We consider metric spaces satisfying a certain four point property, called
the Ptolemy inequality. Our interest in such spaces originates from an
analysis of certain metric boundaries of CAT($k$)-spaces, $k<0$. Such metric
boundaries indeed turn out to be Ptolemy metric spaces. Using this
knowledge, one for instance obtains a characterization of complete
CAT($k$)-spaces, $k<0$, with geodesic Hamenstaedt boundaries, up to isometry.
The correspondig four point property has occasionally been studied here
and there over the years and the results obtained suggest that it somehow
relates to concepts of nonpositive curvature.
Among other things I will prove that indeed, though the Ptolemy condition
is not a nonpositive curvature condition itself, it distinguishes
precisely between the two most common such conditions, namely the one due
to Alexandrov and the one due to Busemann.
Theorem: A metric spaces is CAT($0$) if and only if it is Ptolemy and
Busemann convex.