On degrees in random triangulations of point sets

We study the expected number of interior vertices of degree i in a triangulation of a planar point set S, drawn uniformly at random from the set of all triangulations of S, and derive various bounds and inequalities for these expected values. One of our main results is: For any set S of N points in general position, and for any fixed i, the expected number of vertices of degree i in a random triangulation is at least γ_iN, for some fixed positive constant γi (assuming that N>i and that at least some fixed fraction of the points are interior).We also present a new application for these expected values, using upper bounds on the expected number of interior vertices of degree 3 to get a new lower bound, >(2.4317^N), for the minimal number of triangulations any N-element planar point set in general position must have. This improves the previously best known lower bound of Ω(2.33^N).