Triangulations with very few geometric bistellar neighbors (Q1961849)

A regular (or coherent) triangulation of a configuration \({\mathcal A}\) of \(m\) points which spans \(\mathbb{R}^d\) affinely arises from a particular kind of projection from an \((m-1)\)-simplex, and so has at least \(m-d-1\) bistellar neighbours, in which one of the two ways of triangulating some \(d+1\) of the points is replaced by the other. This property fails for non-regular triangulations. In this paper, the author gives some striking examples. When \(d=3\) there is, for each even \(n\), a configuration with \(m= n^2+ 2n+ 2\) points and only \(4n- 3\) bistellar neighbours. For \(d= 4\), there are configurations with arbitrarily many points but a bounded number of bistellar neighbours.