Nonpositive Curvature and the Ptolemy Inequality

Abstract

We provide examples of nonlocally, 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.

Geodesic Ptolemy spaces and fixed points

“…Now let u, v ∈ Fix(T ) and m = (1 − α)u + αv for some Since by [7,Theorem 1.2] X is uniquely geodesic we obtain that T (m) = m. This completes the proof.…”

“…In [7] it is shown that a geodesic Ptolemy space does not even have to be a uniquely geodesic space.…”

“…We begin with a result stating that geodesic Ptolemy spaces with a continuous midpoint map are strictly convex. The proof of this result has its roots in a construction used in the proof of [7,Theorem 1.2] where it is shown that a proper geodesic Ptolemy space is uniquely geodesic. …”

“…Therefore, a natural question at this point is to know if geodesic Ptolemy spaces with a uniformly continuous mipoint map are uniformly convex. Notice that this question lies somewhere in between what it is known and the open question raised in [7] of whether such spaces are CAT(0).…”

“…However, in [7] a characterization of CAT(0) spaces is given in terms of the Busemann convexity and the Ptolemy inequality. Namely, it is shown that a metric space is CAT(0) if and only if it is Busemann convex and Ptolemy.…”