their structures of thick spherical Tits buildings.

### Spherical geometry {#sec:sphgeom}

Let $S$ be a unit sphere in a euclidean space, and let $\si\subset S$ be a spherical simplex with dihedral angles $\leq\pihalf$. Then $\diam(\si)\leq\pihalf$.

For a face $\tau\subseteq\si$, we define the [*$\tau$-boundary*]{} $\D_{\tau}\si$ as the union of faces of $\si$ which do not contain $\tau$, and the [*$\tau$-interior*]{} $\inte_{\tau}(\si)$ as the union of open faces of $\si$ whose closure contains $\tau$. We obtain the decomposition $$\si=\inte_{\tau}(\si)\sqcup\D_{\tau}\si.$$ If $\tau'\subset\tau$, then $\D_{\tau'}\si\subset\D_{\tau}\si$ and $\inte_{\tau'}(\si)\supset\inte_{\tau}(\si)$. Note that $\D_{\si}\si=\D\si$ and $\inte_{\tau}(\si)=\inte(\si)$.

We need the following fact about projections of spherical simplices to their faces:

\[lem:sphprjfc\] The nearest point projection $\inte_{\tau}(\si)\to\inte(\tau)$ is well-defined.

In other words, for every point $x\in\inte_{\tau}(\si)$ there exists a point $p\in\inte(\tau)$ such that $px\perp\tau$. In view of $\diam(\si)\leq\pihalf$, this point is necessarily unique.

We argue by induction on the dimension of $\si$.

Let $x\in\inte_{\tau}(\si)$. We apply the induction assumption to the link $\Si_v\si$ at a vertex $v$ of $\tau$. Note that $\D_{\Si_v\tau}\Si_v\si =\Si_v \D_{\tau}\si$. Since $\oa{vx}\in\inte_{\Si_v\tau}(\Si_v\si)$, the nearest point projection $\bar\de$ of this direction to $\Si_v\tau$ is contained in $\inte(\Si_v\tau)$ and has angle $<\pihalf$ with $\oa{vx}$. It follows that the nearest point projection $p$ of $x$ to $\tau$ is different 