Triangulations with Very Few Geometric Bistellar Neighbors

We are interested in a notion of elementary change between triangulations of a point connguration, the so-called bistellar ips, introduced by Gel'fand, Kapranov and Zelevinski. We construct sequences of triangulations of point conngurations in dimension 3 with n 2 +2n+2 vertices and only 4n?3 geometric bistellar ips (for every even integer n), and of point conngurations in dimension 4 with arbitrarily many vertices and a bounded number of ips. This drastically improves previous examples and seems to be evidence against the conjecture that any two triangulations of a point connguration can be joined by a sequence of ips.