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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 (see §2). 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 (see §3). Consider a measure on the integral polytopes that is polynomial with respect to the shifts of the polytopes by the vectors of the integral lattice. As is known, such a measure is a polynomial on the semigroup of integral polytopes with respect to the Minkowski addition. For the measures invariant with respect to shifts by the vectors of the integral lattice, this result is due to McMullen [6]. It was extended to the case of polynomial measures with respect to the shifts in [5], where the integral over the Euler characteristic was used. Here we show that the result is the direct consequence of the generalized relation (∗) .

Consistent partitions of polytopes and polynomial measures