Intersections of sets starshaped via paths of bounded length

Let S be a nonempty closed, simply connected set in the plane. For alpha > 0, let M denote the family of all maximal subsets of S which are starshaped via paths of length at most alpha. Then boolean AND{M : M in M} is either starshaped via alpha-paths or empty. The result fails without the simple connectedness condition. However, even with a simple connectedness requirement, there is no Helly theorem for intersections of sets which are starshaped via alpha-paths.