  In the last decade, the problem of characterizing the normability of the
weighted Lorentz spaces has been completely solved (\cite{Sa}, \cite{CaSoA}).
However, the question for multidimensional Lorentz spaces is still open. In
this paper, we consider weights of product type, and give necessary and
sufficient conditions for the Lorentz spaces, defined with respect to the
two-dimensional decreasing rearrangement, to be normable. To this end, it is
also useful to study the mixed norm Lorentz spaces. Finally, we prove
embeddings between all the classical, multidimensional, and mixed norm Lorentz
spaces.
