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

TL;DR

This paper explores a duality between the expected volume and the expected number of vertices of random polytopes, extending Efron's 1965 identity to higher moments and revealing deep geometric relationships.

Contribution

It extends Efron's identity from expected values to higher moments and introduces a transformation that clarifies the duality between volume and vertex count in random polytopes.

Findings

Extended Efron's identity to higher moments

Revealed a duality between volume and vertex count

Applied elementary symmetric polynomial identities

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 $n+1$ 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.