Recurrence Relationships for the Mean Number of Faces and Vertices for Random Convex Hulls
This paper studies the convex hull of n random points in R(d). A recently proved topological identity of the author is used in combination with identities of Efron and Buchta to find the expected number of vertices of the convex hull yielding a new recurrence formula for all dimensions d. A recurrence for the expected number of facets and (d - 2)- faces is also found, this analysis building on a technique of Renyi and Sulanke. Other relationships for the expected count of i-faces (1 <= i < d) are found when d <= 5, by applying the Dehn-Sommerville identities. A general recurrence identity (see (3) below) for this expected count is conjectured.
convex hull sylvester's problem random geometry faces of polytopes