Recall that a density ρ is (ensemble) v-representable when it arises from an N-electron (mixed) ground state with an external potential v. Let me emphasize that there are two notions of vrepresentability, for the interacting and noninteracting cases, respectively. The question is whether a v-representable density (for the interacting system) is v-representable by the noninteracting system. In other words, we need to study the set of densities that are simultaneously v-representable for the two cases. At the moment, nothing is known rigorously about this set, to my knowledge -in principle, it could even be empty! Of course, to properly discuss this problem, it is important to first fix a class of admissible potentials v that we wish to consider in DFT. This class should be large in order to increase the probability of being vrepresentable, but probably not too crazy either. Lieb considered all potentials belonging to L 3/2 (R 3 ) + L ∞ (R 3 ) because the energy is always bounded-below for such potentials. However, many physical cases do not appear in this class, such as the harmonic potential for instance. This is definitely a too small class.