The robustness of the generalized Gini index

In this paper, we introduce a map Φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPhi $$\end{document}, which we call zonoid map, from the space of all non-negative, finite Borel measures on Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^n$$\end{document} with finite first moment to the space of zonoids of Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^n$$\end{document}. This map, connecting Borel measure theory with zonoids theory, allows to slightly generalize the Gini volume introduced, in the context of Industrial Economics, by Dosi (J Ind Econ 4:875–907, 2016). This volume, based on the geometric notion of zonoid, is introduced as a measure of heterogeneity among firms in an industry and it turned out to be a quite interesting index as it is a multidimensional generalization of the well-known and broadly used Gini index. By exploiting the mathematical context offered by our definition, we prove the continuity of the map Φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPhi $$\end{document} which, in turn, allows to prove the validity of a SLLN-type theorem for our generalized Gini index and, hence, for the Gini volume. Both results, the continuity of Φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPhi $$\end{document} and the SLLN theorem, are particularly useful when dealing with a huge amount of multidimensional data.