is the set of points $z$ such that $d(x,y)=d(x,z)+d(z,y)$. Let $A$ be an apartment in an affine building containing two points $x$ and $y$. We write ${\mathrm{seg}}_A(x,y)$ for the intersection of ${\mathrm{seg}}(x,y)$ with $A$.

-   For all triples of points $x,y$ and $z$ in $X$ and all apartments $A$ containing $x$ and $y$ the segment ${\mathrm{seg}}_A(x,y)$ is contained in a finite union of Weyl chambers based at $z$.

Property (EC) was introduced by Bennett [@Bennett2] as an alternative to the sixth axiom. Axiom (A3’) is a stronger version of (A3) and the precise analog of the simplicial condition that two (affine) chambers are always contained in a common apartment. Both, (A3’) and property (GG), were introduced by Parreau [@Parreau]. Property (CO) did as well appear in [@Parreau] first. Axiom (A3”) is ’in between’ (A3) and (A3’) and suffices for one of the implications in \[MainThm\]. In [@Convexity2] we used a slightly stronger version of property (FC”) to prove that certain retractions are distance diminishing. However, Koen Struyve noticed that (FC”) suffices for our purposes.

We say that $(X,{\mathcal{A}})$ is *a space modeled on ${\mathbb{A}}$* if $X$ is a set together with a collection ${\mathcal{A}}$ of injective *charts* $f:{\mathbb{A}}\hookrightarrow X$ such 