A multidimensional Lorenz dominance relation: some corrections

While Proposition 2.2 of Banerjee (2014, p. 182) is valid, its proof, as given there, contains some erroneous statements. The paper referred to above defines a multidimensional Lorenz dominance relation (MLDR), LM, as a binary relation on the set, X, of distribution matrices which (a) satisfies the conditions of continuity (CONT), quasi-ordering (QORD), ratio-scale invariance (RSI), invariance w.r.t. row permutations (IRP) and the Pigou–Dalton bundle principle (PDBP) and (b) coincides with the unidimensional Lorenz dominance relation if the number of attributes is 1. LM is also desired to satisfy two additional conditions, comonotonizing majorization (CM) and prioritization of attributes under comonotonicity (PAC). The Proposition in question considers six dominance relations (which, in this paper, are called L1, L2, L Z , LeZ , L3 and L4) that have been proposed in the existing literature.