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.


52A35 Helly-type theorems and geometric transversal theory
52A30 Variants of convex sets (star-shaped, (\(m, n\))-convex, etc.)
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