Intensity-modulated radiation therapy (IMRT) is a state-of-the-art technique for administering radiation to cancer patients. The goal of a treatment is to deliver a prescribed amount of radiation to the tumor, while limiting the amount absorbed by the surrounding healthy and critical organs. Planning an IMRT treatment requires determining fluence maps, each consisting of hundreds or more beamlet intensities. Since it is difficult or impossible to deliver a sufficient dose to a tumor without irradiating nearby critical organs, radiation oncologists have developed guidelines to allow tradeoffs by introducing so-called dose-volume constraints (DVCs), which specify a given percentage of volume for each critical organ that can be sacrificed if necessary. Such constraints, however, are of combinatorial nature and pose significant challenges to the fluence map optimization problem. In this paper, we describe a new geometric approach to the fluence map optimization problem. Contrary to the traditional view, we regard dose distributions as our primary independent variables, while treating beamlet intensities as secondary ones. We consider two sets in the dose space: (i) the physical set consisting of physically realizable dose distributions, and (ii) the prescription set consisting of dose distributions meeting the prescribed tumor doses and satisfying the given dose-volume constraints. We seek a suitable dose distribution by successively projecting between these two sets. A crucial observation is that the projection onto the prescription set, which is non-convex, can be properly defined and easily computed. The projection onto the physical set, on the other hand, requires solving a nonnegative least squares problem. We show that this alternating projection algorithm is actually equivalent to a greedy algorithm driven by local sensitivity information readily available in our formulation. Moreover, the availability of such local sensitivity information will enable us to devise greedy algorithms to search for a desirable plan even when a ``good and achievable'' prescription is unknown.