A CAT(0) space is a geodesic metric space whose triangles satisfy Euclidean comparison. For a triangle formed by three distance-minimizing paths, take a Euclidean triangle with the same side lengths. The distance between any two points on the original sides is at most the distance between the corresponding points on the Euclidean sides, where correspondence preserves distance along each side (Bridson and Haefliger 1999). This is a metric formulation of nonpositive curvature.
Euclidean spaces and trees with edges realized as intervals, equipped with the path metric, are examples. A geometric group action on a CAT(0) space imposes both properness and cocompactness. Artin groups and hyperbolic groups motivate questions about the existence of such actions.
