The Lorenz zonoid of a multivariate distribution

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, a greater than or equal to 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.