Parallelepipeds and decomposition of ranges of vector measures in Banach spaces

We extend a result of A. NEYMAN about zonoids in R-n to zonoids in Banach s paces. A zonoid in a Banach space is the closed convex hull of the range of a vector measure. We show that the following conditions on a zonoid Z are equivalent: (1) Z determines univocally the associated conical measure; (2) There exists a vector measure defined on (Omega, Sigma) such that every de composition of Z into sum of zonoids can be obtained by decomposing (Omega, Sigma); (3) Z boolean AND (-alpha Z) is a parallelepiped for every alpha > 0. We also Drove other results about decomposition of zonoids; for instanc e, we prove that if Z is a zonoid and Z boolean AND(-Z) is a zonotope, ther e exists a zonoid L such that Z = Z boolean AND(-Z)+L.