Volume of convex polytopes equals mixed volume of simplices

qaaa We provide 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 show the seemingly restrictive problem of computing the mixed volume of simplices is still at least as hard as computing volumes of convex polytopes in general.

Automated summary

In this paper, a simple proof is provided to show the equality between the normalized volume of a convex polytope with m vertices and the mixed volume of m simplices. This demonstrates that computing the mixed volume of simplices, which may appear to be restrictive, is at least as challenging as computing volumes of convex polytopes in general.