$\Delta$ is a polygon, we denote by $\operatorname{\mathrm{D}}: \operatorname{\mathrm{Vert}}\Delta \times \operatorname{\mathrm{Vert}}\Delta \to \operatorname{\mathbf{N}}$ the graph distance in the $1$-dimensional simplicial complex $\Delta$.

\[lemma:gallery\] Let $\Delta$ be a compact polygon and $x, y$ be distinct non-opposite vertices of $\Delta$. If $x_n \to x$, $y_n \to y$ and $\operatorname{\mathrm{D}}(x_n, y_n) = \operatorname{\mathrm{D}}(x, y)$ for all $n \in \operatorname{\mathbf{N}}$, then $[x_n, y_n] \to [x, y]$.

See [@Burns]\*[Lemma 1.12]{}.

We say that two vertices $v$ and $v'$ in an $m$-gon are **almost opposite** if $\operatorname{\mathrm{D}}(v,v') = m-1$ (note that $\operatorname{\mathrm{D}}(v,v') = m$ if and only if $v$ and $v'$ are opposite). We then have a result similar to Proposition \[proposition:opposite\] for almost opposite vertices.

\[proposition:almost-opposite\] Let $\Delta$ be a compact polygon. The set $$\{(v,v') \in (\operatorname{\mathrm{Vert}}\Delta)^2 \mid v \text{ is almost opposite } v'\}$$ is open in $(\operatorname{\mathrm{Vert}}\Delta)^2$.

See [@Burns]\*[Lemma 1.13]{}.

The following result will finally be helpful in the proofs of Theorems \[theorem:Characterization\] and \[theorem:Criterion\].

\[lemma:non-isolated\] Let $\Delta$ be an infinite thick compact $m$-gon with $m \geq 3$. For each $v \in \operatorname{\mathrm{Vert}}\Delta$, no chamber of $\operatorname{\mathrm{Cham}}(v)$ is isolated in $\operatorname{\mathrm{Cham}}(v)$. In particular, $\operatorname{\mathrm{Cham}}(v)$ is infinite.

See [@Burns]\*[Lemma 1.14]{}.

Strong transitivity and the Moufang property {#subsection:transitivity}
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Let $\Delta$ be a spherical building 