2020/07/25 by Michelangelo Franciosi, Franciosi, M., Simona Settepanella +3
Economics, Econometrics and Finance · #28A78 #28B05 #Economic Theory and Policy #Economic theories and models #FOS: Mathematics #Functional Analysis (math.FA) #Housing Market and Economics #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2007.12924
openalex publication_date 2020/07/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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 Φ 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 Φ and Glivenko-Cantelli theorem, are particularly useful when dealing with a huge amount of multi-dimensional data.