Recurrence relationships for the mean number of faces and vertices for random convex hulls

Abstract

This paper studies the convex hull of n random points in Rd. 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 Rényi and Sulanke. Other relationships for the expected count of i-faces (1≤i