Description
Let $P_1,...,P_n$ be polytopes in $n$space. Then the volume of the Minkowski sum $\lambda_1 P_1 + ... + \lambda_n P_n$ is a homogeneous polynomial of degree $n$ in nonnegative variables $\lambda_1,...,\lambda_n$. The coefficient Vol$(P_1,...,P_n)$ of $\lambda_1\lambda_2 ... \lambda_n$ is called the mixed volume of $P_1,...,P_n$. For example, the number of toric solutions to a generic system of /$n$ polynomial equations on $n$space amounts to the mixed volume of the corresponding Newton polytopes.
The function
mixedVolume takes the
List L with $n$ polytopes in $n$space and computes their mixed Volume by using the algorithm by Ioannis Z. Emiris in his paper
Mixed Volume Implementation. Note that this function computes an upper bound by using a random lifting. To reassure the result run the function until it returns the same result.
CAVEAT: So far the input is not checked so use the function with care!
i1 : P = crossPolytope 2
o1 = {ambient dimension => 2 }
dimension of lineality space => 0
dimension of polyhedron => 2
number of facets => 4
number of rays => 0
number of vertices => 4
o1 : Polyhedron

i2 : Q = hypercube 2
o2 = {ambient dimension => 2 }
dimension of lineality space => 0
dimension of polyhedron => 2
number of facets => 4
number of rays => 0
number of vertices => 4
o2 : Polyhedron

i3 : mixedVolume {P,Q}
o3 = 8
o3 : QQ
