which defines an optimal-transport multimarginal problem with the repulsive Coulomb interaction as cost function. Techniques from optimal transport made it possible, for example, to prove 693 that the exact SCE functional for one-dimensional systems is provided by the solution first guessed by Seidl 141 on physical grounds. Another application is on the Lieb-Oxford bound (see contribution (4.5.2)), where optimal-transport methods can be used to improve bounds on the optimal constant. Some open questions that remain on the rigorous side concern the next leading term in the h → 0 expansion, whose form was conjectured in ref. 144. A first step in this direction has been recently taken. 696 I do have the feeling that there is still a lot to learn from the connection with optimal transport and by further analysing this limit, which provides complementary information with respect to perturbation theory. Although it is a semiclassical limit, it may perhaps be corrected for exchange (spin) effects, 697 and could be combined with Kohn-Sham DFT. (4.5.9) Lewin : I fully agree with contribution (4.5.8). Related to the above discussion in contribution (4.5.2), one interesting question could be to understand what can be said about the Kohn-Sham potential v s in the limit h → 0. As I have mentioned, all densities are v-representable in the SCE limit eqn (76). So, more v-representable densities seem to exist when h gets sufficiently small. . . who explicitly computed the functional derivative of the next leading term 144 in the h → 0 expansion, in the special onedimensional case (for which we now know 696 that the functional proposed in 144 is exact). However, Kohn-Sham self-consistent calculations that include the functional derivative of this next leading term 700 make the density poorer relative to the bare Kohn-Sham SCE result -even for very low-density systems, where inclusion of this term improves the energy. This suggests that this is not the right route to take to answer the question on v s , or at least that we need to better understand the expansion.