  The unsolved problem of determining which densities are ground state
densities of an interacting electron system in some external potential is
important to the foundations of density functional theory. A coarse-grained
version of this ensemble V-representability problem is shown to be thoroughly
tractable. Averaging the density of an interacting electron system over the
cells of a regular partition of space produces a coarse-grained density. It is
proved that every strictly positive coarse-grained density is coarse-grained
ensemble V-representable: there is a unique potential, constant over each cell
of the partition, which has a ground state with the prescribed coarse-grained
density. For a system confined to a box, the (coarse-grained) Lieb [Intl. J.
Quantum Chem. 24, 243 (1983)] functional is also shown to be Gateaux
differentiable. All results extend to open systems.
