Regular Metric: Definition and Characterization in the Discrete Plane

Abstract

We say that a metric space is regular if a straight-line (in the metric space sense) passing through the center of a sphere has at least two diametrically opposite points. The normed vector spaces have this property. Nevertheless, this property might not be satisfied in some metric spaces. In this work, we give a characterization of an integer-valued translation-invariant regular metric defined on the discrete plane, in terms of a symmetric subset B that induces through a recursive Minkowski sum, a chain of subsets that are morphologically closed with respect to B.