all $x,y,z$ in $X$ the following axioms are satisfied

1.  $d(x,y)=0$ if and only if $x=y$

2.  $d(x,y)=d(y,x)$ and

3.  the triangle inequality $d(x,z)+d(z,y)\geq d(x,y)$ holds.

There is however a small problem in viewing Euclidean buildings as a subclass of affine buildings. The definition of an affine building is based on the definition of a given model space, which in turn comes with a fixed metric. In case of ${\mathbb{R}}$-buildings one usually uses the Euclidean metric on the model space. Therefore the metric on the affine building $X$ is, when restricted to an apartment, precisely the Euclidean metric. Compare for example [@Parreau] or Kleiner and Leeb [@KleinerLeeb].

The natural metric on the model space of a generalized affine building is however defined in terms of the defining root system ${\Phi}$, compare [@Diss]. It is a generalization of the length of translations in apartments of simplicial affine buildings. This length function on the set of translational elements of the affine Weyl group is defined with respect to the length of certain minimal galleries. The problem is that this natural metric used for “$\Lambda$-buildings” is different from the Euclidean one in case $\Lambda={\mathbb{R}}$. For our purposes it is not necessary to specify 