Optimal cover of points by disksin a simple polygon

Let P be a simple polygon, and let Q be a set of points in P. We present an almostlinear time algorithm for computing a minimum cover of Q by disks that are contained in P. We then generalize the algorithm so that it can compute a minimum cover of Q by homothets of any fixed compact convex set O of constant description complexity that are contained in P. This improves previous results of Katz and Morgenstern [Lecture Notes in Comput. Sci. 5664, 2009, pp. 447-458]. We also consider the minimum disk-cover problem when Q is contained in a (sufficiently narrow) annulus and present a nearly linear algorithm for this case, too.