A Helly theorem for intersections of sets starshaped via staircase \(n\)-paths. (English) Zbl 1157.52303

A nonempty set \(S\) in the plane is an orthogonal polygon if and only if \(S\) is a connected union of finitely many convex polygons whose edges are parallel to coordinate axes. A staircase path is a sequence of alternating connected horizontal and vertical segments such that all horizontal segments have the same direction and all vertical segments have the same direction.
The paper establishes the existence of a (Helly) number \(p(n)\) which is the minimum number such that the following holds: for any finite family of orthogonal polygons, if every \(p(n)\) of them contains a common staircase path of length \(n\), then the intersection of all orthogonal polygons from the family contains a staircase path of length \(n\); exact values of \(p(n)\) are given for \(n=1,2,3,4\) and an upper bound for larger values of \(n\). Other related results are described.


52A30 Variants of convex sets (star-shaped, (\(m, n\))-convex, etc.)
52A35 Helly-type theorems and geometric transversal theory
52A10 Convex sets in \(2\) dimensions (including convex curves)