Unions of Sets Starshaped via Staircase Paths or via Paths of Bounded Length

Let S be an orthogonal polygon in the plane, S simply connected, and let k=2,3. Set S is a union of k sets starshaped via staircase paths if and only if for every F finite, F bdry S, there is a set G bdry S arbitrarily close to F and points si,1 ik, (depending on G) such that each point of G is clearly visible from some si. An analogous result holds for a union of 2 sets starshaped via -paths when S is a closed simply connected set in the plane. Each result is best possible.