list of six axioms. Later, for their proof of the Margulis conjecture in [@KramerTent], Kramer and Tent made use of the theory of generalized affine buildings. Recently they have been studied by the author in [@Diss] and [@Convexity2].

Our purpose is to study equivalent sets of axioms for generalized affine buildings. We will reduce the number of axioms and obtain that a universal definition for both ${\mathbb{R}}$-buildings and affine buildings defined over arbitrary Krull-valuated fields can be given. From our main result we deduce that the building structure does not depend on its metric. In other words, whichever metric one might impose on the model apartment, the induced distance function on the affine building will be a metric. In particular does the induced metric always satisfy the triangle inequality.

In [@Bennett; @Brown] or [@Parreau] the triangle inequality is solely used to prove existence of a spherical building at infinity. We were able to find an equivalent definition of parallelism of Weyl simplices which is purely combinatorial and does not build on the metric structure of the affine building. This helps us to prove that the building structure does not depend on the metric imposed on the apartment level.

Equivalent sets 