Robust convexity in flip-graphs
Authors: Lionel Pournin, Zili Wang
Summary: The triangulations of a floor Σ with a prescribed set of vertices might be endowed with a graph construction F(Σ). Its edges join two triangulations that differ by a single arc. It’s identified that, when Σ is a convex polygon or a topological floor, the subgraph Fε(Σ) induced in F(Σ) by the triangulations that comprise a given arc ε is strongly convex within the sense that every one the geodesic paths between two such triangulations stay in that subgraph. Right here, we offer a associated outcome that entails a triangle as a substitute of an arc, within the case when Σ is a convex polygon. We present that, when the three edges of a triangle τ seem in (probably distinct) triangulations alongside a geodesic path, τ should belong to a triangulation in that path. Extra typically, we show that sure third-dimensional triangulations associated to the geodesics in F(Σ) are flag when Σ is a convex polygon with flat vertices, and supply two penalties. The primary is that Fε(Σ) isn’t at all times strongly convex when Σ is a convex polygon with both two flat vertices or two punctures. The second is that the variety of arc crossings between two triangulations of a topological floor Σ doesn’t permit to approximate their distance in F(Σ) by an element of lower than 3/2.
