Strong convexity in flip-graphs

The triangulations of a surface $\Sigma$ with a prescribed set of vertices can be endowed with a graph structure $\mathcal{F}(\Sigma)$. Its edges connect two triangulations that differ by a single arc. It is known that, when $\Sigma$ is a convex polygon or a topological surface, the subgraph $\mathcal{F}_\varepsilon(\Sigma)$ induced in $\mathcal{F}(\Sigma)$ by the triangulations that contain a given arc $\varepsilon$ is strongly convex in the sense that all the geodesic paths between two such triangulations remain in that subgraph. Here, we provide a related result that involves a triangle instead of an arc, in the case when $\Sigma$ is a convex polygon. We show that, when the three edges of a triangle $\tau$ appear in (possibly distinct) triangulations along a geodesic path, $\tau$ must belong to a triangulation in that path. More generally, we prove that certain $3$-dimensional triangulations related to the geodesics in $\mathcal{F}(\Sigma)$ are flag when $\Sigma$ is a convex polygon with flat vertices, and provide two consequences. The first is that $\mathcal{F}_\varepsilon(\Sigma)$ is not always strongly convex when $\Sigma$ is a convex polygon with either two flat vertices or two punctures. The second is that the number of arc crossings between two triangulations of a topological surface $\Sigma$ does not allow to approximate their distance in $\mathcal{F}(\Sigma)$ by a factor of less than $3/2$.