Volume of convex polytopes equals mixed volume of simplices

TL;DR

This paper presents a straightforward proof establishing that the volume of convex polytopes equals the mixed volume of simplices, highlighting the computational complexity of mixed volume calculations.

Contribution

It provides a simple proof linking convex polytope volume to mixed volume of simplices, emphasizing the computational difficulty of these calculations.

Findings

Volume of convex polytopes equals mixed volume of simplices

Computing mixed volume of simplices is as hard as computing polytope volumes

Highlights the complexity of mixed volume computation

Abstract

This note provides a simple proof for the equality between the normalized volume of a convex polytope with $m$ vertices and the mixed volume of $m$ simplices and thus shows the seemingly restrictive problem of computing mixed volume of simplices is still at least as hard as computing volumes of convex polytopes.