from $v$ and lies on the arc in direction $\bar\de$, $\oa{vp}=\bar\de$. In particular, it is not contained in a face of $\tau$ with vertex $v$. Letting run $v$ through the vertices of $\tau$, we conclude that $p\in\inte(\tau)$.

As a consequence of the lemma, the nearest point projection $\inte_{\tau}(\si)\to\tau$ agrees with the nearest point projection $\inte_{\tau}(\si)\to s$ to the geodesic sphere $s\subset S$ spanned by $\tau$ (i.e. containing $\tau$ as a top-dimensional subset), and its image equals $\inte(\tau)$.

### Spherical Coxeter complexes

A [*spherical Coxeter complex*]{} $(\amod,W)$ consists of a unit sphere (in a euclidean space) $\amod$ and a finite reflection group $W$ acting isometrically on $\amod$. We will refer to $\amod$ as the [*model apartment*]{} (because it will serve as the model for apartments in spherical buildings, see below).

A [*wall*]{} in $\amod$ is the fixed point set of a reflection in $W$. A [*half-apartment*]{} is a closed hemisphere in $\amod$ bounded by a wall. A [*singular sphere*]{} in $\amod$ is an intersection of walls.

A [*chamber*]{} in $\amod$ is the closure of a connected component of the complement of the union of the walls. The group $W$ acts transitively on the set of chambers. The chambers are simplices with 