  In this chapter we first review the Levy-Lieb functional, which gives the
lowest kinetic and interaction energy that can be reached with all possible
quantum states having a given density. We discuss two possible convex
generalizations of this functional, corresponding to using mixed canonical and
grand-canonical states, respectively. We present some recent works about the
local density approximation, in which the functionals get replaced by purely
local functionals constructed using the uniform electron gas energy per unit
volume. We then review the known upper and lower bounds on the Levy-Lieb
functionals. We start with the kinetic energy alone, then turn to the classical
interaction alone, before we are able to put everything together. An appendix
is devoted to the Hohenberg-Kohn theorem and the role of many-body unique
continuation in its proof.
