  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 $n+1$ 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.
