Flips in Two-dimensional Hypertriangulations

We study flips in hypertriangulations of planar points sets. Here a level-$k$ hypertriangulation of $n$ points in the planes is a subdivision induced by the projection of a $k$-hypersimplex, which is the convex hull of the barycenters of the $(k-1)$-dimensional faces of the standard $(n-1)$-simplex. In particular, we introduce four types of flips and prove that the level-2 hypertriangulations are connected by these flips.