Page "Building (mathematics)" Paragraph 15

Every building has a canonical length metric inherited from the geometric realisation obtained by identifying the vertices with an orthonormal basis of a Hilbert space.

For affine buildings, this metric satisfies the CAT ( 0 ) comparison inequality of Alexandrov, known in this setting as the Bruhat-Tits non-positive curvature condition for geodesic triangles: the distance from a vertex to the midpoint of the opposite side is no greater than the distance in the corresponding Euclidean triangle with the same side-lengths ( see ).