We consider the problems of computing maximal points and the convex hull of a set of points in 2D, when the points are “in motion.” We assume that the point locations (or trajectories) are not known precisely and determining these values exactly is feasible, but expensive. In our model, the algorithm only knows areas within which each of the input points lie, and is required to identify the maximal points or points on the convex hull correctly by updating some points (i.e. determining exactly their location). We compare the number of points updated by the algorithm on a given instance to the minimum number of points that must be updated by an omniscient adversary in order to provably compute the answer correctly. We give algorithms for both of the above problems that always update at most 3 times as many points as the adversary, and show that this is the best possible. Our model is similar to that of [5,2].