The downlink of a wireless system with multiple transmit antennas and multiple users can be modeled as a vector Gaussian broadcast channel. Unlike the single-antenna scalar Gaussian broadcast channel, the capacity region of this channel is unknown. Building on an earlier result by Caire and Shamai on the 2-user case, we obtained a simple characterization of the sum capacity of this channel for general number of users. The optimal scheme is based on Costa's "dirty-paper" precoding technique. The proof of the result exploits an intimate four-way connection between the multiple antenna broadcast channel, the corresponding point-to-point channel (where the receivers can cooperate), the multiple access channel (where the role of transmitters and receivers are reversed), and the corresponding point-to-point channel (where the transmitters can cooperate). Convex duality plays a central role in the proof as well.
P. Viswanath and D. Tse, " Sum Capacity of the Multiple Antenna Gaussian Broadcast Channel and Uplink-Downlink Duality" , IEEE Transactions on Information Theory, vol 49(8), August, 2003, pp. 1912-1921.