One can understand v-representability in several other equivalent ways, all described by Lieb. My preference goes towards the Legendre-Frenchel point of view, which requires the use of mixed states and ensemble v-representability and was already discussed above in contributions (2.1.13) and (3.7.2). We know that the corresponding lowest total energy and the lowest kinetic energy for a given density ρ satisfy the duality principles 8