Embedding into Rectilinear Spaces

Abstract

We show that the problem whether a given finite metric space.X; d/ can be embedded into the rectilinear spaceR m can be formulated in terms of m-colorability of a certain hypergraph associated with .X; d/. This is used to close a gap in the proof of an assertion of Bandelt and Chepoi (2) on certain critical metric spaces for this embedding problem. We also consider the question of determining the maximum number of equidistant points that can be placed in the m-dimensional rectilinear space and show that this number is equal to 2m for m• 3.