The Number of Geometric Bistellar Neighbors of a Triangulation

Abstract

The theory of secondary and fiber polytopes implies that regular (also called convex or coherent) triangulations of configurations with n points in R d have at least n-d-1 geometric bistellar neighbors. Here we prove that, in fact, all triangulations of n points in R 2 have at least n-3 geometric bistellar neighbors. In a similar way, we show that for three-dimensional point configurations, in convex position and with no three points collinear, all triangulations have at least n-4 geometric bistellar flips. In contrast, we exhibit three-dimensional point configurations, with a single interior point, having deficiency on the number of geometric bistellar flips. A lifting technique allows us to obtain a triangulation of a simplicial convex 4-polytope with less than n-5 neighbors. We also construct a family of point configurations in R 3 with arbitrarily large flip deficiency.