  Based on measure transportation ideas and the related concepts of
center-outward quantile functions, we propose multiple-output center-outward
generalizations of the traditional univariate concepts of Lorenz and
concentration functions, and the related Gini and Kakwani coefficients. These
new concepts have a natural interpretation, either in terms of contributions of
central ("middle-class") regions to the expectation of some variable of
interest, or in terms of the physical notions of work and energy, which sheds
new light on the nature of economic and social inequalities. Importantly, the
proposed concepts pave the way to statistically sound definitions, based on
multiple variables, of quantiles and quantile regions, and the concept of
"middle class," of high relevance
  in various socio-economic contexts.
