## 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 $$T$$.

### MSC:

 52A30 Variants of convex sets (star-shaped, ($$m, n$$)-convex, etc.) 51A35 Non-Desarguesian affine and projective planes

Zbl 0237.52001
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### 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
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