Some characterizations of city block distance

Abstract It is shown that in the presence of a simple geometrical constraint city block distance between n -dimensional lattice point is characterized by a condition used by Rosenfeld in an investigation of continuous functions on digital images. For the digital plane such a description is obtained in purely metric terms. In any metric space the relation of points being at unit distance is a tolerance, i.e. it is reflexive and symmetric. Necessary and sufficient algebraic conditions are given for a tolerance to be that associated with planar city block distance. An appendix continues the bibliography of papers related to distance in digital geometry begun in a previous report.