Staircase connected sets

A compact set S ⊂ ℝR}² is staircase connected if every two points a,b ∞ S can be connected by a polygonal path with sides parallel to the coordinate axes, which is both x-monotone and y-monotone. ξ(a,b) denotes the smallest number of edges of such a path. ξ(.,.) is an integer-valued metric on S. We investigate this metric and introduce stars and kernels. Our main result is that the r-th kernel is nonempty, compact and staircase connected provided r ≥ 1/2} · stdiam}(S)+1.