Bruhat-Tits buildings associated to algebraic groups over complete non-Archimedean fields.

Let $(\mathcal{B}, d)$ be a Euclidean building endowed with such a metric. Then $(\mathcal{B}, d)$ satisfies moreover a remarkable non-positive curvature property, called the [*${\rm CAT}(0)$-property*]{} (where “CAT” seems to stand for Cartan-Alexandrov-Toponogov). Roughly speaking, this property says that geodesic triangles are at least as thin as in Euclidean planes. More precisely, the point is to compare a geodesic triangle drawn in $\mathcal{B}$ with “the” Euclidean triangle having the same edge lengths. A geodesic space is said to have the [*${\rm CAT}(0)$-property*]{}, or to [*be*]{} CAT(0), if a median segment in each geodesic triangle is at most as long as the corresponding median segment in the comparison triangle drawn in the Euclidean plane ${\bf R}^2$ (this inequality has to be satisfied for all geodesic triangles). Though this property is stated in elementary terms, it has very deep consequences [@RousseauGrenoble §7].

One first consequence is the uniqueness of a geodesic segment between any two points [@BriHae Chap. II.1, Prop. 1.4].

The main consequence is a famous and very useful fixed-point property. The latter statement is itself the consequence of a purely geometric one: any bounded subset in a complete, CAT(0)-space has 