The Lorenz Zonoid of a Multivariate Distribution
将单变量分布的洛伦兹曲线和洛伦兹序推广到多元情形，定义了洛伦兹区域和洛伦兹曲面，并引入基于集合包含的多元差异概念，对研究多商品经济中的分配不平等有参考价值。
Abstract The article extends the usual Lorenz curve and Lorenz order of univariate distributions to the multivariate case. For a given probability distribution in nonnegative d space, d ≥ 1, we define and investigate the Lorenz zonoid and the Lorenz surface, which are sets in (d + 1) space. The surface equals the usual Lorenz curve when d = 1. Included is the definition of the Lorenz surface of a finite number of vectors that may be interpreted as the endowments of economic units in d commodities. As a notion of increasing multivariate disparity, we introduce the set inclusion of Lorenz zonoids and show that it is equivalent to directional majorization.