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Polyhedra :: mixedVolume

mixedVolume -- computes the mixed volume of a list of polytope



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 = P

o1 : Polyhedron
i2 : Q = hypercube 2

o2 = Q

o2 : Polyhedron
i3 : mixedVolume {P,Q}

o3 = 8

o3 : QQ

Ways to use mixedVolume :

For the programmer

The object mixedVolume is a method function.