Density Functional Theory for Open Systems*

We provide a formal proof of the convexity relation, $2E[N] \leq E[N - 1] + E[N + 1]$, characterizing the total energy of interacting $N$-electron systems under the action of a given external potential, hitherto assumed to be true only on experimental grounds. This is used to prove the inequality, $I(N) - A(N) \geq 0$, where $I(N)$ and $A(N)$ are, respectively, the ionization potential and electron affinity of a $N$-electron system.