ZOKHOVITZKY PRIZE LECTURE
A staircase is a polygonal path in R² that is x-monotone and y-monotone and whose edges are alternately horizontal and vertical. A subset S of R² is staircase connected if every two points a,b in S are connected by a staircase within S. Denote by ξ(a,b) the smallest number of edges of such a connecting staircase. If S is a compact and staircase connected subset of R², then ξ(.,.) is a bounded integer valued metric on S. We say that a point x belongs to the staircase k-kernel of S if ξ(x,y)≤ k for all y in S. We study these kernels and determine the possible values of their diameters.
Joint work with M.A. Perles.