The polyhedral model provides a single unified foundation for systolic array synthesis and automatic parallelization of loop programs. We investigate the problem of memory reuse when compiling Alpha (a functional language based on this model). Direct compilation would require unacceptably large memory (for example O(n 3 ) for matrix multiplication). Researchers have previously addressed the problem of memory reuse, and the analysis that this entails for projective memory allocations. This paper addresses, for a given schedule, the choice of the projections so as to minimize the volume of the residual memory. We prove tight bounds on the number of linearly independent projection vectors. Our method is constructive, yielding an optimal memory allocation. We extend the method to modular functions, and deal with the subsequent problems of code generation. Our ideas are illustrated on a number of examples generated by the current version of the Alpha compiler.