Decomposition of ranges of vector measures

The following conditions on a zonoid Z, i.e., a range of a non-atomic vector measure, are equivalent: (i) the extreme set containing 0 in its relative interior is a parallelepiped; (ii) the zonoid Z determines the m-range of any non-atomic vector measure with range Z, where the m-range of a vector measure μ is the set of m-tuples (μ(S ₁), ..., μ(S _m), where S ₁, ...S _m are disjoint measurable sets and (iii) there is avector measure space (X, Σ, μ) such that any finite factorization of Z, Z =ΣZ _i, in the class of zonoids could be achieved by decomposing (X, Σ). In the case of ranges of non-atomic probability measures (i) is automatically satisfied, so (ii) and (iii) hold.