where E N [v] and E 0 N [v] are the interacting and noninteracting ground-state energies in an external potential v, respectively. Again, one should specify the set of potentials v in the two suprema, but any reasonable class will yield the same final value. The question is whether these suprema are attained (our desired potential v s is a maximizer for T s [ρ]) and then the chances that this happens are much higher if the allowed class of potentials v is larger. Let me recall in passing that the existence of a dual potential is well understood in classical DFT. At zero temperature this follows from methods in multimarginal optimal transportation. At positive temperature, this result was proved by The quantum kinetic energy is thus the main obstacle here. Discretized quantum systems are studied in refs. 34,680,681. Another point of view has been mentioned above in contributions (2.2.2) and (3.7.2) and involves a kind of differentiability of F[ρ] and T s [ρ]. Any potential solving a maximum principle such as in eqn ( ) is, formally at least, a derivative of the corresponding functional. To be able to treat the difference F[ρ] -T s [ρ], we thus need both to be differentiable at the same time. Although the notion of differentiability looks natural and intuitive, it is in fact not so easy. The reason is that the natural domain of F[ρ] is the set of densities with a finite von Weizäcker energy 8 , which is not such a nice set.