The robustness of the generalized Gini index

In this paper we introduce a map $\Phi$, which we call \textit{zonoid map}, from the space of all non-negative, finite Borel measures on $\mathbb{R}^n$ with finite first moment to the space of zonoids of $\mathbb{R}^n$. This map, connecting Borel measure theory with zonoids theory, allows us to slightly generalize the Gini volume introduced, in the contest of Industrial Economics, by Dosi, Grazzi, Marengo and second author in 2016. This volume, based on the geometric notion of zonoid, is introduced as a measure of heterogeneity among firms in an industry and turned out to be quite interesting index as it is a multi-dimensional generalization of the well known and broadly used Gini index. By exploiting the mathematical contest offered by our definition, we prove the continuity of the map $\Phi$ which, in turns, allows to prove the validity of a Glivenko-Cantelli theorem for our generalized Gini index and, hence, for the Gini volume. Both results, continuity of $\Phi$ and Glivenko-Cantelli theorem, are particularly useful when dealing with a huge amount of multi-dimensional data.