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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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