The robustness of the generalized Gini index

In this paper, we introduce a map Φ, which we call zonoid map, from the space of all non-negative, finite Borel measures on Rn with finite first moment to the space of zonoids of Rn. 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 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 and the SLLN theorem, are particularly useful when dealing with a huge amount of multidimensional data.