Lieb proved that the two sets of v-representable densities are dense in the space L 1 (R 3 ) ∩ L 3 (R 3 ). The problem is that a dense set can, in principle, be extremely small -think of the rational numbers, which are dense but form a set of zero measure in the set of real numbers. Even worse, we need to look at the intersection of these two dense sets, which can be arbitrarily small or even empty. Very little is thus known mathematically about this problem. Let me now quickly mention the other two problems I had in mind. The second one is to improve existing exact constraints. For instance, I already mentioned the Lieb-Oxford bound, 677 of which the best constant is believed to be that of the uniform electron gas. But at present, we have no idea on how to justify this rigorously.