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

TL;DR

This paper explores the algebraic structure of Lorenz hulls and zonoids, showing they form a commutative, associative algebra under a natural product defined via measure theory, unifying these concepts in convex geometry.

Contribution

It introduces a natural product operation on Lorenz hulls and zonoids, revealing their algebraic properties and unifying their geometric representations.

Findings

Lorenz hulls form a commutative, associative algebra under the product.

Zonoids share the same algebraic structure as Lorenz hulls.

The product is defined via the common notion of a product measure.

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.