The Duality of the Volumes and the Numbers of Vertices of Random Polytopes

Abstract

An identity due to Efron dating from 1965 relates the expected volume of the convex hull of n random points to the expected number of vertices of the convex hull of $$n+1$$ n + 1 random points. Forty years later this identity was extended from expected values to higher moments. The generalized identity has attracted considerable interest. Whereas the left-hand side of the generalized identity—concerning the volume—has an immediate geometric interpretation, this is not the case for the right-hand side—concerning the number of vertices. A transformation of the right-hand side applying an identity for elementary symmetric polynomials overcomes the blemish. The arising formula reveals a duality between the volumes and the numbers of vertices of random polytopes.