Unions of orthogonally convex or orthogonally starshaped polygons
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, \textit{J. F. Lawrence, W. R. Hare jun.} and \textit{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 \(T\).
scientific article; zbMATH DE number 701142