Intersections and Unions of Orthogonal Polygons Starshaped Via Staircase n-Paths

摘要

For n ≥ 1, define p (n) to be the smallest natural number r for which the following is true: For ${cal K}$ any finite family of simply connected orthogonal polygons in the plane and points x and y in $cap {K:K {rm in} {cal K}}$ , if every r (not necessarily distinct) members of ${cal K}$ contain a common staircase n-path from x to y, then $cap {K:K {rm in} {cal K}}$ contains such a path. We show that p(1) = 1 and p(n) = 2 (n − 1) for n ≥ 2. The numbers p(n) yield an improved Helly theorem for intersections of sets starshaped via staircase n-paths.