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An identity due to Efron dating from 1965 relates the expected volume of theconvex hull of $n$ random points to the expected number of vertices of theconvex hull of $n+1$ random points. Forty years later this identity wasextended from expected values to higher moments. The generalized identity hasattracted considerable interest. Whereas the left-hand side of the generalizedidentity -- concerning the volume -- has an immediate geometric interpretation,this is not the case for the right-hand side -- concerning the number ofvertices. A transformation of the right-hand side applying an identity forelementary symmetric polynomials overcomes the blemish. The arising formulareveals a duality between the volumes and the numbers of vertices of randompolytopes.

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