  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.

Algebraic Geometry;Combinatorics