Tropical varieties with polynomial weights and corner loci of piecewise polynomials

We find a relation between mixed volumes of several polytopes and the convex hull of their union, deducing it from the following fact: the mixed volume of a collection of polytopes only depends on the product of their support functions (rather than on the individual support functions). For integer polytopes, this dependence is essentially a certain specialization of the isomorphism between two well-known combinatorial models for the cohomology of toric varieties, however, this construction has not been extended to arbitrary polytopes so far (partially due to the lack of combinatorial tools capable of substituting for toric geometry, when vertices are not rational). We provide such an extension, which leads to an explicit formula for the mixed volume in terms of the product of support functions, and may also be interesting because of the combinatorial tools (tropical varieties with polynomial weights and their corner loci) that appear in our construction. As an example of another possible application of these new objects, we notice that every tropical subvariety in a tropical manifold M can be locally represented as the intersection of M with another tropical variety (possibly with negative weights), and conjecture certain generalizations of this fact to singular M. The above fact about subvarieties of a tropical manifold may be of independent interest, because it implies that the intersection theory on a tropical manifold, which was recently constructed by Allerman, Francois, Rau and Shaw, is locally induced from the ambient vector space.