  We propose a multivariate extension of the Lorenz curve based on multivariate
rearrangements of optimal transport theory. We define a vector Lorenz map as
the integral of the vector quantile map associated with a multivariate resource
allocation. Each component of the Lorenz map is the cumulative share of each
resource, as in the traditional univariate case. The pointwise ordering of such
Lorenz maps defines a new multivariate majorization order, which is equivalent
to preference by any social planner with inequality averse multivariate rank
dependent social evaluation functional. We define a family of multi-attribute
Gini index and complete ordering based on the Lorenz map. We propose the level
sets of an Inverse Lorenz Function as a practical tool to visualize and compare
inequality in two dimensions, and apply it to income-wealth inequality in the
United States between 1989 and 2022.
