chamber opposite $C$. If $R_n \to R$ and $C_n \to C$, then $\operatorname{\mathrm{proj}}_{R_n}C_n \to \operatorname{\mathrm{proj}}_R C$.

See [@Burns]\*[Proposition 1.10]{}.

\[corollary:homeo\] Let $R$ and $R'$ be two opposite residues of a compact spherical building $\Delta$. The projection $\operatorname{\mathrm{proj}}_R \mid_{R'} : \operatorname{\mathrm{Cham}}R' \to \operatorname{\mathrm{Cham}}R$ is a homeomorphism.

This follows from Proposition \[proposition:projections\].

In a polygon (i.e. a spherical building of rank $2$), we will say that a gallery $(C_1, \ldots, C_n)$ **stammers** if $C_i = C_{i-1}$ for some $i \in \{2, \ldots, n\}$. The two following results are true in any usual (non-topological) polygon. Recall that an $m$-gon is a polygon of diameter $m$.

\[lemma:minimal\] Let $\Delta$ be an $m$-gon. A gallery $(C_1, \ldots, C_n)$ in $\Delta$ with $n \leq m+1$ is minimal if and only if it does not stammer. In particular, there is no non-stammering closed gallery in $\Delta$ having less than $2m$ chambers.

This is a direct consequence of [@Burns]\*[Lemma 0.4]{}.

Let $\Delta$ be a polygon and $x, y$ be distinct non-opposite vertices of $\Delta$. There is a unique minimal gallery with initial vertex $x$ and final vertex $y$, which we denote by $[x,y]$.

The existence of two minimal galleries from $x$ to $y$ would contradict Lemma \[lemma:minimal\].

When 