This paper seeks to extend the unidimensional notion of Lorenz dominance to the multidimensional framework in the context of distributions of well-being in an economy. It formulates a definition of a multidimensional Lorenz dominance relation (MLDR) that incorporates two generalizations of the well-known Pigou-Dalton transfer condition as well as a condition relating to the sensitivity of the dominance relation to the correlations between distributions of the different attributes. The paper notes that the existing literature does not seem to contain an example of a relation that is an MLDR as per the definition developed here. It seeks to provide one.