Séance Séminaire

Séminaire Gaston Darboux

The Ptolemy Inequality and Nonpositive Curvature

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.