## 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:

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