The Newton polytope theory provides a beautiful connection between algebraic geometry and convex geometry. One of its central results is the Bernstein-Khovanskii-Kushnirenko (BKK) theorem which expresses the number of complex solutions to a generic polynomial system in terms of the mixed volume of its Newton polytopes. Recently Esterov and Gusev found a multivariate generalization of the Abel-Ruffini theorem on solvability in radicals. They showed that the problem of classifying generic polynomial systems which are solvable reduces to the problem of classifying generic systems with at most 4 solutions. In the view of the BKK theorem, this is equivalent to classifying collections of lattice polytopes of mixed volume up to 4. In the course of solving this (and related) problem one needs to understand relations between the mixed volumes as functions on the space of lattice polytopes or, more generally, convex bodies. The Aleksandrov-Fenchel inequality, although begin a major one, is often not enough. This motivates a systematic study of ``mixed volume configuration spaces''. The first steps in describing them goes back to the 1938's work of Heine where he worked out some small 2-dimensional cases, but there has been not much progress since then. I will talk about mixed volume configuration spaces and their analogs. Besides explicit description we are interested in semialgebraicity and basic measures of these spaces.
This is joint work with Gennadiy Averkov, Tobias Boege, and Christopher Borge