Embedding metric spaces in the rectilinear plane: A six-point criterion

Abstract

We show that a metric space embeds in the rectilinear plane (i.e., isL 1-embeddable in ℝ2) if and only if every subspace with five or six points does. A simple construction shows that for higher dimensionsk of the host rectilinear space the numberc(k) of points that need to be tested grows at least quadratically withk, thus disproving a conjecture of Seth and Jerome Malitz.