Metric betweenness, Ptolemaic spaces, and isometric embeddings of pretangent spaces in R

摘要

We study some properties of pretangent spaces describing the infinitesimal geometry of general metric spaces. The necessary and sufficient conditions characterizing Ptolemaic pretangent spaces and pretangent spaces, where every three points are situated "on a single straight line," are found. As a corollary, we obtain a criterion of embeddability of pretangent spaces into the set K of real numbers endowed by the natural metric.