This paper provides capacity results for multi-user mm-wave hybrid-beamforming, and presents optimal joint beam allocation and user scheduling algorithms. We characterize the downlink capacity of a practical system with quantized analog beamforming code-books under the constraint that users cannot be scheduled at the same time if they are closer together than a beam width in angle. We show that the capacity region is determined by a small number of linear inequality constraints. We also present capacity-achieving scheduling algorithms that provide beam allocations guaranteeing that user rate requirements are met within each resource block. In particular, we propose “sand-filling” algorithms that are provably optimal and which have linear complexity. Intuitively, our schemes can be viewed in terms of filling containers with coloured sand, in such a way that the colours at any given height do not conflict with the colours in the other containers at the same height, where the containers represent the RF chains (i.e. the beamforming resources), and the coloured sand represents the users (and their rate requirements). We show a numerical example where the capacity of our scheme is 82% higher than a traditional resource partitioning scheme.