Center-OutwardMultiple-Output Lorenz Curves and Gini Indices a measure transportation approach

Based on measure transportation ideas and the related concepts of centeroutwardquantile functions,we propose multiple-output center-outward generalizationsof the traditional univariate concepts of Lorenz and concentration functions,and the related Gini and Kakwani coefficients. These new concepts have anatural interpretation, either in terms of contributions of central (“middle-class”)regions to the expectation of some variable of interest, or in terms of the physicalnotions of work and energy, which sheds new light on the nature of economicand social inequalities. Importantly, the proposed concepts pave the way to statisticallysound definitions, based on multiple variables, of quantiles and quantileregions, and the concept of “middle class,” of high relevance in various socioeconomiccontexts.