  A crucial theorem in Reduced Density Matrix Functional Theory (RDMFT)
suggests that the universal pure and ensemble functional coincide on their
common domain of pure N-representable one-matrices. We refute this by a
comprehensive analysis of the geometric picture underlying Levy's constrained
search. Moreover, we then show that the ensemble functional follows instead as
the lower convex envelop of the pure functional. It is particularly remarkable
that the pure functional determines the ensemble functional even outside its
own domain of pure N-representable one-matrices. From a general perspective,
this demonstrates that relaxing pure RDMFT to ensemble RDMFT does not
necessarily circumvent the complexity of the one-body pure N-representability
conditions (generalized Pauli constraints). Instead, the complexity may simply
be transferred from the underlying space of pure N-representable one-matrices
to the structure of the universal one-matrix functional.
