A note on the convergence of lift zonoids of measures

Abstract

The lift zonoid is a convenient representation of an integrable measure by a convex set in a higher‐dimensional space. It is known that, under appropriate conditions, a uniformly integrable sequence of measures converges weakly if and only if the corresponding sequence of lift zonoids converges in the Hausdorff metric. We provide a new proof of this essential result. Our proof technique allows us to eliminate the unnecessary conditions previously considered in the literature. As a by‐product, we obtain a characterization of uniform integrability, and a simple sufficient condition for tightness, of a sequence of integrable measures in terms of their lift zonoids.