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We show that the lift zonoid concept for a probability measure on R^d,introduced in (Koshevoy and Mosler, 1997), leads naturally to a one-to onerepresentation of any interior point of the convex hull of the support of acontinuous measure as the barycenter w.r.t. to this measure of either of ahalf-space, or the whole space. We prove the infinite-dimensionalgeneralization of this representation, which is based on the extension of thelift-zonoid concept for a cylindrical probability measure.

Lift zonoid and barycentric representation on a Banach space with a cylinder measure