The robustness of the generalized Gini index

Abstract

AbstractIn this paper, we introduce a map $$\varPhi $$ Φ , which we call zonoid map, from the space of all non-negative, finite Borel measures on $${\mathbb {R}}^n$$ R n with finite first moment to the space of zonoids of $${\mathbb {R}}^n$$ R n . 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 $$\varPhi $$ Φ 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 $$\varPhi $$ Φ and the SLLN theorem, are particularly useful when dealing with a huge amount of multidimensional data.