Non-positive curvature and the Ptolemy inequality

TL;DR

This paper explores the properties of Ptolemy metric spaces, demonstrating conditions under which they are uniquely geodesic and characterizing CAT(0) spaces through Busemann convexity and Ptolemy inequality.

Contribution

It provides new examples of non-locally compact Ptolemy spaces and establishes a characterization of CAT(0) spaces using Ptolemy and Busemann convexity.

Findings

Non-locally compact Ptolemy spaces can be non-uniquely geodesic.

Locally compact Ptolemy spaces are uniquely geodesic.

CAT(0) spaces are characterized by Busemann convexity and Ptolemy inequality.

Abstract

We provide examples of non-locally compact geodesic Ptolemy metric spaces which are not uniquely geodesic. On the other hand, we show that locally compact, geodesic Ptolemy metric spaces are uniquely geodesic. Moreover, we prove that a metric space is CAT(0) if and only if it is Busemann convex and Ptolemy.