On flips in polyhedral surfaces

Let V be a finite point set in 3-space, and let S(V) be the set of triangulated polyhedral surfaces homeomorphic to a sphere and with vertex set V. Let abc and cbd be two adjacent triangles belonging to a surface SεS(V); the flip of the edge bc would replace these two triangles by the triangles abd and adc. The flip operation is only considered when it does not produce a self-intersecting surface. In this paper we show that given two surfaces S1,S2εS(V), it is possible that there is no sequence of flips transforming S1 into S2, even in the case that V consists of points in convex position.