##
**Unions of orthogonally convex or orthogonally starshaped polygons.**
*(English)*
Zbl 0814.52002

This paper considers orthogonal polygons obtained as union of finitely many convex polygons whose edges are parallel to the coordinate axes. A simple polygonal path in the plane whose edges are finite and parallel to the coordinate axes is called a staircase path iff the associated vectors alternate in direction. A set \(T\) is said to be an orthogonally convex polygon iff for all \([x,y]\) in \(T\) such that \([x,y]\) is either horizontal or vertical, then \([x,y] \subseteq T\). A set \(T\) is starshaped via staircase paths with respect to a point \(p\) iff \(p\) sees each point of \(T\) via staircase paths. By imposing conditions on finite subsets of a set \(T\) in a linear space, J. F. Lawrence, W. R. Hare jun. and John W. Kenelly [Proc. Am. Math. Soc. 34, 225-288 (1972; Zbl 0237.52001)] have characterized \(T\) as union of finite number of convex sets. The author in her first theorem obtains a similar result for simply connected orthogonal polygons in the plane. In the theorem it is stated that the necessary and sufficient condition for a simply connected orthogonal polygon in the plane to be a union of \(k\) orthogonally convex sets is that for every finite subset \(F\) of \(T\) there exists a \(k\)- partition \(\{F_ i\}_{1 \leq i \leq k}\) of \(F\) such that every pair of points in \(F_ i\) can be joined by a staircase path in \(T\). In the second theorem the author proves that if a simply connected orthogonal polygon having that for every three points of \(T\), at least two of these see each other via staircase paths in \(T\), the set is a union of three orthogonally convex polygons. In Theorem 3 the necessary and sufficient condition for a decomposition of a simple connected orthogonal polygon in the plane into two orthogonally convex polygons is given by introducing an odd numbered finite point sequence \(\{\nu_ i\}\) where \(\nu_{n + 1} = \nu_ 1\), \(1 \leq i \leq n\) in the set such that at least one consecutive pair of points see each other via staircase paths belonging to the set. Theorem 4 states that a simply connected orthogonal polygon \(T\) in the plane is a union of \(k\)-orthogonal polygons, each one starshaped via staircase paths, iff there exists a \(k\)-partition \(\{F_ i\}\) of \(F\) such that every pair in \(F_ i\) sees a common point of \(T\) via staircase paths. In the last theorem, the author expresses a necessary and sufficient condition for a simply connected orthogonal polygon to be the union of two orthogonal polygons, each one starshaped via staircase paths, by substituting the \(k\)-partition of a finite subset with point sequences \(\nu_ i,\dots,\nu_{n + 1} = \nu_ 1\), \(n\) odd, such that at least one consecutive pair of points sees a common point via staircase paths in \(T\).

Reviewer: S.Abdoǧan (İstanbul)

### MSC:

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

51A35 | Non-Desarguesian affine and projective planes |

### Citations:

Zbl 0237.52001
Full Text:
DOI

### References:

[1] | Breen, M.: A Krasnosel’skii theorem for staircase paths in orthogonal polygons,J. Geom. (to appear). · Zbl 1298.52008 |

[2] | Breen, M.: An improved Krasnosel’skii-type theorem for orthogonal polygons which are starshaped via staircase paths,J. Geom. (to appear). · Zbl 0815.52005 |

[3] | Breen, M.: Staircase kernels in orthogonal polygons,Arch. Math. 59 (1992), 588-594. · Zbl 0789.52011 |

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

[5] | Harary, F.:Graph Theory, Addison Wesley, Reading, Mass., 1972. · Zbl 0235.05105 |

[6] | Hare, W.R., Jr and Kenelly, J. W.: Sets expressible as unions of two convex sets,Proc. Amer. Math. Soc. 25 (1970), 379-380. · Zbl 0195.51603 |

[7] | Lawrence, J. F., Hare, W. R. Jr and Kenelly, J. W.: Finite unions of convex sets,Proc. Amer. Math. Soc. 34 (1972), 225-228. · Zbl 0237.52001 |

[8] | Lay, S. R.:Convex Sets and Their Applications, Wiley, New York, 1982. · Zbl 0492.52001 |

[9] | McKinney, R. L.: On unions of two convex sets,Canad. J. Math. 18 (1966), 883-886. · Zbl 0173.15305 |

[10] | Molnàr, J.: Über den zweidimensionalen topologischen Satz von Helly,Mat. Lapok 8 (1957), 108-114. · Zbl 0105.16705 |

[11] | Motwani, R., Raghunathan, A. and Saran, H.: Covering orthogonal polygons with star polygons: the perfect graph approach,J. Comput. System Sci. 40 (1990), 19-48. · Zbl 0705.68082 |

[12] | Nadler, S.:Hyperspaces of Sets, Marcel Dekker, New York, 1978. |

[13] | Valentine, F. A.: A three point convexity property,Pacific J. Math. 7 (1957), 1227-1235. · Zbl 0080.15401 |

[14] | Valentine, F. A.:Convex Sets, McGraw-Hill, New York, 1964. · Zbl 0129.37203 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.