Publication at Faculty of Mathematics and Physics |
2022
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 lift zonoids.