. Here an orthogonal polygon is a connected union of finitely many possibly degenerate convex polygons with edges parallel to the coordinate axes.

Let \(n\geq 2\). A staircase \(n\)-path is a polygonal path with at most \(n\) edges, each parallel to the coordinate axes, such that every second edge is in the same direction (and with two different directions occurring). It is shown that for any finite family of simply connected orthogonal polygons in the plane and any points \(x\) and \(y\) in their intersection, if the intersection of every \(2(n-1)\) polygons contains a staircase \(n\)-path, then the intersection of all the polygons contains such a path. The value \(2(n-1)\) is best possible. A set \(S\) is starshaped via staircase \(n\)-paths if for some point \(p\in S\), there exists a staircase \(n\)-path from \(p\) to any other point in \(S\). It is shown that for any finite family of orthogonal polygons with simply connected union, if the union of every three polygons is starshaped via staircase \(n\)-paths, then the union of all the polygons is starshaped via staircase \((n+1)\)-paths. The value \(n+1\) is best possible.

Let \(n\geq 2\). A staircase \(n\)-path is a polygonal path with at most \(n\) edges, each parallel to the coordinate axes, such that every second edge is in the same direction (and with two different directions occurring). It is shown that for any finite family of simply connected orthogonal polygons in the plane and any points \(x\) and \(y\) in their intersection, if the intersection of every \(2(n-1)\) polygons contains a staircase \(n\)-path, then the intersection of all the polygons contains such a path. The value \(2(n-1)\) is best possible. A set \(S\) is starshaped via staircase \(n\)-paths if for some point \(p\in S\), there exists a staircase \(n\)-path from \(p\) to any other point in \(S\). It is shown that for any finite family of orthogonal polygons with simply connected union, if the union of every three polygons is starshaped via staircase \(n\)-paths, then the union of all the polygons is starshaped via staircase \((n+1)\)-paths. The value \(n+1\) is best possible.

Reviewer: Konrad Swanepoel (Chemnitz)

### MSC:

52A35 | Helly-type theorems and geometric transversal theory |

52A30 | Variants of convex sets (star-shaped, (\(m, n\))-convex, etc.) |

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DOI

### References:

[5] | Breen M (2005) A Helly theorem for intersections of sets starshaped via staircase n-paths. Ars Combin (to appear) · Zbl 1080.52006 |

[8] | Danzer L, Grünbaum B, Klee V (1962) Helly’s theorem and its relatives. In: Convexity, Amer Math Soc, pp 101–180. Proc Sympos Pure Math 7: Providence, RI · Zbl 0132.17401 |

[10] | Karimov UK, Repov \(\breve{s}\) D, \(\breve{Z}\) eljko M (2005) On unions and intersections of simply connected planar sets. Monatsh Math 145 239–245 · Zbl 1087.54015 |

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