A proof of the log-concavity conjecture related to the computation of the ergodic capacity of MIMO channels

An upper bound on the ergodic capacity of {\bf MIMO} channels was introduced recently in arXiv:0903.1952. This upper bound amounts to the maximization on the simplex of some multilinear polynomial $p({\lambda }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\lambda }_n)$ with non-negative coefficients. Interestingly, the coefficients are subpermanents of some non-negative matrix. In general, such maximizations problems are {\bf NP-HARD}. But if say, the functional $\log (p)$ is concave on the simplex and can be efficiently evaluated, then the maximization can also be done efficiently. Such log-concavity was conjectured in arXiv:0903.1952. We give in this paper self-contained proof of the conjecture, based on the theory of {\bf H-Stable} polynomials.