Good afternoon,

I would like to know if it is possible to use polymake in order to compute the Kazarnovskii pseudovolume of 4-dimensional polytopes.

If \( \Gamma \) is a polytope in \( \mathbb C^2 \), the Kazarnovskii pseudovolume \( P_2(\Gamma) \) is, by definition, the sum \( \frac{1}{\pi}\sum_\Delta \rho(\Delta)vol_2(\Delta)\psi(\Delta) \), as \( \Delta \) runs in the set of 2-dimensional faces of \( \Gamma \), where:

- \( \rho(\Delta)=1-\langle v_1,v_2\rangle^2 \), with \( \{v_1,v_2\} \) an orthonormal basis (respect to the scalar product \( Re\langle\,,\rangle \) given by the real part of the standard hermitian one) of the plane parallel to \( \Delta \) and passing through the origin;

- \( vol_2(\Delta) \) is the surface area of \( \Delta \);

- \( \psi(\Delta) \) is the outer angle of \( \Gamma \) at \( \Delta \).

So \( P_2(\Gamma) \) is just a weighted version of the 2nd intrinsic volume of \( \Gamma \) taking into account the position of \( \Gamma \) with respect to complex structure of the ambient space. My question is the following: is polymake able to perform the necessary linear algebra computation on the set of the ridges of \( \Gamma \)?

Thank you in advance