non-discrete Euclidean buildings in any dimension, which will also be seen more concretely by studying spaces of non-Archimedean norms on a given vector space – see \[ss - SL(n) Bruhat-Tits\].

\[rk - BrT bis\] Note that given a reductive group ${\rm G}$ over a valued field $k$, Bruhat-Tits theory “often” provides a Euclidean building on which the group ${\rm G}(k)$ acts strongly transitively in a suitable sense (see Sect. \[s - Bruhat-Tits general\] for an introduction to this subject).

### More geometric properties {#sss - geometry of buildings}

We motivated the definitions of buildings by metric considerations, therefore we must mention the metric features of Euclidean buildings once these spaces have been defined. First, a Euclidean building always admits a metric whose restriction to any apartment is a (suitably normalized) Euclidean distance [@RousseauGrenoble Prop. 6.2]. Endowed with such a distance, a Euclidean building is always a geodesic metric space as introduced in the above metric definition of real trees (\[sss - non-simplicial\]).

[*Recall that we use the axioms [(EB)]{} from Def. \[defi - non-simplicial building\] to define a building; moreover we assume that the above metric is complete.*]{} This is sufficient for our purposes since we will eventually deal with 