I consider the tradeoff between the information gained about an initially unknown quantum state, and the disturbance caused to that state by the measurement process. I show that for any distribution of initial states, the information-disturbance frontier is convex, and disturbance is nondecreasing with information gain. I consider the most general model of quantum measurements, and all post-measurement dynamics compatible with a given measurement. For the uniform initial distribution over states, I show that an optimal information-disturbance combination may always be achieved by a measurement procedure which satisfies a generalization of the projection postulate, the ``square-root dynamics.'' I use this to show that the information-disturbance frontier for the uniform ensemble may be achieved with ``isotropic'' (unitarily covariant) dynamics. This results in a significant simplification of the optimization problem for calulating the tradeoff in this case, giving hope for a closed-form solution. Some of the results also apply to certain discrete ensembles relevant to quantum cryptography.