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Coarse median spaces simultaneously generalise the classes of hyperbolicspaces and median algebras, and arise naturally in the study of the mappingclass groups and many other contexts. One issue with their definition asoriginally conceived by Bowditch is the need to establish median approximationsfor all finite subsets of the space, an approach which allowed the definitionof rank (a proxy for dimension) in terms of the dimensions of the approximatingspaces. Here we provide a simplification of the definition in terms of a$4$-point condition analogous to the $4$-point condition defininghyperbolicity. We show how to define rank in this context, and use this to givea direct proof that rank $1$ geodesic coarse median spaces are$\delta$-hyperbolic, bypassing Bowditch's use of asymptotic cones. A keyingredient of the proof is a new definition of intervals in coarse medianspaces and an analysis of their interaction with geodesics.

A four point characterisation for coarse median spaces