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Hyperbolicity, CAT($$-1$$)-spaces and the Ptolemy inequality. (English) Zbl 1219.53042
A four points inequality for the boundary of CAT$$(-1)$$-spaces is proved. This is used in order to study the relation between Gromov hyperbolic spaces and CAT$$(-1)$$-spaces. Especially, the authors obtain a significant partial answer to the following main question:
Given a non-treelike visual Gromov hyperbolic space endowed with its critical metric, is it rough isometric to (embedded into) some CAT$$(-1)$$-space?

##### MSC:
 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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