Staircase kernels

Let S subset of R-2 be a compact staircase connected set with stdiam(S) = n. In [4] we showed that Ker(r)(S) is nonempty if r >= n+1/2, and for r >= n/2+1, Ker(r)(S) is staircase connected. In this paper we determine the possible values of the staircase diameter of Ker(r)(S) for r >= n/2+1 and present interesting facts about Ker(r)(S) when r= n/2 and r = n+1/2.