Consistent partitions of polytopes and polynomial measures

A.Khovanskii

Let $\De_1,\dots,\De_n$ be $n$ polytopes in $\Bbb R^n$. There
are many ways to decompose the polytopes $\De_i$ into smaller
polytopes $\Ga_i^j$ and then choose certain ``consistent''
collections that contain a single polytope $\Ga_i^{j_i}$ from each
polytope $\De_i$ in such a way that the following formula holds:
$$
V(\De_1,\dots,\De_n)=
\sum V(\Ga_1^{j_1},\dots,\Ga_n^{j_n}),\tag"$({*})$"
$$
where $V$ is the mixed volume, and the summation is taken over all
consistent collections. The theory of Newton polytopes provides a lot
of examples of this kind. The present paper arose from
attempts to clarify whether there exists a similar effect for other
finitely additive measures that differ from the volume measure (for
instance, for the number of integral points of a polytope).

In the paper we introduce the notion of consistent partition for
several polytopes (whose number can differ from the dimension of the
underlying space). The main result of the present paper is the
generalization of relation $({*})$ to any consistent partition
and for any finitely additive measure.