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.

翻译：我们提出基于最优传输理论多维重排的洛伦兹曲线多元推广。定义向量洛伦兹映射为多元资源分配对应的向量分位数映射的积分。洛伦兹映射的每个分量表示每种资源的累积份额，与传统单变量情形一致。此类洛伦兹映射的点态排序定义了新的多元优化序，该序等价于任何具有厌恶不等式的多元秩依赖社会评价函数的社会规划者的偏好。基于洛伦兹映射，我们定义了多属性基尼指数族及完全排序。提出逆洛伦兹函数的水平集作为可视化和比较二维不等式的实用工具，并将其应用于1989年至2022年美国收入-财富不平等分析。