A Product on Lorenz Hulls, Zonoids, and Vector Measure Ranges

Abstract

A Lorenz hull is the convex hull of the range of an $n$-dimensional vector of finite signed measures defined on a common measurable space. We show that the set of $n$-dimensional Lorenz hulls is endowed with a natural product that is commutative, associative, and distributive over Minkowski sums. The same holds with "zonoid" in place of "Lorenz hull" as the two concepts give rise to the same set of subsets of $\mathbb{R}^n$. The product is defined via the common notion of a product measure.