  It has long been postulated that within density-functional theory (DFT) the
total energy of a finite electronic system is convex with respect to electron
count, so that 2 E_v[N_0] <= E_v[N_0 - 1] + E_v[N_0 + 1]. Using the
infinite-separation-limit technique, this article proves the convexity
condition for any formulation of DFT that is (1) exact for all v-representable
densities, (2) size-consistent, and (3) translationally invariant. An analogous
result is also proven for one-body reduced density matrix functional theory.
While there are known DFT formulations in which the ground state is not always
accessible, indicating that convexity does not hold in such cases, this proof
nonetheless confirms a stringent constraint on the exact exchange-correlation
functional. We also provide sufficient conditions for convexity in approximate
DFT, which could aid in the development of density-functional approximations.
This result lifts a standing assumption in the proof of the piecewise linearity
condition with respect to electron count, which has proven central to
understanding the Kohn-Sham band-gap and the exchange-correlation derivative
discontinuity of DFT.
