The number of geometric bistellar neighbors of a triangulation
A regular (or coherent) triangulation of a configuration of \(n\) points in \(\mathbb R^d\) arises from the projection of a simplicial polytope with \(n\) vertices in \(\mathbb R^{d+1}\). For straightforward geometric reasons, such a triangulation must have at least \(n-d-1\) bistellar neighbours, obtainable from it by bistellar operations. A non-regular triangulation, though, may have fewer bistellar neighbours. However, the authors show here that the result still holds if \(d=2\), or if \(d=3\) and the points are in convex position (lying on the boundary of their convex hull) with no three collinear. On the other hand, if \(d=3\) and even one point lies in the interior of the convex hull, then the result may fail. Lifting counterexamples to \(\mathbb R^4\) gives triangulations of simplicial 4-polytopes with \(n\) vertices and fewer than \(n-4\) bistellar neighbours.
scientific article; zbMATH DE number 1271013